Datasets:

---

ashishkgpian

/

adspapers_80_85

like 0

[ { "title": "ABSTRACT", "content": "Convergence of the Rosseland Mean Opacity (RMO) is investigated with respect to the equation-of-state (EOS) and the number of atomic levels of iron ions prevalent at the solar radiative/convection boundary. The 'chemical picture' Mihalas-HummerDappen MHD-EOS, and its variant QMHD-EOS, are studied at two representative temperature-density sets at the base of the convection zone (BCZ) and the Sandia Z experiment: (2 × 10 6 K, 10 23 /cc ) and (2 . 11 × 10 6 K, 3 . 16 × 10 22 /cc ), respectively. It is found that whereas the new atomic datasets from accurate R-matrix calculations for opacities (RMOP) are vastly overcomplete, involving hundreds to over a thousand levels of each of the three Fe ions considered - Fe xvii , Fe xviii , Fe xix - the EOS constrains contributions to RMOs by relatively fewer levels. The RMOP iron opacity spectrum is quite different from the Opacity Project distorted wave model and shows considerably more plasma broadening effects. This work points to possible improvements needed in the EOS for opacities in high-energy-density (HED) plasma sources. Key words: Physical Data and Processes, atomic processes", "pages": [ 1 ] }, { "title": "Anil K. Pradhan 1 , 2", "content": "1 Department of Astronomy, 2 Chemical Physics Program, The Ohio State University, Columbus, OH 43210, USA. Accepted xxxxxx Received xxxxxx; in original form xxxxxx", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "calculations, enhanced iron opacity might countenance lower solar abundances (Bailey et al. (2015). As a fundamental quantity in light-matter interaction opacity plays a key role in astrophysics, such as stellar interiors, helioseismology, and asteroseimology, elemental abundance determination, host-star and exoplanetary fluxes, etc. (Christensen-Dalsgaard et al. (2009); Basu et al. ( 2015); Asplund et al. (2009); Carlos et al. (2019); Buldgen et al. (2023a). In addition, radiation transport models of inertial plasma fusion devices requires accurate opacities (Bailey et al. (2015); Perry et al. (2018). In particular, the outstanding uncertainty in the solar chemical composition affects elemental calibration of all astronomical sources. Attempts to employ advances in helioseismology and abundances are an active area of basic research (Basu and Antia (2008); Buldgen et al. (2022), but require enhanced solar opacities by about 10%. That, in turn, depends on two elements, oxygen and iron, that determine about half of the solar opacity at BCZ. However, a downward revision of oxygen abundance by up to 20-40% from earlier solar composition is a major part of the 'solar problem' (Asplund et al. (2021); Pietrow et al. (2023); Li et al. (2023); Buldgen et al. (2023b). Since about 90% of oxygen is either fully ionized or H-like at BCZ, its absorption coefficient is small and unlikely to change from current atomic Opacity computations depend on atomic data on the one hand and the plasma EOS on the other (The Opacity Project Team (1995); Seaton et al. (1994); Pradhan et al. (2023). Voluminous amounts of data are needed for all photon absorption and scattering processes in order to ensure completeness. Recently, accurate and extensive calculations of atomic data for iron ions of importance under BCZ conditions have been carried out using the R-matrix method (Pradhan et al. (2023); Nahar et al. (2023); Pradhan (2023); Zhao et al. (2023). However, the EOS determines how and to what extent the atomic data contribute to monochromatic and mean opacities at a given temperature and density. The Planck and Rosseland Mean Opacity (PMO and RMO respectively) are defined as where g ( u ) = dB ν /dT is the derivative of the Planck weighting function , and κ ν is the monochromatic opacity. Atomic processes and contributions to opacity are from bound-bound ( bb ), bound-free ( bf ), free-free ( ff ), and photon scattering ( sc ) as where a k is the abundance of element k , x j the j ionization fraction, i and i ' are the initial bound and final bound/continuum states of the atomic species, and /epsilon1 represents the electron energy in the continuum. Whereas the ff and sc contributions are small, the opacity is primarily governed by bb and bf atomic data that need to be computed for all atomic species. Existing opacity models generally employ the relatively simple distorted wave (DW) approximation based on atomic structure codes, but higher accuracy requires considerable effort. Originally, the Opacity Project (The Opacity Project Team (1995) (hereafter OP) envisaged using the poweful and highly accurate R-matrix method for improved accuracy. But that turned out to be intractable owing to computational constraints, and also required theoretical developments related to relativistic fine structure and plasma broadening effects. Therefore, the OP opacities were finally computed using similar atomic physics as other existing opacity models, mainly based on the simpler distorted wave (DW) approximation (Seaton OPCD (2003), and later archived in the online database OPserver (Mendoza et al. (2007). However, following several developments since then renewed R-matrix calculations can now be carried out, as discussed below.", "pages": [ 1, 2 ] }, { "title": "2 THEORETICAL FRAMEWORK", "content": "Recently, with several improvements in the extended Rmatrix and opacity codes large-scale data have been computed for Fe ions Fe xvii , Fe xviii and Fe xix , which determine over 80% of iron opacity near BCZ conditions (Pradhan et al. (2023); Nahar et al. (2023); Pradhan (2023); Zhao et al. (2023). The R-matrix (RM) framework and comparison with existing opacity models based on atomic structure codes and the distorted wave (DW) approximation, and associated physical effects, are described in detail. The primary difference between the RM and DW approximations is the treatment of bound-free opacity which is dominated by autoionizing resonances that are included in an ab initio manner in RM calculations, but treated perturbatively as bound-bound transitions in the DW method. Plasma broadening effects are very important, but manifest themselves quite differently in the two methods. Resonances in RM photoionization cross sections are broadened far more than lines as function of temperature and density since autoionization widths, shapes and heights are considered explicitly (Pradhan (2023). Also, the intrinsically asymmetric features of the large Seaton photoexcitation-of-core (PEC) resonances in bound-free cross sections are preserved in RM calculations. The unverified assertion that RM and DWopacities are equivalent is incorrect owing to basic physical effects (Delahaye et al. (2021). On the contrary, the RM method is based on the coupled channel approximation that gives rise to autoionizing resonances, and has historically superseded the DW method which neglects channel coupling. RM calculations for all relevant atomic processes are generally much more accurate than the DW, as for example in the work carried out under the Iron Project, including relativistic effects in the Breit-Pauli R-matrix (BPRM) approximation (Hummer et al. (1993) that is also employed in the present work (Nahar et al. (2023). The interface of atomic data with EOS parameters is implemented through the MHD-EOS (Mihalas et al. (1988), formulated in the 'chemical picture' as designed for OP work. It is based on the concept of occupation probability w of an atomic level being populated in a plasma environment, characterized by a temperature-density (hereafter T-D) related to Boltzmann-Saha equations. The level population is then given as where w ij are the occupation probabilities of levels i in ionization state j , and U j is the atomic internal partition function. The occupation probabilities do not have a sharp cut-off, but approach zero for highn as they are 'dissolved' due to plasma interactions. The partition function is re-defined as E ij is the excitation energy of level i , g ij its statistical weight, and T the temperature. The w ij are determined upon free-energy minimization in the plasma at a given T-D. However, the original MHD-EOS was found to yield w -values that were unrealistically low by up to several orders of magnitude. An improved treatment of microfield distribution and plasma correlations was developed, leading to the so-called QMHD-EOS (Nayfonov et al. (1999) and employed for subsequent OP calculations and results (Seaton OPCD (2003); Mendoza et al. (2007).", "pages": [ 2 ] }, { "title": "3 OPACITY COMPUTATIONS", "content": "The new RMOP data are interfaced with the (Q)MHDEOS to obtain opacities. Computed RM atomic data for bb oscillator strengths and bf photoionization cross sections of all levels up to n (SLJ) = 10 yields datasets for 454 levels for Fe xvii , 1174 levels for Fe xviii and 1626 for Fe xix (Nahar et al. (2023); some results for Fe xvii were reported earlier (Nahar and Pradhan (2016). Monochromatic and mean opacities may then be computed using atomic data for any number of these levels and the EOS . In order to study the behavior of MHD and QMHD, we employ the new RMOP opacity codes (Pradhan et al. (2023), varying the number of atomic levels for each Fe ion, and both sets of EOS parameters at specified temperaturedensity pairs for a particular ion. Monochromatic opacities are computed at the same frequency mesh in the variable and range 0 /lessorequalslant u = hν/kT /lessorequalslant 20, as in OP work (Seaton et al. (1994); Mendoza et al. (2007). Since RMOP calculations were carried out for the three Fe ions that comprise over 80% of total Fe at BCZ, we replace their opacity spectra in OP codes (Seaton OPCD (2003) and recompute RMOP iron opacities. Thus, ∼ 15% contribution is from OP data for other Fe ions; a table of Fe ion fractions at BCZ is given in (Pradhan et al. (2023). To circumvent apparently unphysical behavior of MHDEOS at very high densities, an ad hoc occupation probability cut-off was introduced in OP calculations with w ( i ) /greaterorequalslant 0 . 001 (Badnell and Seaton (2003). We retain the cut-off in the new RMOP opacity codes (Pradhan et al. (2023), since the same EOS is employed, but also tested relaxing the cutoff to smaller values up to w ( i ) /greaterorequalslant 10 -12 . However, no significant effect on RMOs was discernible, indicating that a more fundamental revision of (Q)MHD-EOS might be necessary (Trampedach et al. (2006). Level population fractions are normalized to unity, and therefore including more levels would not necessarily affect opacities in a systematic manner, as discussed in the next section. unless they are modified with inclusion of possibly missing atomic-plasma microphysics of individual levels and associated atomic data.", "pages": [ 2, 3 ] }, { "title": "4 RESULTS AND DISCUSSION", "content": "The EOS determines the contribution to opacity and its cut-off from an atomic level i via the occupation probability w ( i ) depending on density and resulting plasma microfield, and the level population pop ( i ) via the Boltzmann factor exp ( -E i /kT ) at temperature T. Fig. 1 illustrates the behavior of the EOS parameters for Fe xvii at BCZ conditions. The new RMOP data include autoionizing resonances due to several hundred coupled levels, but can not be directly compared with DW bound-free cross sections that neglect channel coupling and are feature-less (Nahar et al. (2023); Zhao et al. (2023). However, a comparison of the total monochromatic opacity spectrum can be done to illustrate differences due to plasma broadening of resonances in the RMOP data vs. lines as in the OP DW data. The primary focus of this work is the interface of EOS with atomic data. As exemplar of the detailed analysis of EOS parameters, Fig. 1 shows the occupation probabilities for Fe xvii at BCZ conditions (red dots, top panel) for all levels with w ( i ) > 0 . 001, and corresponding level populations (black open circles, middle panel). Since the contribution to RMO is limited by significant level populations Pop ( i ), the number of levels with Pop ( i ) > 0 . 1% is found to be much smaller, around 50 or so (blue dots, bottom panel). The reason for the given distribution of w ( i ) (top panel) is because the BPRM calculations are carried out according to total angular momentum quantum number and parity Jπ . Therefore, all BPRM data are produced in order of ascending order in energy within each Jπ symmetry , and descending order due to Stark ionization and dissolution of levels (Mihalas et al. (1988). Converged RMOs with NLEV = NMAX proach near-constant values for NLEV ≈ NMAX = 200, for all three Fe ions and for both the MHD and QMHD; no further significant contribution to RMOs is made due to EOS cut-offs and saturation. Therefore, this 'convergence' should be treated as apparent, and would be real if and only if the EOS is precisely determined . The converged RMOs should be regarded as a lower bound, in case revisions to EOS enable contributions from more levels that are included in the extensive RMOP atomic datasets, and the EOS+data combination may yield higher opacities. Fig. 3 shows a comparison of the new RMOP opacity spectrum (red) with OP (black). The Sandia Z measurements are also shown (cyan), but it should be noted that the experimental values are convolved over instrument resolution and the magnitudes of individual features are not directly compatible. In the top panel in Fig. 3 the monochromatic opacities are plotted on a log 10 -scale, and on a linear scale in the bottom panel to better elucidate the differences. The RMOP and OP opacity spectra differ in detailed energy distribution and magnitude. In general, the RMOP background is higher and the peaks lower than OP due to opacity re-distribution, with significant enhancement around 0.7 1600 keV. The difference is more striking on a linear-scale in Fig. 3 (bottom panel) around 0.9-1.0 keV, where the RMOP peaks are lower by several factors. Fig. 3 also shows that the Sandia Z measurements span only a small energy range relative to the Planck function derivative dB/dT that determines the Rosseland window and therefore the RMO. But the considerable difference between the background RMOP opacity with experiment remains as with the earlier OP and other works (Bailey et al. (2015); Nahar and Pradhan (2016). As we expect, the background non-resonant R-matrix photoionization cross sections are similar to DW results. However, the RMOP results are qualitatively in better agreement with experimental results with shallower 'windows' in opacity than OP, for example at E ≈ 1 . 0 keV (top panel) and several other energies. Nevertheless, there seems to be a source of background opacity in the Z experiment for iron (Nagayama et al. (2019) that is not considered in theoretical calculations. It is also interesting to revisit the only available comparison between and OP and OPAL occupations probabilities for the simple case of H-like C 5+ (Badnell and Seaton (2003). Table 3 gives these parameters, and also the level populations going up to n = 6. However, owing to the fact that the ground state population dominates over all other levels, and Carbon is fully ionized or H-like at given temperature-density, the RMO remains nearly constant at 170.3 cm 2 /g. We might expect similar behavior for Oxygen opacity, though more detailed study is needed, and of course for complex ions such as in this Letter .", "pages": [ 3, 4 ] }, { "title": "5 CONCLUSION", "content": "Whereas improved opacities may now be computed with high precision atomic data using the state-of-the-art Rmatrix method, the EOS remains a source of uncertainty. Therefore, the results presented herein should be considered tentative, pending more studies and comparison of (Q)MHD-EOS parameters with other equations-of-state, as well as newly improved versions (Trampedach et al. (2006). However, preliminary RMOP results indicate considerable differences with OP iron opacity spectrum, and by extension other existing opacity models based on the DW method and plasma broadening treatment of lines vs. resonances. While the present RMOP iron opacities are significantly higher than the OP owing to higher accuracy and enhanced redistribution of resonance strengths in bound-free opacity, final results might yet depend on an improved MHD-EOS resolving issues outlined herein and related to pseudo bound-free continua (Dappen et al. (1987); Seaton et al. (1994). Although the contribution may be relatively small around BCZ, completeness requires R-matrix calculations for other Fe ions (in progress). It is also noted that the Sandia Z experimental data are in a relatively small energy range and therefore inconclusive as to determination of RMOs. Although differences in background opacity with experimental data remain unexplained, there appears to be better agreement in detailed features. Finally, the atomic-plasma issues described in this Letter need to be resolved accurately in order to obtain astrophysical opacities to solve the outstanding solar problem.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I would like to thank Sultana Nahar for atomic data for Fe ions and discussions. The computational work was carried out at the Ohio Supercomputer Center in Columbus Ohio, and the Unity cluster in the College of Arts and Sciences at the Ohio State University.", "pages": [ 5 ] }, { "title": "DATA AVAILABILITY", "content": "The data presented herein are available upon request from the author.", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "We have shown in a recent study, using 3D climate simulations, that dayside land cover has a substantial impact on the climate of a synchronously rotating temperate rocky planet such as Proxima Centauri b. Building on that result, we generate synthetic transit spectra from our simulations to assess the impact of these land-induced climate uncertainties on water vapour transit signals. We find that distinct climate regimes will likely be very difficult to differentiate in transit spectra, even under the more favourable conditions of smaller planets orbiting ultracool dwarfs. Further, we show that additional climate ambiguities arise when both land cover and atmosphere mass are unknown, as is likely to be the case for transiting planets. While water vapour may be detectable under favourable conditions, it may be nearly impossible to infer a rocky planet's surface conditions or climate state from its transit spectrum due to the interdependent effects of land cover and atmosphere mass on surface temperature, humidity, and terminator cloud cover. Key words: planets and satellites: atmospheres - Planetary Systems, planets and satellites: terrestrial planets - Planetary Systems, software: simulations - Software, exoplanets - Planetary Systems", "pages": [ 1 ] }, { "title": "Water Vapour Transit Ambiguities for Habitable M-Earths", "content": "Evelyn Macdonald 1 ★ Kristen Menou, 1 , 2 , 3 Christopher Lee, 1 and Adiv Paradise 3 Accepted XXX. Received YYY; in original form ZZZ", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Temperate rocky planets orbiting M-dwarfs, or M-Earths, may have climates suitable for hosting life. JWST has started to produce transit spectra of small planets (Greene et al. 2023; Zieba et al. 2023), with more expected in the coming years. These spectra will provide some clues about the atmospheres - or lack thereof - of these planets, but their surfaces will be difficult to observe due to instrumental limitations. However, surface conditions are known to affect a planet's climate and habitability (e.g. Shields & Carns 2018; Lewis et al. 2018; Rushby et al. 2020; Salazar et al. 2020). In particular, we showed in Macdonald et al. (2022) that the configuration and amount of land on a synchronously rotating M-Earth can significantly affect its humidity and temperature. We will now attempt to map these landrelated climate differences to differences in the corresponding transit spectra. In this paper, we use a general circulation model (GCM) combined with a radiative transfer model to generate synthetic transit spectra over a large parameter space for synchronously rotating, habitablezoneM-Earths.Wevarytheplanet'slandfraction, land configuration, and atmosphere mass, all of which will be difficult to independently measure. We find that while the planet's temperature and humidity are heavily dependent on the parameters varied, there are significant degeneracies between climate states in synthetic transit spectra, especially in the presence of clouds.", "pages": [ 1 ] }, { "title": "2.1 General circulation model", "content": "We use ExoPlaSim (Paradise et al. 2022), a fast, intermediate complexity GCM which is able to simulate the climates of a diverse range of habitable planets. Our simulations are separated into three groups, summarized in Table 1 and described below. All are synchronously rotating. Group A is our ExoPlaSim simulations from Macdonald et al. (2022), which have the parameters of Proxima Centauri b (AngladaEscudé et al. 2016). These simulations fall under two landmap classes: substellar continent (SubCont), with a circular continent at the substellar point and ocean covering the nightside and the rest of the dayside, and substellar ocean (SubOcean), with land everywhere except for a circular dayside ocean centred at the substellar point (Fig. 1 of Macdonald et al. 2022). The land fraction is varied from 0 to 100% dayside land cover for both landmap classes. Group B are variations on Group A. Parameters are varied one at a time and land cover is systematically varied in each configuration, as in Group A. Group A and B climates are summarized in Figs. 7 and 10 of Macdonald et al. (2022). emperature and water vapour are highest when ice-free ocean area is maximized, which occurs at low land fraction for SubCont climates and at partial dayside land cover for SubOcean climates, similar to the 1 bar Group C trends seen in Figure 5 below. Group C is a new set of simulations optimized for larger atmosphere transit depth. We use a smaller star to increase the planet-star size ratio, and a smaller planet to increase the scale height of its atmosphere while keeping the planet realistic, as per the mass-radius relation of Otegi et al. (2020). We also shorten the planet's syn- chronous rotation period in a physically consistent way, such that it receives the same flux as Proxima Centauri b despite its cooler star. We use the SubCont and SubOcean landmap classes described above, again systematically varying the dayside land fraction.", "pages": [ 1, 2 ] }, { "title": "2.2 Radiative transfer model", "content": "To generate synthetic transmission spectra from our GCM simulations, we use the radiative transfer model petitRADTRANS (Mollière et al. 2019), which uses opacity data from the Exomol database (Tennyson et al. 2016). For each ExoPlaSim simulation, we use pressure, temperature, and specific humidity profiles to calculate transmission through each column of the atmosphere along the terminator. We disregard other gases in order to emphasize water vapour differences between models; water vapour can safely be isolated from the spectrum in this way because the effects of different molecules are additive to first order. We construct the planet's water vapour transit spectrum as the average of all of the terminator columns, weighted by cross-sectional area (Figure 2). For comparison, we also show dry CO 2 spectra in Figure 1.", "pages": [ 2 ] }, { "title": "3.1 Groups A and B", "content": "Substellar and terminator-averaged specific humidity profiles and cloud-free synthetic water vapour transit spectra for Group A are shown in Figure 2. All simulations have less water at the terminator than at the substellar point. There is significantly more variation in both profiles and spectra of SubCont than SubOcean models due to the higher variability in ice-free ocean area in the former. Because the water in SubOcean simulations is centred at the warmest part of the planet, at least some of this ocean is always ice-free, so evaporation can take place. As a result, water vapour can enter the atmosphere even when the substellar ocean is small, so profiles and spectra of SubOcean simulations do not depend heavily on dayside land fraction. Low-land-fraction SubCont models have spectra that fall within the range of the SubOcean models, meaning that it would be difficult to differentiate between an ocean planet with a small substellar continent and a planet with ocean covering only the central 10%ofitsdayside. For SubCont models, the substellar and terminator humidities and the amplitudes decrease steadily as the land fraction increases. The lack of clear bimodality between spectra of models with and without ice-free ocean will make it difficult to determine from a transit spectrum whether a planet has surface liquid water. To facilitate comparison between simulations, we show in Figure 3 the amplitude of the ∼ 6 𝜇 m water vapour spectral feature as a function of dayside land fraction for Groups A and B. This amplitude is defined as the maximum differential transit depth ( Δ ppm) of the spectrum. We find that amplitude has a similar land fraction dependence to temperature and water vapour (Macdonald et al. 2022, Figs. 7 and 10) in these simulations. This implies that the substantial climate differences caused by landmap changes are recovered in transit spectra, albeit at too small of a scale for detection with existing instruments.", "pages": [ 2 ] }, { "title": "3.2 Group C", "content": "Toexplore a more observationally favourable system for water vapour detection, we include a new set of simulations of a 0.2 M ⊕ planet orbiting a 2600 K M-dwarf (Group C in Table 1). Note that 0.2 M ⊕ is approximately 2 Mars masses, and larger than multiple Kepler planets, so this is a reasonable size for a terrestrial planet with an atmosphere. The planet's small size is chosen to maximize the amplitude of spectral features, since the atmosphere's scale height is inversely related to the planet's surface gravity. In most cases, the shorter rotation period does not cause a qualitative shift in dynamic regime relative to Groups A and B. Figure 4 shows the zonal mean zonal wind and tidally locked streamfunction for a sample simulation. This streamfunction describes the circulation in tidally locked coordinates, with 90 · at the substellar point, 0 · at the terminator, and -90 · at the antistellar point (Koll & Abbot 2015; Hammond & Lewis 2021; Paradise et al. 2022). Our simulated planets are Rhines rotators (Haqq-Misra et al. 2018) with two zonal jets and overturning circulation from the substellar point toward the nightside. Most precipitation in this circulation regime falls near the substellar point regardless of land configuration. Group C climate trends are shown in Figure 5. The land fraction and configuration dependence is qualitatively similar to that of Groups A and B, with a shift to higher temperatures and humidities as pN 2 increases. This pN 2 -induced warming is expected for cool stars, because warming occurs due to pressure broadening, but the cooling effect of Rayleigh scattering is minimal since the Rayleigh scattering cross section scales as 𝜆 -4 , and thus is very small at the wavelengths at which M-dwarfs emit most of their light (Paradise et al. 2021, 2022). The top left panel of Figure 6 shows the amplitude of the water vapour transit signal as a function of dayside land fraction for Group C. The trends are qualitatively similar to the Group A and B trends (Figure 3), but with significantly larger amplitudes. Each of land fraction, land configuration, and pN 2 can vary the expected peak amplitude by more than 10 ppm, and their combined effect brings this variation closer to 20 ppm, or more in the case of very dry, high-land-fraction, low-pN 2 planets.", "pages": [ 2, 3 ] }, { "title": "3.3 Clouds", "content": "We include clouds in our petitRADTRANS spectra by prescribing profiles using the cloud liquid water field of our ExoPlaSim outputs and a temperature-dependent cloud particle size parameterization from Edwards et al. (2007). The bottom left panel of Figure 6 shows the effect of clouds on the maximum water vapour transit amplitude for Group C simulations. Clouds obscure spectral features, especially for simulations with low dayside land fraction and high pN 2 , which have the most water vapour and the cloudiest terminators. As a result, some high-land-fraction spectra display water vapour the most prominently, despite not actually having the moistest atmospheres. Thefour right columns of Figure 6 show water vapour amplitude as a function of dayside ice-free ocean fraction (defined as the fraction of the dayside that is neither land nor ice), average water vapour and surface temperature, and maximum surface temperature. Without clouds (top row), the amplitude is positively correlated to the first three, except in the driest cases; maximum surface temperature is more obscure because dry models have larger day-night temperature contrasts. The bottom row shows the same variables when clouds are included. Clouds largely obscure the correlations, but do not make water vapour entirely undetectable.", "pages": [ 3 ] }, { "title": "4 DISCUSSION AND CONCLUSIONS", "content": "Using ExoPlaSim and petitRADTRANS, we have generated synthetic transit spectra for a range of synchronously rotating, potentially habitable planets with a range of land configurations. We have", "pages": [ 3 ] }, { "title": "Zonal mean zonal wind", "content": "found that for the Proxima b-sized planet from Macdonald et al. (2022), terminator water vapour is consistently high in simulations with substellar oceans, and is much more dependent on land fraction in models with substellar continents. The water vapour amplitudes of the corresponding spectra correlate to globally averaged surface temperature and atmosphere water vapour content; however, there is no clear separation between the transit spectra of the two landmap classes or of planets with and without ice-free ocean, and the transit signal is weak in all cases. We have shown that these relationships hold for a more observationally favourable system with a smaller planet orbiting a smaller star. The addition of atmosphere mass as an independent variable creates further ambiguity. There is a trend toward higher temperature, humidity, and transit depth with increasing pN 2 which is largely obscured by the inclusion of clouds. The differential transit depth of our simulations ranges from 12-47 ppm in the clear case and 12- 42 ppm when clouds are included, with all but the driest planets above 28 ppm. Aquaplanets and low-land-fraction planets are spread widely in the upper part of the range in the former case and clustered around 30-35 ppm in the latter because of their cloudier terminators. The interpretation of exoplanet transit spectra will depend heavily on clouds. Terminator cloud cover, which significantly affects a planet's spectrum, is difficult to measure because a cloud deck can be indistinguishable from the planet's surface; consequently, the altitude of the minimum transit depth is uncertain. Further, clouds are a major source of modelling uncertainty. Komacek et al. (2020) found using ExoCAM simulations that clouds can cause a drastic reduction in spectral feature amplitude. Wolf et al. (2022) note that ExoCAM clouds are sensitive to changes in the model's parameterizations. Sergeev et al. (2022); Fauchez et al. (2022) found significant differences in cloud-related climate variables between four GCMs (ExoCAM, ROCKE-3D, LMD-G, and UM) in simulations of TRAPPIST-1e. ROCKE-3D generally produced the most cloud cover, but ExoCAM had the most cloud liquid water on the nightside and the highest-altitude terminator clouds, which resulted in the largest impact on transit spectra. These differences in cloud and convection parameterizations resulted in inter-GCM cloud-related uncertainties of up to 50% in the number of transits required to detect molecules in the transmission spectrum of TRAPPIST-1e. By running ExoPlaSim simulations of these cases and the slowerrotating planet of Yang et al. (2019)'s intercomparison, Paradise et al. (2022) found that ExoPlaSim was broadly consistent with these other GCMs, but produced lower nightside cloud cover than ExoCAM. It is therefore possible that ExoPlaSim is underestimating terminator cloudiness, in which case our results would represent an optimistic estimate of water vapour detectability. Our synthetic transit spectra do not show the uncertainties that will be present in JWST data, which will have contributions from photon noise (e.g., Cowan et al. 2015) and stellar variability. The noise floor of JWST, estimated at around 10 ppm (Schlawin et al. 2021; Rustamkulov et al. 2022), will add to the challenge of detecting small spectral features. The parameter space explored in this study covers a large range of potentially habitable climates. Although water vapour and other spectral features may be detectable in some cases, the combination of unknown land fraction, land configuration, and atmosphere mass will make it difficult or impossible to precisely infer an M-Earth's climate or surface conditions from its transit spectrum in the near future. Imperfectly modelled clouds add considerable uncertainty. More work is needed to improve cloud models so that ambiguities in future data can be better understood.", "pages": [ 4 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "EMissupported by a Natural Science & Engineering Research Council (NSERC) Post-Graduate Scholarship and by the University of Toronto Department of Physics. KM is supported by NSERC. CL is supported by the Department of Physics. We would like to thank an anonymous reviewer whose comments improved this paper. TheUniversity of Toronto, where most of this work was performed, is situated on the traditional land of the Huron-Wendat, the Seneca, and the Mississaugas of the Credit. This study made substantial use of supercomputing resources; although most of the energy used comes from low-carbon sources, we acknowledge that building and operating these facilities has a significant environmental impact.", "pages": [ 4 ] }, { "title": "DATA AVAILABILITY", "content": "The simulation outputs for this study and the files needed to reproduce them are available in Borealis repositories (Macdonald et al. 2021, 2024), as per Paradise et al. (2020). The ExoPlaSim source code is available at https://github.com/alphaparrot/ ExoPlaSim/ . The petitRADTRANS source code is available at https://gitlab.com/mauricemolli/petitRADTRANS .", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Edwards J., Havemann S., Thelen J.-C., Baran A., 2007, Atmospheric Re- Anglada-Escudé G., et al., 2016, Nature, 536, 437 Cowan N. B., et al., 2015, PASP, 127, 311 search, 83, 19 Fauchez T. J., et al., 2022, Planetary Science Journal, 3, 213 Greene T. P., Bell T. J., Ducrot E., Dyrek A., Lagage P.-O., Fortney J. J., 2023, Nature, 618, 39 Hammond M., Lewis N. T., 2021, Proceedings of the National Academy of Science, 118, 2022705118 Haqq-Misra J., Wolf E. T., Joshi M., Zhang X., Kopparapu R. K., 2018, ApJ, 852, 67 Koll D. D. B., Abbot D. S., 2015, ApJ, 802, 21 Komacek T. D., Fauchez T. J., Wolf E. T., Abbot D. S., 2020, ApJ, 888, L20 Lewis N. T., Lambert F. H., Boutle I. A., Mayne N. J., Manners J., Acreman D. M., 2018, ApJ, 854, 171 Macdonald E., Paradise A., Menou K., Lee C., 2021, ExoPlaSim models for 'Climate uncertainties caused by unknown land distribution on habitable M-Earths', doi:10.5683/SP3/JCLM0O, https://doi.org/10.5683/ SP3/JCLM0O Macdonald E., Paradise A., Menou K., Lee C., 2022, MNRAS, Macdonald E., Menou K., Lee C., Paradise A., 2024, ExoPlaSim models for 'Water Vapour Transit Ambiguities for Habitable MEarths', doi:10.5683/SP3/MQVUFQ, https://doi.org/10.5683/ SP3/MQVUFQ This paper has been typeset from a T E X/L A T E X file prepared by the author.", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "Liquid-water oceans likely underlie the ice shells of Europa and Enceladus, but ocean properties are challenging to measure due to the overlying ice. Here, we consider gravity-driven flow of the ice shells of icy satellites and relate this to ocean freeze and melt rates. We employ a first-principles approach applicable to conductive ice shells in a Cartesian geometry. We derive a scaling law under which ocean freeze/melt rates can be estimated from shell-thickness measurements. Under a steady-state assumption, ocean freeze/melt rates can be inferred from measurements of ice thickness, given a basal viscosity. Depending on a characteristic thickness scale and basal viscosity, characteristic freeze/melt rates range from around O(10 -1 ) to O(10 -5 ) mm/year. Our scaling is validated with ice-penetrating radar measurements of ice thickness and modelled snow accumulation for Roosevelt Island, Antarctica. Our model, coupled with observations of shell thickness, could help estimate the magnitudes of ocean freeze/melt rates on icy satellites. Key words: planets and satellites: surfaces - planets and satellites: oceans - methods: analytical - methods: numerical", "pages": [ 1 ] }, { "title": "Nicole C. Shibley, 1 , 2 , 3 /uni2605 Ching-Yao Lai, 4 , 5 and Riley Culberg 4 , 6", "content": "1 Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 5 (currently) Department of Geophysics, Stanford University, Stanford, CA 94305, USA 4 Department of Geosciences, Princeton University, Princeton, NJ 08544, USA 6 (currently) Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY 14853, USA Accepted XXX. Received YYY; in original form ZZZ", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Several icy satellites exist in the solar system. Europa and Enceladus, in particular, have generated significant interest due to their young ice covers thought to be overlying liquid-water oceans (e.g., Cassen et al. 1979; Carr et al. 1998; Pappalardo et al. 1999; Kivelson et al. 2000; Porco et al. 2006; Postberg et al. 2009; Roth et al. 2014). Such speculation has prompted interest in these satellites as possible locations for extraterrestrial life (e.g., Hand et al. 2009; Cable et al. 2021). However, there are significant first-order questions which have yet to be answered about both the ice shells and oceans of these satellites, which may help constrain future questions about astrobiology. One key question is the thickness of the satellites' ice shells and whether or not this thickness varies spatially. On Europa, generally, the ice shell is thought to be between ∼ 3 km (e.g., Hoppa et al. 1999; Schenk 2002) and ∼ 30 km thick (e.g., Ojakangas & Stevenson 1989; Schenk 2002; Pappalardo et al. 1998; Howell 2021). Due to the lower surface temperature at the pole than at the equator, it is expected that the ice shell may be thicker near the poles than near the equator (Ojakangas & Stevenson 1989); this gradient in ice thickness may result in spatially-varying ocean stratification (Zhu et al. 2017). The presence of a lateral ice thickness gradient is also thought to occur on Enceladus (e.g., Hemingway & Mittal 2019; Beuthe 2018), with modelling results based on inferences from observations suggesting pole-to-equator thickness di ff erences of between 5 to 30 km (Hemingway & Mittal 2019). The ability of the ocean to transport heat meridionally can ultimately homogenize the ice thickness in either scenario (Kang & Jansen 2022). When ice exhibits horizontal gradients in thickness, on long enough timescales, it can flow as a viscous fluid (Pegler & Worster 2012; Worster 2014), similar to how syrup spreads on a pancake. This is due to a gravity-driven flow from regions of high pressure (thick ice) to regions of low pressure (thin ice), known as a gravity current (e.g., Huppert 1982). Gravity currents are ubiquitous in nature and describe many natural phenomena ranging from cold fronts (Simpson & Britter 1980), to mantle intrusions (Kerr & Lister 1987) to glacial flow (Kowal & Worster 2015). In the context of icy satellites, several past studies have considered how ice flow may be invoked to understand surface topography (e.g., Stevenson 2000; Nimmo 2004; Nimmo & Bills 2010; ˘ Cadek et al. 2019) and the underlying ocean (Kamata & Nimmo 2017; Ashkenazy et al. 2018; ˘ Cadek et al. 2019; Kang et al. 2022). In particular, the two-dimensional and three-dimensional general circulation modelling studies of Ashkenazy et al. (2018); Kang & Jansen (2022); Kang (2022); Kang et al. (2022) have related the lateral ice flow on icy satellites to ocean dynamics (and vice versa), considering cases with both meridional ocean heat transport and ice convection (Ashkenazy et al. 2018), tidal heating in the shell (e.g., Ashkenazy et al. 2018; Kang et al. 2022), the e ff ects of gravity (Kang & Jansen 2022), and oceanic eddy transport (Kang 2022). Such general cir- lation models have the advantage of simulating multiple physical processes of a complex system in a global setup. However, a limitation of such models is the number of free parameters inherent to the system, making it challenging to invert for any single parameter given a set of observations. More specifically, the modeling study of Ashkenazy et al. (2018) consider a conductive (along with convective) spherical setup for Europa's ice shell. In their seminal work, they describe how Europan ice thicknesses would look under various ice-ocean configurations (for example, a conductive versus convective ice shell, with varying e ff ects of internal heating, with/without ocean heat transport, and with di ff erent ocean di ff usivities), incorporating the e ff ects of ocean freezing/melting into their model equations. In our study, we consider the converse: if one knows the ice-thickness distribution from observations, what can realistically be said about the ocean, specifically about freeze and melt rates? Thus, here we attempt to distill the governing physics for a purelyconductive shell with a temperature-dependent viscosity to provide an understanding of the simplest ocean parameters which can realistically be inferred from future, expected ice-thickness measurements. A two-dimensional floating viscous gravity current is considered, where ice flows from pole to equator and under which an ice thickness gradient can be sustained by spatially-varying freezing and melting. The value of our O(0) simplified model is that it requires only one choice of free parameter (the basal viscosity, /u1D702 /u1D44F ), while containing all information about the ocean into a freezing and melting term, /u1D44F ( /u1D465 ) . We examine how freeze and melt rates can be inferred from lateral thickness gradients in a steady-state, and relate these to di ff erent viscosity regimes. We further explicate a scaling law which describes how freeze/melt rates can be estimated from ice thickness scales. Finally, our simplified model and scaling are compared to Earth-based radar observations of Antarctica to corroborate our results.", "pages": [ 1, 2 ] }, { "title": "A SIMPLIFIED ICE-OCEAN MODEL", "content": "We consider a simplified setup, with an 'inviscid' ocean underlying the ice shell. The ice shell experiences a temperature gradient across it since the surface temperature is much colder than the basal temperature (e.g., Ashkenazy 2019), leading to a depth(temperature)dependent viscosity (e.g., Goldsby & Kohlstedt 2001). This leads to an upper, brittle ice lid under which sits a flowing, viscous ice layer (Figure 1). To illustrate the dynamics, we consider a two-dimensional setup, where ice flows laterally from pole to equator (Figure 1). This setup is predicated on the assumption that there is thick ice at the pole and thin ice at the equator, as in Ojakangas & Stevenson (1989), for example; such a gradient would be set up by a pole-to-equator temperature di ff erence with colder temperatures at the pole and warmer temperatures at the equator (see e.g., Ashkenazy et al. 2018; Ashkenazy 2019). We note that studies which have considered the influence of ice convection have suggested that it may be possible to setup the reverse gradient, with thicker ice at the equator and thinner ice at the pole (Ashkenazy et al. 2018). Our analysis applies only to a conductive system. Thus, we restrict our values of basal viscosity and thickness to values which can generally be expected to be in a conductive heat-transport regime (see e.g., Shibley & Goodman 2024; McKinnon 1999). We further note that we do not consider an ice-pump mechanism (Lewis & Perkin 1986) here.", "pages": [ 2 ] }, { "title": "Mathematical Formulation", "content": "Thesystem can be described by the following equations, which generally follow the standard gravity current equations of Huppert (1982) and which we extend to include the e ff ect of a temperature-dependent viscosity. The total thickness of the ice shell is /u1D43B = /u1D43B /u1D446 + /u1D43B /u1D435 , where /u1D43B /u1D446 falls above the /u1D467 = 0 sea level, and /u1D43B /u1D435 falls below the /u1D467 = 0 line. Then, /u1D43B /u1D446 = ( 1 -/u1D70C /u1D456 / /u1D70C /u1D464 ) /u1D43B , where /u1D70C /u1D456 = 920 kg m -3 is the density of ice, and /u1D70C /u1D464 = 1000 kg m -3 is the density of the ocean. We start with conservation of mass: /u1D715/u1D462 /u1D715/u1D465 + /u1D715/u1D464 /u1D715/u1D467 = 0 , and conservation of momentum for Stokes flow, combined with the fact that the horizontal length scale is much larger than the vertical length scale, leading to: where /u1D45D is pressure, /u1D462 is the velocity in the /u1D465 -direction, /u1D464 is the velocity in the /u1D467 -direction, /u1D702 ( /u1D447 ) is a temperature-dependent viscosity, /u1D70C /u1D456 is ice density, and /u1D454 is gravity. We present our method in Cartesian coordinates here for ease of understanding, but the same physics holds regardless of coordinate system. A temperature-dependent (or equivalently depth-dependent) viscosity is appropriate since the upper surface of the ice shell will be significantly colder than the base. Here we employ the FrankKamenetskii approximation (e.g., Jain & Solomatov 2022), defined as: where /u1D702 /u1D44F is the specified basal viscosity, ˜ /u1D447 = /u1D447 -/u1D447 /u1D446 /uni0394 /u1D447 , /uni0394 /u1D447 = /u1D447 /u1D435 -/u1D447 /u1D446 , /u1D447 /u1D435 = 273 /u1D43E is the ice temperature at the ice-ocean interface, /u1D447 /u1D446 = 93 /u1D43E is an ice surface temperature (appropriate for Europa, Ashkenazy 2019), and /u1D459 = /u1D445/u1D447 2 /u1D435 /u1D444 /uni0394 /u1D447 , where /u1D445 is the gas constant and /u1D444 = 60 kJ mol -1 is the activation energy. For Enceladus, an appropriate ice surface temperature would be ∼ 60 K (e.g., Hemingway & Mittal 2019). We note here that we take /uni0394 /u1D447 to be constant across the entire ice shell, though lateral variations in the temperature jump will exist; these are what give rise to variations in ice thickness in the first place. This approximation allows for the equation to be solved analytically. We note, however, that if we consider the case where the change in temperature jump is ∼ 223 K (i.e., taking the surface temperature to be a Europan polar surface temperature, 50 K) and which would really only be relevant near the pole, the main results do not vary significantly. The freeze-melt scaling law discussed later returns slightly lower values in the polar surface temperature case, but approximately the same order of magnitude regardless. This is likely because the e ff ect of the increased e ff ective viscosity (due to the larger temperature jump) is most pronounced closest to the surface, where the ice shell is already not dynamically very interesting. Then, taking /u1D45D = /u1D45D /u1D45C + /u1D70C /u1D456 /u1D454 ( /u1D43B /u1D460 -/u1D467 ) , where /u1D45D 0 is a reference pressure, yields: The relevant boundary conditions are no-slip at the top surface (brittle lid), and free-slip (no-stress) at the ocean-ice interface, given by: /u1D462 ( /u1D467 = /u1D43B /u1D446 ) = 0 , and /u1D715/u1D462 /u1D715/u1D467 | /u1D467 = -/u1D43B /u1D435 = 0 . where /u1D43B /u1D446 -/u1D467 /u1D43B = /u1D447 -/u1D447 /u1D446 /uni0394 /u1D447 for a linear temperature gradient across the shell. /u1D462 = ˜ - Then, nondimensionalizing where ˜ /u1D467 = ( /u1D43B /u1D460 -/u1D467 ) /u1D43B , /u1D451 ˜ /u1D467 = -/u1D451/u1D467 /u1D43B , and /u1D462/u1D702 /u1D44F /u1D452 1 / /u1D459 /u1D70C /u1D456 /u1D454/u1D43B 2 /u1D715/u1D43B /u1D446 /u1D715/u1D465 gives: ( / ) with boundary conditions: /u1D715 ˜ /u1D462 /u1D715 ˜ /u1D467 | ˜ /u1D467 = 1 = 0 , and ˜ /u1D462 ( ˜ /u1D467 = 0 ) = 0 . This can be solved to give an analytical dimensionless velocity profile: Finally, depth-integrating the mass conservation equation gives: This can be rewritten as: where /u1D45E = /uni222B /u1D43B /u1D446 -/u1D43B /u1D435 /u1D462/u1D451/u1D467 and /u1D44F ( /u1D465 ) is the source/sink term that describes background freeze and melt rates from the ocean. Finally, defining /u1D6FE = /u1D6FD/u1D452 -1 / /u1D459 andrecalling that /u1D43B /u1D446 = ( 1 -/u1D70C /u1D456 / /u1D70C /u1D464 ) /u1D43B (note this balance assumes isostasy, relevant to large-scale/longwavelength thickness changes, see e.g., Nimmo et al. (2007)), we obtain the following: This equation describes how the thickness of an ice shell with a temperature-dependent viscosity changes in time as it flows laterally and is modified by spatially-varying freezing and melting.", "pages": [ 2, 3 ] }, { "title": "RESULTS", "content": "Changes in shell thickness due to gravity-driven flattening compete with thickening and thinning of the shell due to freezing and melting of ice driven by ocean heat fluxes (see also Kang & Jansen 2022; Kang 2022). Given the temperature-dependence of viscosity and the temperature gradient across the shell, the flowing portion of the shell is confined to the bottom of the shell (Figure 2a); this can also be seen from the functional form of Equation 5. The flow rate varies in space and in time depending on the evolving local thickness gradient. It is also larger for lower basal viscosities, and smaller for higher basal viscosities. For example, for the ice shell shown here after 100 million years with a basal viscosity of 10 14 Pa s, flow rates vary from approximately 0 to 2.6 mm/year (Figure 2a). Unless sustained by ocean freezing and melting, the ice thickness will homogenize over time (Figure 2b); ice shells with lower basal viscosities flatten faster than those with larger viscosities (Figure 2c). Recent work (Kihoulou et al. 2023) has also shown how larger e ff ective viscosities can lead to steeper ice shell topography.", "pages": [ 3 ] }, { "title": "Spatial Variation of Freeze and Melt Rates", "content": "Freezing and melting concurrent with ice flow modifies the shellthickness distribution (Figure 3a). If ice freezes at the pole and melts at the equator at exactly the same rate as ice moves laterally between the two regions, then there will be no temporal change in ice thickness, and the ice will remain in steady-state. If an ice cover can be considered to be in steady state (a common assumption for planetary ice shells, e.g., Ojakangas & Stevenson 1989; Hussmann et al. 2002; Tobie et al. 2003; Ashkenazy et al. 2018; Hemingway & Mittal 2019; Akiba et al. 2022; Kang & Jansen 2022; Kang 2022), this then provides a method by which to calculate both the spatial distribution of ocean freeze and melt rates based on global measurements of ice thickness, as well as a way to infer the magnitudes of maximum ocean freeze/melt rates and their spatial locations. Ocean freezing or melting results when the ice-ocean heat flux at the ice-ocean interface and the vertical conductive flux out of the base of ice shell are out of balance (see e.g., Shibley & Goodman 2024, for a simplified vertical model). The magnitude of ice-ocean heat flux arises from ocean heat-transport processes (such as turbulent mixing or molecular di ff usion) which redistribute heat from the ocean towards the ice-ocean interface. Several studies incorporate ocean mixing parametrizations into their modeling of planetary ice or ocean dynamics (Kang 2022, 2023; Zeng & Jansen 2024). Here, we instead infer ocean freeze and melt rates, which result from ocean mixing/heat-transport processes, from the ice thickness measurements. This term /u1D44F ( /u1D465 ) captures the e ff ect of ocean heat transport, which causes the freezing and melting, without requiring a need for a particular knowledge of specific ocean processes. In fact, these inferred freezing and melting rates could o ff er insight or validation of hypothesized subsurface mixing processes in the ocean. We describe how steady-state measurements of ice thickness can be used to infer information about ocean freezing and melting next. The freeze/melt rate necessary to result in a steady-state ice thickness can be calculated via the following equation (see Equation 8, Figure 3b): The spatial location of the maximum or minimum freeze/melt rate can then be found by solving: The magnitude of the freeze and melt rate can then be determined via Equation 9. This means that a su ffi ciently-resolved global icethickness distribution /u1D43B ( /u1D465 ) would indicate the locations ( /u1D465 ) of maximum/minimum melt rates in an icy satellite ocean.", "pages": [ 3 ] }, { "title": "Scalings of Ocean Dynamics", "content": "Further, how the freeze and melt rate depends on both ice shell basal viscosity, as well as ice shell thickness, can be described by a scaling law. Such a law arises by considering characteristic scales for each of the terms in Equation 8 and balancing them against each other; the scaling has the advantage of necessitating sparse ice-thickness measurements to make estimates of the freeze/melt rate and does not require a steady-state assumption. This means that even if upcoming space missions do not return a robust spatial map of observations, a scaling may still be used to estimate freeze/melt rates on an icy satellite, provided that /u1D715/u1D43B / /u1D715/u1D461 is not much larger than the other terms in Equation 8. Disadvantages of a scaling approach are that this does not give precise location measurements as to freezing and melting, and rather a single order-of-magnitude estimate, which furthermore is dependent upon knowledge of the basal viscosity (a parameter currently unknown for icy satellites). A characteristic freeze and melt rate follows: where /u1D44F 0 is a characteristic freeze-melt scale. Here we take a characteristic horizontal scale /u1D43F 0 to be the pole-to-equator distance and a characteristic vertical scale /u1D43B 0 to be the thickness of ice at the pole (i.e., the maximum ice thickness at any space or time). Further, recall that /u1D6FE = /u1D459/u1D452 -1 / /u1D459 [ 2 /u1D459 2 ( /u1D452 1 / /u1D459 -1 ) -2 /u1D459 -1 ] , where /u1D459 = ( /u1D445/u1D447 2 /u1D435 )/( /u1D444 /uni0394 /u1D447 ) . If a scaling prefactor /u1D458 , whose value depends on the shape of /u1D43B ( /u1D465 ) , is included, this gives /u1D44F 0 = /u1D458 /u1D6FE/u1D70C /u1D456 /u1D454 /u1D702 /u1D44F ( /u1D70C /u1D464 -/u1D70C /u1D456 /u1D70C /u1D464 ) /u1D43B 4 0 /u1D43F 2 0 . For the particular thickness profile examined here, we find /u1D458 = 1 . 1 , indicating a good fit between the scaling law and the estimated freeze and melt rate. This is taken from a linear fit between /u1D6FE/u1D70C /u1D456 /u1D454 /u1D702 /u1D44F ( /u1D70C /u1D464 -/u1D70C /u1D456 /u1D70C /u1D464 ) /u1D43B 4 0 /u1D43F 2 0 and the mean of | /u1D44F ( /u1D465 )| calculated from Equation 9, using the initial thickness profile (described in the Materials and Methods section) shown in Figure 3a. Note that the characteristic vertical scale /u1D43B 0 is taken to be a vertical thickness rather than a thickness di ff erence as for the profiles tested this yielded a calculated prefactor /u1D458 closer to 1. Although we are considering a simplified Cartesian problem here, the extension to a spherical coordinate system can be done without a ff ecting the scaling in Equation 11. The exact magnitude of freeze and melt depends on the shape of the thickness profile. In spherical coordinates, following the derivation described in Ashkenazy et al. (2018) assuming zonal symmetry, integrating from the base to the surface (instead of to a transition temperature, as in their work) and taking a relationship between stress /u1D70F and strain rate /dotacc /u1D716 as /u1D70F /u1D456/u1D457 = 2 /u1D702 ( /u1D447 ) /dotacc /u1D716 /u1D456/u1D457 (instead of /u1D70F /u1D456/u1D457 = /u1D702 ( /u1D447 ) /dotacc /u1D716 /u1D456/u1D457 ), it can be shown that the governing equation in the spherical setup is: where /u1D703 is co-latitude. Note the similarities between this equation and equation 8 in the Cartesian system. This gives a characteristic scale for melt/freeze rate of /u1D44F 0 ∼ /u1D6FE/u1D70C /u1D456 /u1D454 /u1D702 /u1D44F /u1D70C /u1D464 -/u1D70C /u1D456 /u1D70C /u1D464 /u1D43B 4 0 /u1D445 2 0 , the same functional form as the Cartesian scaling, where now the length scale /u1D445 0 is taken to be the radius of the satellite. Note that these scalings apply to a system approximated as two-dimensional, where the shell thickness varies in one of the dimensions. If a scaling prefactor /u1D458 is included as in /u1D44F 0 = /u1D458 /u1D6FE/u1D70C /u1D456 /u1D454 /u1D702 /u1D44F /u1D70C /u1D464 -/u1D70C /u1D456 /u1D70C /u1D464 /u1D43B 4 0 /u1D445 2 0 , it will be a scaling prefactor particular to the spherical setup (any coordinate system will have its own prefactor). We choose to present our methodology in a Cartesian basis for ease of explanation. However, the ability to infer the correct magnitude of ice shell freezing and melting is dependent on a knowledge of the ice viscosity whichcontrols how quickly the ice spreads; this is not well-prescribed (see e.g., Shibley & Goodman 2024). For the same initial thickness profile, a shell with a lower basal viscosity will require a larger freeze/melt rate in order to sustain a steady-state shell thickness than a shell at larger viscosity (Figure 3b). For example, for the profile shown in Figure 3b, the maximum freeze/melt rate necessary to maintain a steady-state thickness is about 0.07 mm/year for a shell with /u1D702 /u1D44F = 10 14 Pa s, whereas for a shell with basal viscosity /u1D702 /u1D44F = 10 15 Pa s, the magnitude of the maximum freeze/melt rate necessary to maintain a steady-state is one order of magnitude smaller (about 0.007 mm/year). This is because the viscous portion of the shell will flow more slowly in the case of larger basal viscosity; thus, the rate of freezing and melting needed to o ff set this flow is smaller than for shells with a lower basal viscosity. The scaling (Equation 11) indicates that with a characteristic shell thickness scale and with a knowledge of ice viscosity, a characteristic freeze/melt rate can be inferred for the ocean. For a larger characteristic vertical thickness /u1D43B 0 , the characteristic freeze and melt rate /u1D44F 0 will be larger than for smaller vertical thicknesses at the same viscosity (Figure 3c); this is because shells for which the value | 3 /u1D43B 2 ( /u1D715/u1D43B /u1D715/u1D465 ) 2 + /u1D43B 3 /u1D715 2 /u1D43B /u1D715/u1D465 | is larger accumulate/flatten ice faster than at lower values. It is important to note that the ice shell viscosity does not physically determine the freeze and melt rate of the ice shell (this is governed by ocean dynamics, e.g., Ashkenazy et al. 2018; Kang 2022), but rather that a knowledge of the ice rheology is required in order to infer an ocean freeze/melt rate based on measurements of ice thickness. Depending on the basal ice shell viscosity, varying here from 10 14 to 10 16 Pa s and the characteristic vertical thickness scale (which will depend on the shell profile), characteristic freeze and melt rates vary between approximately 10 -1 and 10 -5 mm/year (Figure 3c); these are representative of a Europan setup. This means that with a spatial map of ice-thickness observations, such as is expected from Clipper or JUICE , the spatial distribution and magnitudes of ocean freeze and melt rates should be calculable under a steady-state assumption, with a knowledge of basal viscosity. Further, the locations of maximum and minimum ocean freeze and melt rates are calculable regardless of a knowledge of basal viscosity. Finally, in the absence of a steady-state assumption, and provided that /u1D715/u1D43B / /u1D715/u1D461 is not much larger than the other terms in Equation 8, a scaling for freeze and melt rate can still be inferred from icethickness observations. One advantage of using this method to infer freeze/melt rate is that it may prove useful for making inferences of ocean stratification, a control on ocean dynamics of icy satellites. This is because regions of large melt rate are regions of high rates of freshwater input; here, it may be expected that a strong two-layer ocean stratification would exist (i.e., Zhu et al. 2017).", "pages": [ 4, 5 ] }, { "title": "EARTH ANALOG RADAR VALIDATION", "content": "In order to validate our methodology, we consider radar observations of an Earth analog. Radar sounding is an active remote sensing technique that has been used extensively on both Earth and Mars to measure the thickness of ice sheets and ice shelves, leveraging the relative radio-transparency of ice (Gogineni et al. 2001; Holt et al. 2006; Vaughan et al. 2006; Plaut et al. 2007; Phillips et al. 2008). Figure 4a shows an example of data collected over Roosevelt Island, Antarctica as part of NASA's Operation IceBridge that resolves the ice surface, internal reflecting horizon, and the bedrock on which the ice rise is grounded. Given the electromagnetic wave velocity in ice, spatial and temporal variations in ice thickness can be measured directly from such data. NASA's Europa Clipper and ESA's JUICE mission will carry similar radar sounding instruments intended to study the subsurface structure and dynamics of Europa and Ganymede's ice shells, including ice shell thickness (Blankenship et al. 2017; Bruzzone et al. 2011). We show that a simple scaling of the type we propose can be used to reasonably infer accumulation rates on Roosevelt Island from ice-penetrating radar measurements of ice thickness.", "pages": [ 5 ] }, { "title": "Earth-analog equations", "content": "Our icy satellite case is governed by a shear flow with a no-slip upper surface, due to the extremely cold temperatures, and a freeslip base, due to the presence of the ocean. The case of Roosevelt Island, Antarctica is similar, but with a no-slip base in contact with bedrock and a free-slip surface in contact with the atmosphere. On Earth, the relevant flow to consider is that of an ice sheet (governed by shear flow), rather than an ice shelf (which is an extensional flow, e.g., Pegler & Worster 2012; Worster 2014), even though both the Earth-based ice shelf and the ice shell of an icy satellite are floating. Here the analog to the required freezing and melting term of the ice shell needed to maintain the shell in thermodynamic balance is the accumulation of snow at the surface of the ice rise. The equivalent governing equation for our Earth analog is then: where now /u1D6FE /u1D452 = /u1D459 [ 2 /u1D459 2 ( 1 -/u1D452 -1 / /u1D459 ) -2 /u1D459 + 1 ] . Note that the di ff erence between /u1D6FE /u1D452 in Equation 13 and /u1D6FE in Equation 8 for the icy satellite setup arises from the inverted boundary conditions. In the limit of /u1D447 /u1D44F = /u1D447 /u1D460 , /u1D6FE /u1D452 and /u1D6FE both reduce to 1/3.", "pages": [ 5 ] }, { "title": "Validation of Scaling", "content": "On Earth, the freeze/melt rate scaling goes as: Note the di ff erence between this scaling and the scaling for an icy satellite, which contains a hydrostatic component related to the floating shell. Further, while freezing at the surface of Antarctica would result in meteoric ice, and freezing under an ice shell on Europa would result in congelation or marine ice (e.g., Wolfenbarger et al. 2022; Lawrence et al. 2023), it is the rate of accumulation and viscosity of ice which governs the flow dynamics, though it does seem likely that the basal viscosity, an unknown, may be related to the type of ice formed. Estimates from Roosevelt Island radar observations suggest scalings of /u1D43B 0 = 750 m and /u1D43F 0 = 35 km (Figure 4a,b). We take /u1D70C /u1D456 = 920 kg m -3 , and /u1D6FE /u1D452 ∼ 0 . 1 -0 . 3 . (If the surface and basal temperatures vary by 10 Kelvin, for /u1D447 /u1D44F = 273 K, /u1D6FE /u1D452 ∼ 0 . 3 , taking an activation energy /u1D444 of about 60 kJ mol -1 ). The viscosity of ice sheets on Earth is also not particularly well-known and much research has investigated appropriate rheology profiles for Antarctic ice sheets (e.g., Larour et al. 2005; Millstein et al. 2022; Wang et al. 2023). Here, we estimate an e ff ective viscosity of /u1D442 ( 10 14 ) -/u1D442 ( 10 15 ) Pa s (see Materials & Methods); we assign this value to /u1D702 /u1D44F . Using these parameters, we find an estimated freeze/melt scaling of ∼ 70 mm/year for /u1D6FE /u1D452 ∼ 0 . 1 to ∼ 220 mm/year for /u1D6FE /u1D452 ∼ 0 . 3 , taking /u1D702 /u1D44F ∼ /u1D442 ( 10 14 ) . These rates fall in line with the estimated accumulation rates expected for Roosevelt Island (between about 100-200 mm year -1 , Bertler et al. 2018; Winstrup et al. 2019, Figure 4b). This indicates that a scaling based on gravity-driven flow dynamics and thickness/lengthscale estimates may be useful for estimating accumulation rates both on Earth and on icy satellites. Similar mass-balance methodologies have also been invoked to infer melt rates at the base of Antarctic ice shelves (e.g., Wen et al. 2010; Padman et al. 2012; Adusumilli et al. 2020), further supporting this approach.", "pages": [ 6 ] }, { "title": "Summary", "content": "On long time scales, ice flows as a viscous fluid (e.g., Pegler & Worster 2012; Worster 2014). Here we describe the dynamics of gravity-driven ice shell flow and describe how this can be used to infer melt and freeze rates under the ice shells of icy satellites, which are currently impossible to measure directly. Since the ice shell of an icy satellite experiences a steep temperature drop from its base to its surface, and viscosity depends exponentially on temperature, we formulate the viscous gravity current equations to include a temperaturedependent viscosity. This shows analytically how the viscous flow of the ice shell is confined to its base; the thickness of the viscous portion of the shell depends on the across-shell temperature di ff erence. In the limit of no temperature jump, our equation reduces to the classical gravity current equation with no-slip and free-slip boundary conditions (Huppert 1982). We describe how a balance between ice flow and freeze and melt rate implies a scaling for freeze and melt rate which depends on a vertical ice thickness scale and a horizontal length scale. We further describe how in steady state, ocean freeze and melt rates and the spatial locations of maximum freeze and melt rates can be calculated from ice-thickness measurements. Inferences from radar observations from Roosevelt Island, Antarctica are used to corroborate our scaling and give credence to the methodology we propose. We expect that with this methodology, ocean parameters such as freeze and melt rates may be inferred with future measurements of ice thickness from upcoming space missions.", "pages": [ 6 ] }, { "title": "Are Europa and Enceladus in Steady State?", "content": "Much past work, whether considering the dynamics or thermodynamics (or both) of Europan and Enceladean ice shells, assumes that the ice shells have reached steady state (e.g., Ojakangas & Stevenson 1989; Hussmann et al. 2002; Tobie et al. 2003; Ashkenazy et al. 2018; Hemingway & Mittal 2019; Akiba et al. 2022; Kang & Jansen 2022; Kang 2022). Nonetheless, it is not actually clear if these shells can be considered to be in equilibrium (see Shibley & Goodman 2024, for a thermodynamic case on Europa) and ( ˘ Cadek et al. 2019, for a dynamic case on Enceladus). If the steady-state assumption does not apply, then it would not be possible to directly calculate a spatially-varying freeze and melt rate as described. However, the freeze and melt rate scaling will still be applicable provided that the term /u1D715/u1D43B / /u1D715/u1D461 is not much larger than the nonlinear di ff usive term or the freeze and melt term in Equation 8 (See Section B of Materials & Methods for more discussion). Such a system is exemplified via the radar validation of Roosevelt Island, Antarctica in the previous section which is not strictly in steady state, but for which the scaling holds. We note that at timescales appropriate for the surface age of Europa, /u1D715/u1D43B / /u1D715/u1D461 is a similar order of magnitude to the nonlinear di ff usive term for lower values of viscosity (i.e., ∼ 10 14 Pa s) and larger thickness scales /u1D43B 0 ; higher /u1D702 /u1D44F and lower /u1D43B 0 increase the timescale of flattening for the system (see Section B of Materials & Methods).", "pages": [ 7 ] }, { "title": "What does Freeze and Melt Rate Tell You?", "content": "Ultimately, an understanding of ocean freeze and melt may give insight into ocean stratification. Stratification is a key control on an expected ocean circulation and thus the transport of nutrients and other tracers that may be of astrobiological interest (see Lobo et al. 2021, who describe this for the case of Enceladus). Consider for example a region of high melt rate. This would result in a large influx of freshwater into a particular region of the ocean, likely generating a strong stratification and depressing isopycnals (surfaces of constant density). A similar description of a freshwater-stratified region and the circulation it implies via conservation of salt and heat has been described in Zhu et al. (2017). Our methodology provides a framework to infer freeze and melt rates, which are otherwise challenging to measure under kilometers of ice, from observations of ice thickness; these rates ultimately relate to stratification. An important point for future work is to help constrain the basal viscosity of the ice shell. This has been a limitation of modeling studies of ice shells of icy satellites (including our own), as the rheology controls the relevant dynamics but is not well-defined (e.g., Shibley &Goodman 2024). A particular issue is that the elastic processes on the surface can not be easily inverted via a temperature-dependence to constrain the viscous processes of the base. Developing a theoretical or observational way to constrain the basal viscosity is a key area for future research.", "pages": [ 7 ] }, { "title": "A. Numerical Method", "content": "Equation (8) is a one-dimensional partial di ff erential equation that is solved numerically subject to the following conditions: We approximate the initial thickness of the ice shell as a complementary error function profile: where /u1D43F is the end of the horizontal domain. We use a forward di ff erence scheme in time and a centered-di ff erence scheme in space. The spatial grid step is 10 km, and the time step is 5 × 10 8 seconds. We set up a grid of 241 points, equivalent to a horizontal length of 2.4 × 10 6 m, approximately the distance from Europa's pole to equator. where /uni210E 0 is some initial background thickness, /u1D44E = /uni210E 0 / 8 , and /u1D450 is 2.8 × 10 -6 m -1 . The constants /uni210E 0 , /u1D44E , and /u1D450 can be changed to approximate di ff erent initial thickness profiles. In the figures presented here (aside from Figure 3c, where /uni210E 0 varies), we take /uni210E 0 = 8 km. This functional form is chosen to simulate an ice shell which was thicker at the pole and thinner at the equator, whose thickness decreased between pole to equator in some reasonable way. An initial pole thickness of about 10 km is taken since this is expected to be a plausible estimate for a conductive ice shell on Europa.", "pages": [ 7 ] }, { "title": "B. Flattening Timescale", "content": "The timescale at which an ice sheet flattens depends on the viscosity and on the thickness scale of the ice shell. This follows the scaling law: where /u1D447 is a characteristic time scale, /u1D43B is a characteristic thickness, Our assumption of a linear temperature profile implies that possible internal shell heating does not significantly a ff ect the temperature profile; we expect that the e ff ect of including internal heating in our conductive setup would essentially reduce a depth-integrated viscosity in the lower portion of the shell (by keeping temperatures higher), likely leading to somewhat faster ice flow and thus increased freeze/melt rates. 0 0 /u1D43F 0 is a characteristic horizontal length scale, /u1D70C /u1D464 is the density of ocean water, /u1D70C /u1D456 is the density of ice, /u1D454 is gravity, and /u1D702 /u1D44F is the basal ice viscosity. In order for the steady-state assumption to apply here requires /u1D447 0 >> /u1D43B 0 / /u1D44F 0 (or equivalently /u1D447 0 >> /u1D702 /u1D44F /u1D70C /u1D456 /u1D454/u1D6FE ( /u1D70C /u1D464 /u1D70C /u1D464 -/u1D70C /u1D456 ) /u1D43F 2 0 /u1D43B 3 0 ). Thus we can find a scaling for both the time at which a system subject to freezing and melting can be approximated by a steady-state, and a scaling for freeze and melt rate which would maintain the ice thickness gradient.", "pages": [ 7 ] }, { "title": "C. Estimate of Viscosity", "content": "Following the established convention in terrestrial glaciology (Glen 1958; Nye 1957), we define an e ff ective viscosity as: where /dotacc /u1D716 is the strain rate, and /u1D435 is defined as /u1D435 ( /u1D447 ) = 2 . 207 exp ( 3155 /u1D447 -0 . 16612 ( 273 . 39 -/u1D447 ) 1 . 17 ) (Hooke 1981; van der Veen 1998), where /u1D447 is temperature. Estimates for for two extreme temperatures are as follows: /u1D435 ( /u1D447 ) Finally, the order of magnitude of the e ff ective strain rate /dotacc /u1D716 on Roosevelt Island as derived from satellite measurements of surface ice velocities is about 5 × 10 -4 year -1 . Based on the upper and lower bounds of /u1D435 , the e ff ective viscosity can vary in the range of /u1D702 = 6 × 10 14 -2 × 10 15 Pa s.", "pages": [ 7, 8 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "N.C.S., C.-Y.L., and R.C. acknowledge internal funding support from Princeton University. N.C.S. acknowledges a postdoctoral fellowship from the Princeton Center for Theoretical Science. R.C. acknowledges the Harry Hess Postdoctoral Fellowship from the Princeton Department of Geosciences. C.-Y.L. acknowledges internal funding from Stanford University, and R.C. acknowledges support from Cornell University. N.C.S. acknowledges helpful conversations with Jeremy Goodman and Glenn Flierl. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. This work was partially supported by a grant from the Simons Foundation.", "pages": [ 8 ] }, { "title": "DATA AVAILABILITY", "content": "The numerical method to solve Equation 8 is given in the Materials and Methods appendix of the text. from the University of Kansas Center for Remote Sensing and Integrated Systems (last access: 2023-12-08, Leuschen et al. (2023)). (We acknowledge the use of data and/or data products from CReSIS generated with support from the University of Kansas, NASA Operation IceBridge grant NNX16AH54G, NSF grants ACI-1443054, OPP-1739003,andIIS-1838230,Lilly Endowment Incorporated, and Indiana METACyt Initiative.) The ice-penetrating radar data shown in Figure 4 are available The rate of ice-thickness change ( /u1D715/u1D43B / /u1D715/u1D461 ) can be inferred from the data at University of Washington Research Works (last access: 2023-12-08) (Smith et al. 2020). MARregional climate simulations are available from Zenodo (last access: 2023-12-08) (Kittel et al. 2021; Mottram et al. 2021). We thank the MAR team which make available the model outputs, as well agencies (F.R.S - FNRS, CÉCI, and the Walloon Region) that provided computational resources for MAR simulations. The accumulation range in Figure 4b was generated in part using ERA5-Land reanalysis products, which are available from the Copernicus Climate Data Store (Muñoz-Sabater 2019, last access: 2023-12-08). This contains modified Copernicus Climate Change Service information 2023. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains. RACMO regional climate simulations are available from Zenodo (last access: 2023-12-8) (van Wessem et al. 2018; van Wessem et al. 2023). In situ inferred accumulation rates come from Bertler et al. (2018) and Winstrup et al. (2019).", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Adusumilli S., Fricker H. A., Medley B., Padman L., Siegfried M. R., 2020, Nature Geoscience, 13, 616 Kang W., Mittal T., Bire S., Campin J.-M., Marshall J., 2022, Science Ad- vances, 8, eabm4665 Kerr R. C., Lister J. R., 1987, Earth and Planetary Science Letters, 85, 241 Kihoulou M., Č adek O., Kvorka J., Kalousová K., Choblet G., Tobie G., 2023, Icarus, 391, 115337 present, Copernicus Climate Change Service (C3S) Climate Data Store This paper has been typeset from a T E X/L A T E X file prepared by the author.", "pages": [ 8, 9 ] } ]

[ { "title": "Akshat Pandey ∗", "content": "Department of Physics, Shiv Nadar Institution of Eminence Greater Noida, Uttar Pradesh-201314, India.", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the recent gravitational analogue of the Aharonov-Bohm effect for a classical system, namely a complex scalar field. We use this example to demonstrate that the Aharonov-Bohm effect in principle has nothing to do with quantum-mechanics. We then discuss how this classical field description can be connected to the standard one particle quantum description of the AharonovBohm effect.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In 1959, Aharonov and Bohm in their seminal paper [1] proposed a phenomenon within quantum mechanics which demonstrated that potentials and field intensities are not equivalent descriptions [2]. They studied the effects of an external electromagnetic potential on the quantum wave-function of a charged particle. In particular they showed that the presence of a non-zero vector potential can lead to observable effects, even for cases where the corresponding field strength vanished. Ever since, this Aharonov-Bohm (AB) effect has become a textbook example to emphasise that potentials are more fundamental than field intensities. Subsequently, there have been several generalisations of the Aharonov-Bohm effect. These include relativistic generalisations [3-6] and the non-stationary AB effect [7]. Recently there have been papers exploring the electric AB effect as opposed to the magnetic AB effect initially [8] and an alternative scalar electric AB effect studied by Chiaov et al. [9]. Gravitational analogues for the AB effect were first studied in the 1980s [10, 11]. More recently, following [9] Chiaov et al. proposed a novel model for gravitational AB effect [12]. The usual analysis of the AB effect involves starting with a single particle wavefunction in the presence of an external potential which is responsible for the AB effect. Due to reasons related to history and practice, the AB effect is thought to be a consequence of the 'weirdness' of quantum mechanics. However as Wald has emphasised [14], the Aharonov-Bohm effect has nothing to do with quantum mechanics -the same effect can be observed for a classical field. Thought of this way, the AB effect is a consequence of the non-trivial topology of the background space the field pervades in; field strengths and potentials are equivalent in any simply-connected space. To demonstrate this, in this paper we shall study the Gravitational AB effect in a purely classical system. As an example we will analyse a toy-model classical complex scalar field that is weakly coupled to gravity. We shall show that assuming the form of the gravitational potential used in [12], the dynamics of the classical field are influenced in a way similar to the quantum wave-function. Before we move on, there is a point worth mentioning. Previously Ford et al. had proposed a gravitational analogue model for the quantum AB effect within classical general relativity [10]. Subsequently, this was studied in the context of cosmic strings [15]. In contrast to these articles which study a classical gravitational analogue to the electromagnetic AB effect for a quantum wave-function, in the present paper we emphasise that the AB effect, in general, is not an intrinsically 'quantum' effect and demonstrate it for the case of the gravitational AB effect. The paper is organised as follows. In the next section, we begin with a scalar field coupled with gravity. After obtaining a non-relativistic limit, we study the effects of the gravitational potential on the dynamics of the field, thus sketching the AB effect for a classical system. We then discuss how this classical system would give rise to the quantum wave-function description of [12], emphasising the ubiquity of the AB effect. We end with a summary in section 3.", "pages": [ 1, 2 ] }, { "title": "2 Gravitational AB effect", "content": "We start with the equation of motion of a minimally coupled complex scalar field in curved spacetime We are working in units with /planckover2pi1 = c = 1. We will work in within the linearised gravity regime which is analogous to coupling scalar fields to electrodynamics [17]. Within linearised gravity Here the effects of gravity are encoded in h µν which can be thought as a symmetric rank-2 tensor field propagating in Minkowski spacetime and interacting with other fields in it. This is also the weak field limit of GR, O ( h 2 ) terms can be neglected. Plugging equation (2) into (1), we get In order to obtain a Newtonian limit, we choose a particular coordinate system ( /vectorx, t ) and make use the metric corresponding that corresponds to Newtonian gravity which is with all the other components of h µν -→ 0. Here Φ = -h 00 / 2 is similar to the Newtonian gravitational potential except that it is not time independent. Equation (4) can be thought of as the 'electrostatic' limit of linearised gravity although this is not as precise as electrostatics is [16]. Equation (3) takes the form We make the following field redefinition Expanding equation (5) We can now study the non-relativistic limit of the field ψ . For point particles the non-relativistic limit is obtained by taking | /vector p | /lessmuch m . Similarly for a scalar field, the non-relativistic limit would correspond to | ∂ 2 t ψ | /lessmuch m | ∂ t ψ | and | ∂ t ψ | /lessmuch m | ψ | [13]. Therefore we end up with In pure Newtonian gravity the term ∂ t h 00 vanishes but the gravitational AB effect requires a time dependent potential. We shall thus assume that ∂ t h 00 although nonvanishing, is small enough to be neglected in equation (8). We thus end up with We see that this equation has the same form as the Schrodinger equation in [12]. However the interpretation is quite different. Unlike the Schrodinger equation which in this case represents a quantum particle in a gravitational potential, equation (8) is an equation for a classical scalar field coupled to the same potential. We can now study the gravitational AB effect. Following [12], we impose Φ( /vectorx, t ) = Φ( t ). This can be achieved, for example, via the Equivalence Principle in a uniformly accelerated system which at least, locally is the same for all /vectorx . This ensures that the gravitational field intensity ∇ Φ = 0, see [12]. In order to make the Φ dependence on the field dynamics explicit, we perform the following separation of variables This lets us write the equation of motion of τ in the following form The χ term here is related to the gradient energy density of the field. Since it is time-independent, it is a constant in the above equation and is represented by E ∇ . Therefore we end up with Integrating this equation with respect to t we gives The full solution becomes The time evolution due to the E ∇ exponential term is simply a periodic phase and is thus trivial. The rightmost term is responsible for the non-trivial time dependence. It is, in fact, the source of the gravitational AB effect; the dynamics of the classical field depends on the gravitational potential even though the gravitational field intensity ∇ Φ vanishes. For further details, see [12].", "pages": [ 2, 3, 4 ] }, { "title": "AB Effect for the wave-function", "content": "We briefly mention how the AB effect as evidenced in equation (14) relates to the AB effect for a single particle wave-function, for a detailed discussion, see appendix. Starting with the classical field ψ and its corresponding Lagrangian, upon defining the conjugate momentum, we can quantise the field by imposing the field commutation relations. Note, the gravitational potential still remains classical. Now within this quantum field theory we focus on the one particle states obtained by the action of the field (conjugate) on the vacuum These are in fact the position Eigenstates of one-particle quantum mechanics. A general state can be constructed by taking superpositions of these The Ψ( /vectorx ) as introduced above is in fact the wave-function; the time independence is due to the Schrodinger picture description of the quantum field. We can see that the wave function having been obtained from quantising the complex scalar field and focussing on a subspace of states, namely the one particle states. One can thus show (see appendix), that Ψ in fact obeys the Schrodinger equation Thus equation (17) was the starting point in the analysis of [9,12] and of several other AB effect studies.", "pages": [ 5 ] }, { "title": "3 Discussions and Summary", "content": "Equation (14) governs the gravitational AB effect for a classical field; this tells us that there is nothing strictly quantum-mechanical about the gravitational AB effect. Additionally, we want to emphasise that the gravitational AB effect is not only possible for classical systems but is in fact generic, since gravity universally couples to everything. Further, it is the point particles are an exception to this. Considering the fields to be localised at individual points leads to simpler description of coupling to gravity; there are coupling terms between the classical field and the gravitational potential that do not show up in the coupling of a point particle to the gravitational field. However, one must note that for several reasons, the notion of point-particles in General Relativity is not well defined [17]. Often times, careful analyses of matter coupled to gravity involve replacing point-particles with classical fields. It is not difficult to see that as soon as one replaces point particles with classical fields, the gravitational AB effect, this also holds true for the traditional AB effect within electrodynamics. Therefore, in more general scenarios -where the field intensities do not necessarily vanish or when one is working with more arbitrary gauge fields, the AB effects can be thought of interactions that vanish upon localising the classical fields to classical point particles. The simplest generalisation along these lines would be to relax the assumption about the gravitational potential being slowly evolving that was made to obtain equation (9). It would be interesting to study the consequences of the ∂ t h 00 in equation (8) which we neglected for our purposes as we wished to draw a correspondence to the standard Schrodinger equation result. Further, we must mention there are certain aspects of Aharonov Bohm effect for quantum systems that have no classical counterparts, particularly the non-local effects due to entanglement [18]. This should not be surprising as entanglement is a purely quantum phenomena. To summarise, in this paper we worked out the gravitational Aharonov-Bohm effect for a classical complex scalar field. We showed an example of how the AB effect is not inherently a quantum mechanical result as is often thought about.", "pages": [ 5, 6 ] }, { "title": "Appendix", "content": "In this appendix, we explicitly show how the Schrodinger equation for a single particle in a gravitational potential can be obtained from the classical Schrodinger field. We begin with the Hamiltonian density corresponding to equation (9) The conjugate momentum π ψ is The Hamiltonian density H is We now quantise the field by imposing the following quantum conditions Therefore the commutation relations between the fields is The field ψ is similar to the Dirac spinor in the sense that the field equation is first order in time. The field can be expanded as where the corresponding commutation relations become For this quantum field, the vacuum state | 0 〉 is defined by A one particle state | /vector p 〉 is defined by The energy of the one particle state can be obtained by A localised one particle state can be obtained by Fourier transforming | /vector p 〉 This is in fact A general state can be constructed by superposing the one particle position Eigenstates Ψ here is the wave-function. Position and momentum operators corresponding to the one-particle state can be defined and Acting on the general state | Ψ 〉 we get For the momentum operator Combining /vector p and e -i/vectorp./vectorx into a derivative term we get We see that /vector X and /vector P satisfy the non-relativistic particle commutaion relations Similarly the one particle Hamiltonian operator is defined as Acting on | Ψ 〉 we end up with We thus end up with the one-particle Schrodinger equation placed in a gravitational potential.", "pages": [ 6, 7, 8 ] } ]

[ { "title": "ABSTRACT", "content": "To better assess the role that electrons play in exosphere production on icy-rich bodies, we measured the total and O 2 sputtering yields from H 2 O-ice for electrons with energies between 0.75 and 10 keV and temperatures between 15 and 124.5 K. We find that both total and O 2 yields increase with decreasing energy over our studied range, increase rapidly at temperatures above 60 K, and that the relative amount of H 2 O in the sputtered flux decreases quickly with increasing energy. Combining our data with other electron data in literature, we show that the accuracy of a widely used sputtering model can be improved significantly for electrons by adjusting some of the intrinsic parameter values. Applying our results to Europa, we estimate that electrons contribute to the production of the O 2 exosphere equally to all ion types combined. In contrast, sputtering of O 2 from Ganymede and Callisto appears to be dominated by irradiating ions, though electrons still likely contribute a non-negligible amount. While our estimates could be further refined by examining the importance of spatial variations in electron flux, we conclude that, at the very least, electrons seem to be important for exosphere production on icy surfaces and should be included in future modeling efforts.", "pages": [ 1 ] }, { "title": "Energy and temperature dependencies for electron-induced sputtering from H 2 O-ice: Implications for the icy Galilean moons", "content": "Rebecca A. Carmack 1 and Mark J. Loeffler 1, 2 1 Department of Astronomy and Planetary Science, Northern Arizona University, Box 6010, Flagstaff, AZ 86011, USA 2 Center for Materials Interfaces in Research and Applications, Northern Arizona University, Flagstaff, AZ 86011, USA", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Planetary bodies in our solar system that lack protection from a significant atmosphere are subjected to a number of irradiating particles, such as ions, electrons, photons, and cosmic rays. These particles alter the surface composition and/or structure, as well as eject surface material in a process known as sputtering. The sputtering of surface material can create surface bound exospheres on both rocky (Stern 1999; Wurz et al. 2007, 2010; Gamborino et al. 2019) and icy bodies (Hall et al. 1995; Ip et al. 1997; Cunningham et al. 2015; Ligier et al. 2019; Carnielli et al. 2020; Liuzzo et al. 2020; Plainaki et al. 2020; Paranicas et al. 2022; Carberry Mogan et al. 2023; De Kleer et al. 2023). Hall et al. (1995) identified an exosphere on Europa containing atomic oxygen and hypothesized that incoming energetic particles cause the dissociation and excitation of molecular O 2 in the atmosphere, which in turn is predicted to be sputtered off Europa's icy surface along with molecular hydrogen and H 2 O (Cunningham et al. 2015). Since atomic and molecular hydrogen are Corresponding author: Rebecca Carmack [email protected] light enough to dissipate into space and H 2 O falls back onto the surface, the main component of Europa's exosphere is oxygen (Johnson et al. 1982, 2009). Similar sputtering processes may occur on Ganymede (Ligier et al. 2019; Paranicas et al. 2022) and Callisto (Cunningham et al. 2015; Carberry Mogan et al. 2023), although the interactions of irradiating particles with those surfaces are more complex. While both ions and electrons can cause sputtering from icy surfaces, ions have been the main focus of previous experimental (see Baragiola et al. 2003 and Teolis et al. 2017 for a summary) and sputtering/exosphere modeling studies (Fam'a et al. 2008; Cassidy et al. 2013; Teolis et al. 2017; Addison et al. 2022; Pontoni et al. 2022). The lack of prior attention to electrons is at least partially due to early laboratory data showing that the sputtering yield ( Y ; the average number of molecules removed from a target material per incident particle) for a single 100 keV electron (Heide 1984) is 1000 to 10,000 times lower than the sputtering yield for a hydrogen or oxygen ion at similar energies (Shi et al. 1995). However, this difference in sputtering yields may not be that extreme, as the stopping cross section, a parameter which correlates with sputtering, is very low for 100 keV electrons and increases with decreasing energy until it peaks near 0.12 keV (Castillo-Rico et al. 2021). Regardless, electrons contribute ∼ 90% of particles and ∼ 80% of total energy measured near Europa, and smaller but still significant portions of particles/energy measured near Ganymede and Callisto (Cooper 2001). The large flux of electrons near these icy moons could make them important for exosphere production even if electrons are individually less efficient at sputtering than ions. Previous experiments irradiating H 2 O-ice with very low-energy (5 to 100 eV) electrons found that sputtering occurs for energies greater than ∼ 10 eV (Sieger et al. 1998; Orlando & Sieger 2003), and that O 2 sputtering yields increase with increasing electron energy between ∼ 10 and 100 eV (Sieger et al. 1998; Orlando & Sieger 2003), remain relatively constant at low temperatures ( /uni2272 80 K; Petrik & Kimmel 2005; Davis et al. 2021), and increase with increasing temperature above 80 K (Sieger et al. 1998; Orlando & Sieger 2003; Petrik & Kimmel 2005; Davis et al. 2021). Three groups have investigated the composition of material sputtered by higher electron energies (Abdulgalil et al. 2017; Galli et al. 2018; Davis et al. 2021). Both Abdulgalil et al. (2017) and Galli et al. (2018) detected little to no H 2 O sputtered by 0.2 to 10 keV electrons near 100 K, while our group (Davis et al. 2021) determined H 2 O dominates material sputtered by 0.5 keV electrons at low temperatures ( ≤ 60 K) and constitutes ∼ 1/5 of sputtered molecules at 100 K. Whether these differences between laboratory groups are mainly a consequence of the composition of sputtered material depending on electron energy, as has been observed for ions (Brown et al. 1984; Bar-Nun et al. 1985; Baragiola et al. 2002), or due to other factors is currently unclear. Quantifying the composition of material sputtered from H 2 O-ice as a function of electron energy and temperature is critical to properly model sputtering rates and exosphere production on icy bodies. Recently, we estimated Europa's global production of O 2 due to electrons by combining our laboratory data (Davis et al. 2021) with the scaled down ion sputtering model from Teolis et al. (2017). We found that electrons could be responsible for sputtering as much or more O 2 as all incoming ions combined (Davis et al. 2021). However, due to a lack of experimental data, we assumed that the composition of sputtered material did not change with electron energy in our calculation. Thus, here we measure the composition of the sputtering yield as a function of both electron energy and irradiation temperature, using a combination of microbalance gravimetry and mass spectrometry. With our new data, we use Markov chain Monte Carlo methods to determine electron versions of intrinsic model values that Teolis et al. (2017) determined for ions. Lastly, we use our optimized electron sputtering model to recalculate our previous estimate of the global production rate of O 2 by electrons irradiating Europa (Davis et al. 2021) and compare our updated model to additional estimates in literature for sputtering of O 2 from Europa, Ganymede, and Callisto, allowing us to better assess the role of electrons in icy satellite exosphere production.", "pages": [ 1, 2, 3 ] }, { "title": "2. EXPERIMENTAL METHODS", "content": "We performed all experiments within a stainless steel ultra-high vacuum chamber at a base pressure of ∼ 3 x 10 -9 Torr (Meier & Loeffler 2020; Davis et al. 2021). We estimate that the pressure near the sample is 10 to 100 times lower due to a thermal-radiation shield in place around the sample. An Inficon IC6 quartz-crystal microbalance (QCM) with an optically flat gold mirror electrode served as the sample substrate and is mounted onto a rotatable closed-cycle helium cryostat centered inside of the experimental chamber. The cryostat is capable of maintaining temperatures between ∼ 10 and 300 K. We prepared H 2 O (HPLC grade) samples in a separate manifold attached to the chamber and grew samples at 100 K at near normal incidence with a deposition rate of ∼ 2 x 10 15 H 2 O cm -2 s -1 to an average column density of (5.4 + 1 . 3 -0 . 4 ) x 10 18 H 2 O cm -2 ( ∼ 2 µ m), with the error representing the full range of column densities used in this study. The resulting sample thickness is sufficient to avoid any enhancement in our measured yields for all electron energies studied here (Meier & Loeffler 2020). We grew fresh films for all electron energies and irradiation temperatures reported here, since sample irradiation history can affect sputtering yields (Meier & Loeffler 2020). After growth, we changed the sample temperature to the irradiation temperature of interest (between 14 and 125 K). The lower limit ensured we could consistently stabilize the temperature and the higher limit is below the temperature ( ∼ 130 K) where H 2 O begins to sublimate (Sack & Baragiola 1993) and out diffusion of radiolytically O 2 produced below the near-surface becomes important (Teolis et al. 2005). We irradiated the sample with an EGG-3103C Kimball Physics electron gun at an incident angle of 12.5 · with respect to the surface normal with 0.75 to 10 keV electrons. In all experiments, we rastered the beam in an approximately 1 x 1 cm square, which is larger than the exposed surface of our QCM ( ∼ 8 mm diameter). We measured the electron flux before and after irradiation using a retractable Faraday cup to be (2.7 ± 0.8) x 10 13 electrons cm -2 s -1 . During irradiation, the flux varied by /uni2272 2% for all energies except for 10 keV which varied up to ∼ 9%. We analyzed any gases present in the chamber, including residual background and material sputtered from the sample during irradiation, using an Ametek Dymaxion Mass Spectrometer (DYMAX-100) aligned 12.5 · from the sample normal opposite the electron gun. After each irradiation, we desorbed our ice by turning off the cryostat and allowing the substrate to return to ∼ 300 K overnight. In our analysis, we include previous work done by our group in Davis et al. (2021) with 0.5 keV electrons, as they used the same setup and approach as we do here.", "pages": [ 3 ] }, { "title": "3. RESULTS", "content": "During irradiation, we see a clear increase in the partial pressure of H 2 O, O 2 , and H 2 for each electron energy and temperature studied. However, the background signals for H 2 O and H 2 are 1 to 2 orders of magnitude higher than the background for O 2 , and therefore are highly affected by baseline changes. Additionally, the cooled thermal-radiation shield around our sample acts as a potential cold trap for H 2 O but is less likely to trap more volatile species like O 2 and H 2 (Davis et al. 2021). Because of these barriers to accurately interpreting our partial pressure signals for H 2 O and H 2 , we only consider the partial pressure signal for O 2 (PPO 2 ) in our data analysis. Figure 1 shows the areal mass loss as monitored by the QCM alongside the baseline subtracted PPO 2 for a sample irradiated with 1 keV electrons at 115 K, with the highlighted area showing when the electron beam was irradiating the sample. When irradiation begins, there is an initial period when the PPO 2 rises until it reaches a peak, after which it levels out at equilibrium for the remainder of the irradiation. In cases where we see a peak ( /uni2273 115 K), the fluence required to reach the peak and subsequent equilibrium is energy and temperature dependent. However, all experiments reached equilibrium between fluences of ∼ (0.4 - 3) x 10 17 electrons cm -2 . Generally, we used the equilibrium value to determine the PPO 2 , but in cases where we observed a peak we took the average of the peak and equilibrium values. We incorporated the differences between the peak and equilibrium PPO 2 values into our error. Regardless of irradiation temperature, when the electron gun is blocked the PPO 2 takes time to return to zero. This could be due to any sputtered O 2 remaining in the chamber slowly being pumped out of our system. Figure 2 shows the total sputtering yield ( Y T ; in terms of the sample's total mass loss) for 0.5 to 10 keV electrons at temperatures between 14 and 124.5 K. Y T is approximately constant below 60 K for all energies, although electron energies below ≤ 2 keV show a small ( /uni2272 10%) increase in Y T between 15 and 60 K. For higher energies, we observe a similar trend but cannot say definitively due to increased variation in Y T . Above 60 K, Y T clearly increases with temperature. For example, Y T increases for all energies by a factor of ∼ 1.5 between 60 and 100 K, and a factor of 2 to 3 between 60 and 120 K. In order to determine the composition of sputtered material, we use the same approach described in Davis et al. (2021) for each electron energy studied. We assume the amount of H 2 Osputtered from ice for a given electron energy is constant with temperature, previously shown to be true for temperatures /uni2272 130 K (Boring et al. 1983; Petrik & Kimmel 2005). Figure 3 (top) shows Y T versus the PPO 2 for a error from the spread in values from repeated experiments b molecular yield of H 2 is twice that of O 2 ( Y H 2 = 2 ∗ Y O 2 ) all experiments where we irradiated our sample with 1 keV electrons above 60 K. Each data point is an experiment completed at a different temperature (if a temperature was repeated more than once, the average data point for that temperature is shown). We do not include experiments performed at temperatures below 60 K in the analysis, because baseline variations in the mass spectrometer signal occur more frequently at lower temperatures and because Y T is ∼ constant below 60 K. We calculate the sputtering yield of H 2 O ( Y H 2 O ) for a given electron energy by extrapolating the PPO 2 to zero (i.e. the y-intercept in the top of Figure 3), implying no O 2 (or consequentially H 2 ) is sputtered from the sample. We tested whether the y-intercept was unique for a given electron energy by repeating a suite of experiments under the same conditions (energy, temperatures, etc.) but using a different mass spectrometer multiplier voltage. In those experiments, we find that the data remains linear (although the slope changes) and the y-intercept remains the same. The difference between Y T and Y H 2 O gives the portion of sputtered material that is comprised of radiolytic products O 2 and H 2 . The sputtering yields for O 2 ( Y O 2 ) and H 2 ( Y H 2 ) are then differentiated from each other by multiplying the sputtered mass of radiolytic products by the mass fraction of O 2 and H 2 in the relation 2 · H 2 O → O 2 + 2 · H 2 (Brown et al. 1980b). We show the compositional breakdown of molecules sputtered by 1 keV electrons in Figure 3 (bottom) for each irradiation temperature studied. While we did not measure Y H 2 directly, we assume it is twice that of Y O 2 (see above). We apply the same analysis to 0.75, 2, 4, 6, and 10 keV electrons and provide a sampling of representative total mass yields, H 2 O mass yields and H 2 O and O 2 molecular yields in Table 1 (for the entirety of our data see Table 4 in Appendix A). We find that the composition of sputtered material varies strongly across 0.5 to 10 keV and 60 to 125 K. At low temperatures ( ≤ 60 K), H 2 O makes up as much as 65% of the sputtered flux for 0.5 keV electrons, about 40% for 1 keV electrons, but only comprises about 20% for 2 keV electrons. Above 4 keV, the contribution of H 2 O to the sputtered flux is essentially zero within our error. At higher temperatures ( > 60 K), H 2 O yields trend similarly with energy as they do at low temperatures, however the relative contribution of H 2 O at each temperature is lower due to the increased production of radiolytic O 2 and H 2 . For instance, at 120 K H 2 O makes up about 30% of the sputtered flux at 0.5 keV, about 20% at 1 keV, but drops to about 6% of the flux at 2 keV. /s54/s49/s46/s53/s54/s56/s51/s50 Figure 4 shows the total, H 2 O, and O 2 mass yields versus electron stopping cross section ( S e ) for irradiation at 60 K. Y H 2 O (in g/e -, for all temperatures) is nearly quadratic with S e , and well fit to y = a ( S e ) n where a =6.65 x 10 -28 and n =2.17. At lower temperatures, the total sputtering yield is superlinearly related to S e , which is consistent with previous studies (Meier & Loeffler 2020). As temperature increases, the trend of Y T with S e becomes more linear, likely because Y O 2 (and Y H 2 ) appear to increase linearly with S e for all temperatures, though given the error on O 2 yields this is hard to state definitively.", "pages": [ 3, 4, 6, 7, 8 ] }, { "title": "4. COMPARISON TO OTHER EXPERIMENTS", "content": "In this study, we expanded on our previous work (Meier & Loeffler 2020; Davis et al. 2021) to investigate the composition of the sputtering yield as a function of irradiation temperature and electron energy. Below, we compare and discuss our results with previous ion and electron work measuring total sputtering yields, as well as studies that have made estimates of the main species sputtered from H 2 O-ice.", "pages": [ 8 ] }, { "title": "4.1. Total Yields", "content": "Previous work on the sputtering of H 2 O-ice with light ions at low temperatures ( /uni2272 80 K) found Y T to be proportional to S e following a superlinear and in some cases quadratic dependence (Brown et al. 1980a,b; Shi et al. 1995; Baragiola et al. 2003). Our group finds a similar superlinear dependence on Y T with S e for electron energies between 0.5 and 10 keV irradiating H 2 O-ice at lower temperatures (this work, Meier & Loeffler 2020), but the trend progressively becomes more linear with increasing temperature. The dependence of Y T with S e ranging from quadratic to linear is likely due to changes in the composition of the sputtered flux (see Section 4.2). In contrast, Galli et al. (2018) found Y T was independent of S e between 0.2 and 3 keV for thin films irradiated with electrons at 90 K. We suspect that their observed constancy of Y T with energy is likely a consequence of using previously irradiated samples, as processed samples can show enhancements in Y T by a factor of ∼ 3 to 6 at 60 K (Meier & Loeffler 2020), which we attribute to the buildup of O 2 beneath the sample's surface. At low temperatures ( ≤ 60 K), we observe a slight ( /uni2272 10%) increase in Y T between 15 and 60 K, which is consistent with previous studies for electrons (Petrik & Kimmel 2005; Davis et al. 2021). For higher temperatures, we find that Y T increases rapidly above ∼ 60 K for all electron energies studied, consistent with our previous work with 0.5 keV electrons (Davis et al. 2021) and with previous ion irradiation studies (Brown et al. 1984; Baragiola et al. 2002, 2003; Fam'a et al. 2008).", "pages": [ 8 ] }, { "title": "4.2. Composition of the Sputtered Flux", "content": "We find that the composition of the sputtered flux depends on both electron energy and irradiation temperature. Changes in the composition of our sputtered flux with electron energy are consistent with previously observed experimental trends for ions, which have shown the composition changes with ion energy and ion type. More specifically, experiments with 1.5 MeV He + found that H 2 O makes up ∼ 90% of the sputtered flux at low temperatures (Brown et al. 1984), while only about ∼ 30% is H 2 O for 1 to 5 keV H + (Bar-Nun et al. 1985). Variations in composition are also observed with heavier ions: samples irradiated with 1 to 5 keV Ne + found H 2 O comprises about 60% of the sputtered flux at 1 keV but only about 30% at 5 keV. Additionally, studies using 100 keV Ar + show that ∼ 75% of the sputtered flux is H 2 O (Baragiola et al. 2002). In addition, we find that H 2 O, and possibly also O 2 , has a quantifiable dependence on S e (Figure 4). Our observed quadratic dependence for Y H 2 O is consistent with what has been seen for Y T in previous studies with fast ions (Brown et al. 1980b; Baragiola et al. 2003). Given that H 2 O is the dominant component sputtered by fast ions (Brown et al. 1984), we speculate that the quadratic dependence for Y H 2 O observed in our experiments is also a result of excitation pairs overlapping at the surface (Brown et al. 1980b; Baragiola et al. 2003). For O 2 , the possible linear relation with S e suggests that the multiple reactions required to form O 2 from H 2 O (Boring et al. 1983; Teolis et al. 2005) may occur from a single electron breaking multiple bonds as it travels into the ice. We can also compare our results to the two other groups who estimate the composition of flux sputtered from H 2 O-ice by ∼ keV electrons. Abdulgalil et al. (2017) irradiated films coated with islands of C 6 H 6 with ∼ 0.25 keV electrons at 112 K. During irradiation, they observed a H 2 signal, a much weaker O signal, but no H 2 O signal above the noise level; no measurement of O 2 was reported. They conclude that H 2 and O 2 are the dominant species removed during irradiation, which is inconsistent with our findings that H 2 O makes up 45% of our total sputtering yield at 110 K for 0.5 keV electrons. Interestingly, Galli et al. (2018) irradiated several H 2 O-ice types (thin films, frost, etc.) with 0.2 to 10 keV electrons at ∼ 90 K and, similar to Abdulgalil et al. (2017), did not see a rise in H 2 O above their detection limit while irradiating. They report an average composition between 0.4 and 10 keV for their frost and fine-grained ice samples, estimating the contribution of H 2 O to the sputtering yield to be < 10%. Although it is unclear what energies were averaged, this upper limit may be in-line with our findings. For instance, H 2 O only contributes ∼ 13% for 2 keV electrons at 90 K and subsequently less at energies approaching 10 keV. Using processed H 2 O-ice films, as in Galli et al. (2018), may act to suppress the relative contribution of H 2 O further, as the total yield can be enhanced temporarily due to the presence of O 2 below the sample's surface (Meier & Loeffler 2020). Besides the compositional dependence on energy ( S e ), we also see a clear increase in the O 2 yield with temperature. For ions, this increase with temperature has been attributed to the ability of radiolytically produced radicals to diffuse and increase production of H 2 and O 2 near the surface (Brown et al. 1980b; Baragiola et al. 2003; Teolis et al. 2009). Our findings support a similar process for electrons, as expected from previous low-energy ( ∼ eV) electron irradiation studies showing that Y H 2 O remains constant with irradiation temperature (Petrik & Kimmel 2005) while Y O 2 is relatively constant (but still increases slightly) below ∼ 60 K and increases rapidly as temperature increases above ∼ 60 K (Petrik & Kimmel 2005; Petrik et al. 2006; Orlando & Sieger 2003).", "pages": [ 8, 9 ] }, { "title": "5. MODELING O 2 SPUTTERING", "content": "As noted in the Introduction, between ions and electrons, ions have been the main focus of previous sputtering/exosphere modeling studies (Marconi 2007; Fam'a et al. 2008; Teolis et al. 2010; Cassidy et al. 2013; Teolis et al. 2017). The most comprehensive model for predicting O 2 sputtering yields for any particle irradiating an icy surface is Teolis et al. (2017) which builds off their work in Teolis et al. (2010). Teolis et al. (2017) calculates the sputtering yield of O 2 as where /epsilon1 is the effective particle energy contributing to sputtering (total energy E = /epsilon1 for electrons), T is the irradiation temperature, β is the particle's incident angle, g 0 O 2 is the surface radiolytic yield of O 2 ( Y O 2 /slash.left E when r 0 cos β /uni226A x 0 ), x 0 is the optimal depth for O 2 production, r 0 cos β is the particle's range, q 0 is the exponential prefactor for the temperature dependence, k B is the Boltzmann constant, and Q is the 'activation' energy (noted in Teolis et al. 2017 to not have a determined physical significance). Teolis et al. (2017) fit Equation 1 to existing laboratory data for ions and determined intrinsic parameter values for g 0 O 2 , x 0 , q 0 , and Q (listed in Table 2). In Tribbett & Loeffler (2021), we determined that Equation 1 overestimates O 2 production from ions with ranges r 0 cos β >> x 0 by as much as an order of magnitude and explored how this could be caused by the assumption in Teolis et al. (2017) that energy is deposited uniformly over the ion's range. Upon further investigation, we noticed a mistake in Teolis et al. (2017) regarding the angle of incidence for data taken by Bar-Nun et al. (1985) 1 for highly penetrating ions, which could also be contributing to the discrepancy between Equation 1 and experimental data. We are hoping to revisit the effects of these corrections in a future study.", "pages": [ 9, 10 ] }, { "title": "5.1. Scaling the Model to Electrons", "content": "The model's predicted O 2 sputtering trends are generally consistent with what has been observed for electrons (see Introduction). Thus, it seems reasonable that first attempts to model electron sputtering simply scale Equation 1, calculated with parameter values derived using ion data, down by a constant factor ( C ∗ Y O 2 ) since at the time there was a lack of electron data with which to determine electron specific parameter values. Teolis et al. (2010, 2017) uses a factor of C =0.29 to fit C ∗ Y O 2 to experimental O 2 yields for low-energy (5 to 30 eV) electrons provided by Petrik, Kavesky, and Kimmel at Pacific Northwest National Laboratory (supplemental Figure S9 in Teolis et al. 2010), although their measured yields are an order of magnitude higher than the values reported by Sieger et al. (1998) 2 for similar electron energies. In two of our recent studies, we applied Equation 1 to our electron sputtering data and found best-fit scaling factors of C =0.25 (Meier & Loeffler 2020) and 0.14 (Davis et al. 2021), keeping in mind that Meier & Loeffler (2020) only measured Y T and not Y O 2 . Following this precedent, we find the scaling factor C =0.12 minimizes chi-squared between C ∗ Y O 2 and all of our group's data listed in Table 4. When calculating Y O 2 , we interpolate our electron ranges from the newly published model predicting the S e and range of electrons in liquid H 2 O by Castillo-Rico et al. (2021), which differs slightly from Grun (1957) and ESTAR (Berger et al. 2017) estimates used in Meier & Loeffler (2020) and Davis et al. (2021), and from estimates by LaVerne & Mozumder (1983) used by Teolis et al. (2010, 2017). To be consistent with the derived electron ranges in Castillo-Rico et al. (2021), we assume the density of H 2 O-ice is the same as liquid H 2 O (1 g cm -3 ). As seen in Figure 5, scaling Y O 2 down by a constant value results in a reasonable fit above 1 keV for higher temperatures, but underestimates our data at lower energies and lower temperatures, suggesting that the energy and temperature dependencies for ions are not accurately describing trends in all electron data currently available. Thus, as we now have new data for the sputtered component of O 2 , we reevaluate intrinsic parameter values ( g 0 O 2 , x 0 , q 0 , and Q ) in Equation 1 using Markov chain Monte Carlo (MCMC) methods to determine whether we can improve the model's overall fit while removing the need for a constant scaling factor.", "pages": [ 10, 11 ] }, { "title": "5.2. Updating Parameter Values for Electrons", "content": "Here we present a brief summary of our modeling methods (see Appendix B for additional details). Due to the conflicting O 2 yields for low-energy ( ∼ 10 to 30 eV) electrons (Sieger et al. 1998; Teolis et al. 2010), we excluded both data sets from our initial MCMC analysis. However, we re-ran our MCMC optimization process using a combination of data from our group and Sieger et al. (1998) and from our group and Teolis et al. (2010). We assume an error of 100% for data from both Sieger et al. (1998) and Teolis et al. (2010). Table 2 shows each version of our MCMC optimized values for g 0 O 2 , x 0 , q 0 , and Q compared with the values determined in Teolis et al. (2017) for ions. Regardless of the electron data set used in the optimization, the resulting g 0 O 2 value is an order of magnitude smaller than what has been determined for ions. Because g 0 O 2 is defined as the radiolytic yield at the surface, experimental data for lower energy ( ∼ eV) electrons, which do not travel very deep beneath the surface, heavily influence the optimized g 0 O 2 value. This explains the variation in g 0 O 2 values with the three electron data sets, which would likely be larger if we had stricter error for the low-energy data sets. Additionally, g 0 O 2 and x 0 are inversely correlated to each other, which explains x 0 increasing when g 0 O 2 decreases. Our optimized values for q 0 and Q for the three data sets show less variation than did g 0 O 2 and x 0 , and all overlap with each other and with the values obtained from ions within error. As shown in Figure 6, the assumption that Y O 2 is approximately constant at temperatures ≤ 60 K ignores the observed weak increase in Y O 2 at low temperatures, which results in discrepancies between the data and model for temperatures /uni2272 100 K. Future modeling efforts could potentially modify the structure of Equation 1 to better fit electron data over the entire temperature range. We plot the energy dependence of the resulting model fits and data at a single representative temperature in Figure 7, showing Y O 2 calculated using each set of optimized parameter values for electrons listed in Table 2 compared to C ∗ Y O 2 for C =0.12 (our best-fit scaling factor) and C =0.29 (Teolis et al. 2010, 2017) with Y O 2 calculated using parameter values for ions from Teolis et al. (2017). The C =0.29 fit used by Teolis et al. (2010, 2017) overestimates all of our measured O 2 sputtering rates for keV electrons. Conversely, our best fit scaling factor C =0.12 underestimates the yields for all of our data (with the exception of 10 keV). Equation 1 optimized to data from only our group and data combined from our group and Teolis et al. (2010) are very similar (and practically overlap in Figures 6 and 7), although with better constrained error for data from Teolis et al. (2010) differences in the resulting curves may be greater. Finally, we note that the electron energy associated with the peak of Y O 2 ( ∼ 0.65 keV) does not match the electron energy associated with the peak of electron stopping cross section ( ∼ 0.12 keV, Castillo-Rico et al. 2021) even though Y T is expected to trend with electron stopping cross section. Assuming the peak position of Y O 2 should match the peak position of Y T , this difference could be due the overlapping Y O 2 values in our 0.5 and 0.75 keV data. Different models also shift the peak position of electron stopping cross section (Ashley 1982; LaVerne & Mozumder 1983; Luo et al. 1991; Gumu¸s 2008; Castillo-Rico et al. 2021) which could also contribute to this difference, although to a lesser extent.", "pages": [ 11, 12, 13 ] }, { "title": "6. ASTROPHYSICAL IMPLICATIONS", "content": "Below, we apply our newly optimized O 2 sputtering model for electron irradiation of H 2 O-ice to three Jovian icy satellites: Europa, Ganymede, and Callisto. For Europa, we compare how the inclusion of low-energy electron experimental sputtering data affects our calculated production yields by calculating Equation 1 with each set of parameter values in Table 2. When comparing our calculations of Europa to other values in literature, and when discussing Ganymede and Callisto, we calculate Equation 1 with the parameter values optimized to our group's data only.", "pages": [ 13 ] }, { "title": "6.1. Europa", "content": "As done in Davis et al. (2021), we calculate the flux of sputtered O 2 from Europa as where J ( E ) is the differential flux of electrons (e -cm -2 s -1 sr -1 MeV -1 ) near Europa, assuming a uniform electron flux striking the surface. We adopt the same differential electron flux that Davis et al. (2021) estimated by combining measurements from the Galileo Energetic Particle Detector (Cooper 2001) and Voyager Plasma Spectrometer (Scudder et al. 1981; Sittler & Strobel 1987). We integrate Equation 2 between 10 eV (the minimum energy required for electron sputtering; Sieger et al. 1998; Orlando & Sieger 2003) and 1 MeV, calculating Y O 2 with Equation 1, intrinsic parameter values from Table 2, and assuming an average β of 45 · . Additionally, since Castillo-Rico et al. (2021) only calculate electron ranges up to ∼ 430 keV, we used Castillo-Rico et al. (2021) ranges for energies ≤ 425 keV and scaled ESTAR estimated electron ranges from 425 keV to 1 MeV by a factor of 1.011 so that ranges from Castillo-Rico et al. (2021) and ESTAR matched at 425 keV. Table 3 shows the flux of sputtered O 2 and global production rates from Europa found by scaling the sputtered flux to Europa's surface area (mean radius from Showman & Malhotra 1999) for relevant surface temperatures (Spencer et al. 1999; Ashkenazy 2019). Interestingly, with the exception of scaling Y O 2 for ions down by C =0.29 which effectively doubles the production rate of O 2 , the choice of parameter values used to calculate the O 2 production rate does not appear to matter significantly. For instance, there is only ∼ 5% difference between O 2 production rates at 125 K found by multiplying Y O 2 for ions down by C =0.12 (our best fit scaling factor) and calculating Y O 2 with parameter values found by optimizing Y O 2 to our group's data. The O 2 production rates found by calculating Y O 2 with parameter values optimized to data from our group and Sieger et al. (1998) or data from our group and Teolis et al. (2010) differ from one another by ∼ 17%. While a ∼ 17% difference in the O 2 production rate for the two low-energy data sets is not seemingly large, as noted in Section 5.2, having better constrained error for the low-energy data sets would increase the difference in the resulting integrated yield. Further refining these discrepancies would require additional measurements with low-energy ( ∼ eV) electrons, which would enable a more precise estimate of the O 2 surface radiolytic yield ( g 0 O 2 ). While we calculated the values in Table 3 assuming a uniform electron flux striking the surface of Europa, this is an oversimplification of the radiation environment (Paranicas et al. 2001, 2009; Patterson et al. 2012; Dalton et al. 2013; Addison et al. 2023). Future studies investigating to what degree spatial variations in electron flux alter our estimates are important for properly applying our optimized electron model to Europa. Regardless, we find a global production rate of (0.5 - 1.9) x 10 26 O 2 s -1 for 80 to 125 K using the parameter values optimized to our group's data, which is slightly lower than our previous less-refined estimate (Davis et al. 2021). Additionally, our estimate for 125 K is a factor of ∼ 1.6 times higher than the estimate made in Vorburger & Wurz (2018) (1.15 x 10 26 O 2 s -1 , found by multiplying the sum of the O 2 yields from both hot and cold electrons listed in their Table 5 by the surface area of Europa). Considering that, at the time of their study, the only measurement for Y O 2 suggested that Y O 2 was constant above 200 eV and about an order of magnitude higher than what we have measured at 1 keV (Galli et al. 2017), the similarity of the estimates may seem surprising. However, Vorburger & Wurz (2018) also assumed that only 20% of the electron flux below 1 keV reaches Europa's surface. Recently, Addison et al. (2023) combined the low-energy (thermal) electron sputtering rate estimated in Vorburger & Wurz (2018) ( ∼ 2.3 x 10 25 O 2 s -1 ) with a new sputtering rate estimate for 5 keV to 10 MeV electrons taking into account interactions between Jupiter's magnetosphere and Europa's induced magnetic field, and found the total sputtering contribution from electrons to be only ∼ 2.4 x 10 25 O 2 s -1 . While our assumption that all thermal electrons reach Europa's surface is unlikely, it is also unlikely that there is a constant 80% reduction in flux for all electron energies below 1 keV (Vorburger & Wurz 2018; Addison et al. 2023). Until there is better understanding of what portion of the lower energy ( /uni2272 5 keV) electron flux reaches Europa's surface, we consider our estimates to be an upper limit, as we have suggested previously based on another recent, but lower, flux estimate (Jun et al. 2019). In fact, using fluxes from Jun et al. (2019) results in a rate of (1.0 - 3.7) x 10 25 O 2 s -1 , which is a factor of 5 lower than our production rate, putting it in range of the value estimated by Addison et al. (2023). This similarity is a bit surprising, considering the estimate from Addison et al. (2023) is considerably more refined than ours with the inclusion of spatially resolved energetic electron fluxes, surface temperature differences, and various incident particle angles. Our estimated range for the global electron sputtering rate of O 2 from Europa encompasses the total production rate for all ions combined of ∼ 1 x 10 26 O 2 s -1 estimated in both Cassidy et al. (2013) and Addison et al. (2021, 2022) using the unmodified Equation 1 from Teolis et al. (2017). While the effects of ions and electrons are unlikely to simply be additive, it is interesting that the sum of the estimate for ions and our electron estimate is similar to an estimate of O 2 production from Europa's surface via radiation processing ((2.2 ± 1.2) x 10 26 O 2 s -1 ), which was extrapolated from measurements of atmospheric H 2 loss rates during Juno's recent fly-by of Europa (Szalay et al. 2024). Considering the possible reduction of our electron sputtering rate estimates from the deflection of thermal electrons near Europa and that we also recently found Equation 1 likely overestimates Y O 2 by a factor of 5 to 8 at 120 K for 0.5 to 5 keV ions (Tribbett & Loeffler 2021), which are representative of the cold/thermal ion component near Europa, more rigorous investigation is needed to determine whether the apparent agreement with the Juno-derived data is fortuitous. Nonetheless, we expect that, at the very least, electrons are significant contributors to the sputter-produced O 2 exosphere around Europa and need to be considered in any future modeling efforts.", "pages": [ 13, 14, 15 ] }, { "title": "6.2. Ganymede", "content": "Ganymede has an exosphere predominately composed of O 2 , atomic O, and H 2 O (Hall et al. 1998; De Kleer et al. 2023) hypothesized to be produced via sputtering and sublimation. Sputtering from Ganymede by Jupiter's magnetospheric particles is complicated by Ganymede's intrinsic magnetic field deflecting certain energetic particles away from the moon's surface (Delitsky & Lane 1998; Plainaki 2015; Fatemi et al. 2016; Poppe et al. 2018; Liuzzo et al. 2020). A recent study (Liuzzo et al. 2020) showed Ganymede's closed field lines around its equator completely shield the moon's equato- rial region from irradiating electrons with energies /uni2272 40 MeV, while electrons of all energies reach the surface of Ganymede's polar regions (Frank et al. 1997; Cooper 2001; Liuzzo et al. 2020). We estimate the flux of O 2 sputtered from Ganymede by electrons with Equation 2, calculating Y O 2 with Equation 1, parameter values optimized to our group's data, and assuming J ( E ) for electrons near Ganymede's orbital radius (Paranicas et al. 2021) contributes to a uniform electron flux striking Ganymede's polar regions. We extrapolate the differential electron fluxes given in Paranicas et al. (2021) down to 10 eV in order to integrate from 10 eV to 1 MeV. We find the flux of sputtered O 2 to be (3 - 20) x 10 7 cm -2 s -1 for 65 to 140 K (limits for the temperature range at Ganymede's poles; Squyres 1980). Using the mean radius for Ganymede (Showman & Malhotra 1999) and scaling our O 2 sputtered flux estimate by the area of the polar regions where electrons reach the surface (latitudes ≥ 40 · ; Liuzzo et al. 2020) yields a production rate of (0.9 - 6.2) x 10 25 O 2 s -1 for 65 to 140 K, which, to our knowledge, is the first estimate for O 2 production rates from Ganymede by irradiating electrons. While our estimate could be refined further, it appears to be significantly lower than the most recent estimate for O 2 production by ions of 2.4 x 10 26 O 2 s -1 by Pontoni et al. (2022), and even lower than earlier estimates for ions by Marconi (2007) (1.2 x 10 27 O 2 s -1 ) and Plainaki (2015) (2.6 x 10 28 O 2 s -1 ). However, given that Pontoni et al. (2022) used Equation 1 and found that low-energy ( ∼ keV) ions were the major contributor to the O 2 sputtering flux, it is possible that the production rates for ions were overestimated (see above). Regardless, given the simplicity of our estimate, as well as the wide range of estimates for sputtering from ions, more studies investigating sputtering from Ganymede's surface may be merited.", "pages": [ 15, 16 ] }, { "title": "6.3. Callisto", "content": "O 2 has also been detected in Callisto's atmosphere (Cunningham et al. 2015; De Kleer et al. 2023). While sputtering has been speculated to play a key role for decades (Kliore et al. 2002), surface particle fluxes are difficult to assess due to the presence of Callisto's atmosphere (Strobel et al. 2002). Recently, Carberry Mogan et al. (2023) modeled the spatial variation in temperature and particle (electron and hydrogen, oxygen, and sulfur ion) fluxes across the surface of Callisto. They then calculated sputtering rates for ions using both the model developed by Fam'a et al. (2008) and modified by Johnson et al. (2009) as well Equation 1 (Teolis et al. 2017) with parameter values for ions modified in Tribbett & Loeffler (2021), and for electrons using Equation 1 with parameter values derived for ions (Teolis et al. 2017) and no scaling factor. They determine that, although sputtering from Callisto's surface is not enough to account for the observed column densities of O 2 in Callisto's atmosphere, electrons contribute between 24 to 32% of the total O 2 sputtered from Callisto's icy patches, which is more than the contribution from hydrogen ( ∼ 0.5 to 7%) and oxygen (18 to 20%) ions, but less than from sulfur ions (57 to 41%). While recalculating the spatially resolved sputtering yields from Callisto's ice patches is beyond the scope of this work, here we estimate the effect that our new electron parameter values have on the predicted O 2 sputtered from Callisto's surface. Although we cannot use Equation 2, due to the complication of the impinging flux interacting with the moon's atmosphere, we make a rough estimate by integrating the radiolytic yield ( G O 2 = Y O 2 /slash.left E ) at temperatures relevant for Callisto (80 to 144 K; Grundy 1999; Carberry Mogan et al. 2023) with our new electron parameter values and compare that result with what we obtain when we perform the integration with the parameters values used in Carberry Mogan et al. (2023). Integrating from 10 eV to 1 MeV, we find that our new electron parameter values reduce the yield by an order magnitude, suggesting that the contribution of electrons to sputtering of O 2 from Callisto is likely significantly less than as estimated by Carberry Mogan et al. (2023).", "pages": [ 16, 17 ] }, { "title": "7. CONCLUSIONS", "content": "In this paper, we measured the total, H 2 O, and O 2 sputtering yields for electrons with energies between 0.75 and 10 keV and for irradiation temperatures between 15 and 124.5 K. Over our studied energies, we found that both total and O 2 yields increase with decreasing energy (increasing S e ) and increase rapidly at temperatures above 60 K, which is in agreement with our previous electron work, as well as previous studies with electrons and ions. In addition, we find that the yield of H 2 O has a nearly quadratic relation with S e while the yield of O 2 appears to trend approximately linearly with S e (although the slope changes with temperature). These different dependencies could explain why the trend of Y T with S e ranges from quadratic to linear in previous studies for light ions and electrons. Additionally, the composition of the sputtered flux has a strong dependence on electron energy with the relative amount of H 2 O decreasing rapidly with decreasing S e over the electron energies studied. In fact, we find that above 4 keV, the contribution from H 2 O is essentially zero within the limits of our error. Combining our data with O 2 sputtering yields for 0.5 keV electrons from Davis et al. (2021) and other low-energy ( ∼ eV) electron data from literature (Sieger et al. 1998; Teolis et al. 2010), we reevaluated intrinsic parameters in the sputtering model from Teolis et al. (2017), finding that we can provide a more satisfying fit while also removing the arbitrary scaling factor. Having better constraints on low-energy electron O 2 yields in literature or restructuring the energy and/or temperature dependent model components may improve the fit further. Combining our newly optimized sputtering model with incoming electron fluxes near Europa, we calculate that electrons may contribute to the production of Europa's O 2 exosphere at a rate similar to all ion types combined. Thus, although electrons may, in most cases, have significantly lower individual sputtering yields than ions, the higher electron fluxes at the surface of icy bodies like Europa may be large enough for electrons to be a major contributor to exospheric O 2 production. In contrast, we find electrons contribute less to O 2 sputtering from Ganymede and Callisto, although the contribution of electrons is still likely non-negligible. Of course, future studies for all moons examining spatial variations in the incoming electron flux are needed to refine our estimates. Regardless, at the very least, it seems clear that the contribution of electrons needs to be included in sputtering and exosphere modeling of icy bodies going forward. We would like to thank T.M. Orlando and B.D. Teolis for explaining their group's published electron sputtering data as well as P.D. Tribbett for contributing to the MCMC methods discussion. This research was supported by NASA Solar System Workings Award #80NSSC20K0464. Data can be found in Northern Arizona University's long-term repository https://openknowledge.nau.edu/id/eprint/6258).", "pages": [ 17 ] }, { "title": "A. DATA", "content": "In Table 4, we list our measured total, H 2 O, and O 2 sputtering yields for each electron energy and temperature studied in this paper and in Davis et al. (2021). Molecular H 2 sputtering yields are calculated as Y H 2 = 2 ∗ Y O 2 . Table 4 continued on next page Table 4 continued on next page", "pages": [ 18, 19 ] }, { "title": "B. MODEL OPTIMIZATION", "content": "To determine the values of g 0 O 2 , x 0 , q 0 and Q in Equation 1, Teolis et al. (2010, 2017) split the equation into separate energy and temperature dependencies. They fit the temperature-independent data ( ≤ 80 K) to the energy-dependent component to determine g 0 O 2 and x 0 and then found q 0 and Q by fitting the temperature-dependent component to O 2 yields for all energies normalized to unity at 150 K. Our use of MCMC methods to optimize the model to electron data enables us to determine values for all parameters ( g 0 O 2 , x 0 , q 0 , and Q ) without splitting Equation 1 into energy and temperature components. This allows us to optimize the model to every data point available regardless of energy or temperature. We use 'emcee,' an open-source software package in Python (Foreman-Mackey et al. 2013) to optimize Equation 1 from Teolis et al. (2017) to electron laboratory data. While more commonly used for observational astronomy (Dunkley et al. 2005; Line et al. 2015; Tribbett et al. 2021; and many others), our group has previously used emcee to match existing models/equations to experimental data (Behr et al. 2020; Carmack et al. 2023). Briefly, emcee uses Markov chain Monte Carlo (MCMC) methods with Bayesian inference (as described in Foreman-Mackey et al. 2013 and Behr et al. 2020) in order to explore the probability distribution of parameters in a model when compared to an observed data set. It does this by utilizing 'walkers' which move around the model's multi-dimensional parameter space along a Markov chain. Each 'step' in the Markov chain, or change in parameter values, depends only on how the probability of the current walker values compare to a random sampling of possible new values. With enough steps in the chain, MCMC forgets the user-specified initial parameter values and is able to escape local solutions. Moreover, no additional knowledge (other than defining the likelihood function) is needed to run emcee, eliminating the constraints of grid searches such as user defined spacing/resolution and limiting values. Furthermore, while the points in a grid search scale exponentially with dimensionality, this is not necessarily the case with MCMC, potentially making computational times with MCMC faster. This puts MCMC above other commonly used fitting methods (e.g. by eye or using a grid method, see Speagle 2019 for more details) by thoroughly exploring the probability of observed data being described by the model throughout the multi-dimensional parameter space. We gave all parameters ( g 0 O 2 , x 0 , q 0 , and Q ) flat priors limiting them to physical values (i.e. ≥ 0), and we gave x 0 an additional Gaussian prior of 5 ± 4 nm in order to encompass estimates of efficient O 2 production depths from Petrik et al. (2006) and Meier & Loeffler (2020). Walkers in MCMC can and will stray away from the mean of the specified Gaussian prior if the likelihood of the observed data given the model prefers it. We randomly distributed initial parameter values for walkers around an estimate made by fitting Equation 1 to the data by eye. While not necessary, starting walkers with an educated guess will reduce the number of steps and therefore computational time the walkers need in order to constrain the posterior distribution and best fit parameter values (unless of course the educated guess was a poor one or a local solution). We ran our MCMC optimization process with three Y O 2 data sets: from our group, from our group and Sieger et al. (1998), and from our group and Teolis et al. (2010). We modeled the data from Sieger et al. (1998) and Teolis et al. (2010) separately because of their large differences in Y O 2 for similar electron energies (see Figure 7). We scaled their data by cos 1 . 3 ( β )/slash.left cos 1 . 3 ( 12 . 5 · ) (Vidal et al. 2005) to account for the differences in incidence angle ( β ) and assumed 100% error for data from both Sieger et al. (1998) and Teolis et al. (2010) due to no error being provided in the original manuscripts and due to possible differences in sample thickness affecting the reported sputtering yields (Petrik & Kimmel 2005). Like we did for our own data, we interpolated electron ranges from Castillo-Rico et al. (2021) for the relevant electron energies used by Sieger et al. (1998) and Teolis et al. (2010). We list the optimized values for g 0 O 2 , x 0 , q 0 , and Q for all data sets in Table 2 and provide additional discussion in Section 5.2.", "pages": [ 21, 22 ] }, { "title": "REFERENCES", "content": "Abdulgalil, A. G. M., Rosu-Finsen, A., Marchione, D., et al. 2017, ACS Earth and Space Chemistry, 1, 209, doi: 10.1021/acsearthspacechem.7b00028 Addison, P., Liuzzo, L., Arnold, H., & Simon, S. 2021, Journal of Geophysical Research: Space Physics, 126, e2020JA029087, doi: 10.1029/2020JA029087", "pages": [ 22 ] } ]

[ { "title": "Solar neutrinos and ν 2 visible decays to ν 1", "content": "Andr'e de Gouvˆea, 1, ∗ Jean Weill, 1, † and Manibrata Sen 2, ‡ 1 Northwestern University, Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA 2 Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Experimental bounds on the neutrino lifetime depend on the nature of the neutrinos and the details of the potentially new physics responsible for neutrino decay. In the case where the decays involve active neutrinos in the final state, the neutrino masses also qualitatively impact how these manifest themselves experimentally. In order to further understand the impact of nonzero neutrino masses, we explore how observations of solar neutrinos constrain a very simple toy model. We assume that neutrinos are Dirac fermions and there is a new massless scalar that couples to neutrinos such that a heavy neutrino ν 2 with mass m 2 - can decay into a lighter neutrino ν 1 with mass m 1 - and a massless scalar. We find that the constraints on the new physics coupling depend, sometimes significantly, on the ratio of the daughter-to-parent neutrino masses, and that, for largeenough values of the new physics coupling, the 'dark side' of the solar neutrino parameter space - sin 2 θ 12 ∼ 0 . 7 - provides a reasonable fit to solar neutrino data. Our results generalize to other neutrino-decay scenarios, including those that mediate ν 2 → ν 1 ¯ ν 3 ν 3 when the neutrino mass ordering is inverted mass and m 2 > m 1 ≫ m 3 , the mass of ν 3 .", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Since the discovery of nonzero, distinct neutrino masses and nontrivial lepton mixing, one can unambiguously conclude that the two heavier neutrinos have finite lifetimes. The weak interactions dictate that these will decay into three lighter neutrinos, assuming the decay is kinematically allowed, or into a lighter neutrino and a photon [1], always kinematically allowed. Quantitatively, however, the weak interactions translate into lifetimes that are many orders of magnitude longer than the age of the universe, exceeding 10 37 years for all values of the neutrino masses and mixing parameters that satisfy existing experimental and observational constraints [2]. Not surprisingly, the presence of new neutrino interactions and new light states can easily translate into much shorter neutrino lifetimes. On the other hand, experimental constraints on the lifetimes of neutrinos - see, for example, [3-38] - are absurdly far from the expectations of the standard model plus massive neutrinos. These rely on experiments with neutrinos that travel long distances before they are detected, ranging from earth bound reactor and accelerator neutrino experiments (1 to 1,000 km), solar neutrino experiments (500 light-seconds), neutrinos from SN1987A (170,000 light-years), to indirect inferences regarding the properties of the cosmic neutrino background. All experimental bounds on the neutrino lifetime are model dependent. They depend on the nature of the neutrinos - are neutrinos Majorana fermions or Dirac fermions? - the decay mode - are there visible particles, such as neutrinos or photons, in the final state? - and the dynamics of the interaction responsible for the decay - does it involve left-chiral or right-chiral neutrino fields? Furthermore, as we explored in [38], in the case where the decay involves active neutrinos in the final state, the neutrino masses qualitatively impact the neutrino decay and how it manifests itself experimentally. Solar neutrinos provide robust, reliable bounds on the neutrino lifetime. Given everything we know about neutrino masses and neutrino mixing, the solar neutrino spectrum is well known and, it turns out, it is characterized by an incoherent mixture of the neutrino mass eigenstates, so the impact of neutrino decay is easy to visualize. There is also a wealth of solar neutrino data collected in the last several decades. Here we will concentrate on data from SuperKamiokande [39] and SNO [40, 41] - on 8 B neutrinos - and on data from Borexino [42] - on 7 Be neutrinos - in order to explore how observations of solar neutrinos constrain a very simple toy model, taking finite neutrino masses into account. We assume that neutrinos are Dirac fermions and there is a new massless scalar that couples to neutrinos such that a heavy neutrino can decay into a lighter neutrino and a massless scalar. We find that the constraints on the new physics coupling depend, sometimes significantly, on the ratio of the daughter-to-parent neutrino masses, and that, for specific values of the new physics coupling, the 'dark side' of the solar neutrino parameter space [43] provides a reasonable fit to solar neutrino data. We also find that 'high-energy' solar neutrino data complement the data on 'low-energy' solar neutrinos in a very impactful manner. In Section II, we discuss the model under investigation and the characteristics of the neutrino decay processes mediated by the model. In Section III, we briefly summarize the effects of neutrino decay on neutrino flavor evolution, highlighting solar neutrinos. We discuss the different experimental data and constraints in Sections IV A, IV B, and IVC while combined results are presented in Section IV D. Section V contains a summary of our findings along with generalizations and some parting thoughts.", "pages": [ 1, 2 ] }, { "title": "II. THE MODEL", "content": "We assume the neutrino mass eigenstates ν i with mass m i , i = 1 , 2, interact with a massless scalar boson φ via the following Lagrangian, where the neutrinos are Dirac fermions and P L is the left-chiral projection operator. Neutrino mass eigenstates are defined in the usual way, m 2 > m 1 , and we do not consider similar interactions involving ν 3 . This operator mediates the decay of a ν 2 into a ν 1 . The analysis of this decay in the case m 1 = 0 was performed, e.g., in [14], where it was argued that a subset of solar neutrino data, as well as KamLAND data, can be used to constrain the invisible decays of ν 2 : m 2 Γ 2 < 9 . 3 × 10 -13 eV 2 . A consistent but more precise bound was later obtained by the SNO collaboration [44]. We expect different results once the daughter neutrinos have non-zero masses. Other consequences of Eq. (II.1) will be briefly discussed in Sec. V. ̸ Eq. (II.1) only contains the right-chiral component of the ν 1 field. In the limit m 1 → 0, in a ν 2 decay process, only right-handed helicity ν 1 are produced, ∗ independent from the polarization state of the ν 2 . For all practical purposes, right-handed helicity neutrinos are inert and cannot be detected. For m 1 = 0, there is a nonzero probability for the production of left-handed helicity - hence detectable - daughter ν 1 . This probability grows as the daughter neutrino mass approaches the parent neutrino mass. In this limit, in fact, the decay into left-handed helicity ν 1 dominates over the decay into right-handed ν 1 . Fig. 1 depicts the differential decay width of a left-handed helicity ν 2 with energy E 2 into a left-handed helicity ν 1 with energy E 1 , normalized to the total decay width, as a function of x = E 1 /E 2 , for different values of m 1 /m 2 . Clearly, as m 1 → m 2 , there is a significant increase in the contribution of the helicity preserving - left-handed daughter - channel. Furthermore, as m 1 → m 2 , the decay spectrum is compressed; energymomentum conservation implies that the heavy daughter inherits most of the parent energy while the massless φ comes out with only a tiny fraction of the allowed energy. For many more details and discussions, see [38]. The neutrino-decay physics mediated by Eq. (II.1) is governed by three parameters: the dimensionless coupling g and the neutrino masses m 1 and m 2 . The difference of the neutrino masses squared ∆ m 2 21 is experimentally well constrained, mostly by the KamLAND reactor neutrino experiment [45], so we use g and the ratio of the neutrino masses m 1 /m 2 to define the remaining two-dimensional parameter space of the model. The decay width of a ν 2 at rest multiplied by its mass, † is [38] For fixed ∆ m 2 21 , m 2 Γ 2 depends linearly on g 2 and only very weakly on the ratio of the neutrino masses, varying by a factor of two as the value of m 1 /m 2 covers its entire allowed range from zero to one. There is, however, an experimental upper bound to m 1 /m 2 . It is trivial to compute and note that, for a fixed ∆ m 2 21 , both the values of m 1 and m 2 diverge as m 1 /m 2 → 1. Nonetheless m 1 /m 2 values very close to one are experimentally allowed. Consider, for example, ∆ m 2 21 = 7 . 54 × 10 -5 eV 2 and an upper bound of 0 . 1 eV for m 2 . Using Eq. (II.3), this upper bound translates into m 1 /m 2 ≤ 0 . 996. On the other hand, arbitrarily small values of m 1 /m 2 are allowed as long as the neutrino mass ordering is normal ( m 3 > m 2 > m 1 ). For the inverted neutrino mass ordering ( m 2 > m 1 > m 3 ), m 1 /m 2 ≥ 0 . 985. In summary, virtually all values of m 1 /m 2 are allowed by the data, including values very close to one. In the case of the inverted neutrino mass ordering, only values of m 1 /m 2 close to one are allowed.", "pages": [ 2, 3 ] }, { "title": "III. ANALYSIS STRATEGY", "content": "In vacuum, allowing for the possibility that ν 2 with helicity r and energy E h can decay into a ν 1 with helicity s and energy E l with associated partial differential decay width d Γ rs /dE l , the differential probability (per unit E l ) for a ν α with helicity r and energy E h to behave as a ν β with helicity s and energy E l after it has traveled a distance L is [8, 25] where U αi , α = e, µ, τ , i = 1 , 2 , 3 are the elements of the leptonic mixing matrix and Γ 2 is the ν 2 total decay width. The first term encodes the contribution from the surviving parent neutrino, including oscillations, while the second term includes the contribution from the daughter neutrino. Solar neutrinos, instead, are well described as incoherent mixtures of the mass eigenstates. Hence, the initial state produced inside the sun with energy E h exits the sun as a ν i with probability P i ( E h ), i = 1 , 2 , 3, and all neutrinos are left-handed ( r = -1). The differential probability that the neutrino arriving at the earth with energy E l is potentially detected as a ν β with helicity s is The impact of the decay is as follows. The ν 2 population decays exponentially and is, instead, replaced by a ν 1 population with a softer energy spectrum and with positive and negative helicities. Furthermore, the daughter energy spectrum is also distorted relative to the parent one by the energy dependency of the exponential decay; higher energy parents decay more slowly than lower energy ones. It is pertinent to make a few comments regarding the ν 3 component of the solar neutrino flux. P 3 ∼ 0 . 02 for all E h of interest so the original ν 3 contribution to the flux is very small. Had we allowed for interactions involving ν 3 , these would not lead to especially interesting effects for solar neutrinos. In more detail, if the neutrino mass ordering were normal ( m 3 > m 2 > m 1 ), the new interaction involving ν 3 would mediate potentially visible ν 3 decays. In this case, however, the impact of the decay-daughter population - equivalent to the second line in Eq. (III.5) - would be suppressed by P 3 and hence small relative to the dominant ν 2 and ν 1 original populations. Instead, if the neutrino mass ordering were inverted ( m 2 > m 1 > m 3 ), the new interaction involving ν 3 would mediate potentially visible ν 2 and ν 1 decays into ν 3 . In this case, at least when it comes to detectors predominantly sensitive to the ν e component of the beam, the daughter population would be almost invisible since | U e 3 | 2 ∼ 0 . 02 is very small relative to | U e 1 | 2 , | U e 2 | 2 . The differential number of events at a detector that is sensitive to ν β via the weak interactions, including visible decays, is [8] where Φ( E h ) denotes the neutrino energy spectrum at production and E max = E l m 2 2 /m 2 1 is the kinematical upper bound on E l . The resolution function connecting the true energy E l and the detected energy ˜ E l is R ( ˜ E l , E l ). The total cross-section for detecting a ν β wih helicity s is σ s ( E l ). For right-handed helicity neutrinos, s = 1, the weak cross-section is suppressed by m 2 1 /E 2 l , and is set to zero throughout.", "pages": [ 3, 4 ] }, { "title": "IV. SIMULATIONS AND RESULTS", "content": "Here, we consider in turn the solar neutrino data from Borexino, Super-Kamiokande, and SNO, and estimate their sensitivity to visible solar neutrino decays. When simulating event rates at Borexino, we considered 1,072 days of Borexino Phase-II data taking [42]. For Super-Kamiokande, we consider 504 days of data taking, corresponding to Super-Kamiokande Phase I [39], and for SNO, we consider 365 days of data taking, which corresponds to roughly the first two phases of SNO [41]. We make use of the PDG parameterization for the elements of the mixing matrix and, when applicable, use the following values for the oscillation parameters of interest [2]: Throughout, our main goal is to understand the impact of the daughter neutrino mass m 1 and explore whether nontrivial neutrino decays allow for a different fit to the solar neutrino data.", "pages": [ 4 ] }, { "title": "A. Borexino", "content": "Borexino [42] is a 280 ton liquid scintillator detector located underground at the Laboratori Nazionali del Gran Sasso (LGNS) in Italy. Its main focus is the detection of solar neutrinos, in particular 7 Be neutrinos, through neutrino-electron scattering. Neutrinos are detected via the scintillation light which is emitted isotropically during the propagation of the recoil electron and detected by 2212 photo-multiplier tubes, allowing for the measurement of the recoil-electron energy. When simulating event rates at Borexino, we considered 1,072 days of Borexino Phase-II data taking, N tar = 3 × 10 31 targets coming from the 100 tons of fiducial mass. We approximated the 7 Be neutrino differential energy flux by a delta function. The kinematical parameter most relevant to the experiment is the electron recoil energy, which follows a continuous distribution governed by the neutrino-electron scattering process. The experiment succeeds at detecting 7 Be neutrinos by achieving the strictest radio-purity levels. A detailed understanding of the main backgrounds was therefore necessary to properly estimate the sensitivity of Borexino to neutrino decays. Fig. 2, from [42], depicts the main backgrounds for the solar neutrino measurement. These come from radioactive processes involving 210 Bi, 85 Kr, and 210 Po [42]. In our analyses, we treat the different background components independently. Using Fig. 2, we fit for the shape of the different background components, which we hold fixed. For different values of the decay and mixing parameters g, r ≡ m 1 /m 2 , and sin 2 θ 12 we compute the equivalent of the red curve in Fig. 2. We simplified our analyses by considering a Gaussian energy resolution function for the 7 Be spectrum and assuming 100% efficiency. We restricted our analyses to recoil energies between 200 keV and 665 keV. 665 keV is the maximum kinetic energy of the recoil electron for 862 keV 7 Be neutrinos. For higher recoil energies, we did not have enough information on the Borexino energy resolution in order to perform a trustworthy analysis and decided, conservatively, to exclude these data points from the analysis. The 665 keV threshold is highlighted in Fig. 2 with a red vertical dashed line. We bin both the background and signal curves in order to perform a χ 2 fit to the data in Fig. 2. The value of the unoscillated 7 Be flux, which is rather well known, is held fixed. We first analyze the data assuming neutrinos are stable ( g = 0) and fit for the normalization of each background component along with that of the 7 Be neutrino contribution. We further constrain the 210 Bi background by including the data associated to recoil kinetic energy bins between 740 keV and 800 keV, making the simplifying assumption that only 210 Bi events contribute inside that window. Having done that, henceforth we fix the normalization of the different background components to these extracted best-fit values. Taking all of this into account, we compute χ 2 ( g, r, sin 2 θ 12 ), find χ 2 min , the minimum value of χ 2 , and define the boundaries of 'allowed' and 'excluded' regions of parameter space using fixed values of ∆ χ 2 ≡ χ 2 -χ 2 min . In our analyses, we marginalize over the value of sin 2 θ 12 and add a Gaussian prior in order to include external constraints on this mixing angle. We first make use of the following prior: sin 2 θ 12 = 0 . 30 ± 0 . 05, selected from the current best fit value for sin 2 θ 12 and consistent with the uncertainty reported by KamLAND [45]. Fig. 3(left) depicts the regions of the g × r parameter space allowed at the one-, two-, and three-sigma levels (∆ χ 2 = 2 . 30, 6 . 18, and 11.83, respectively). Using the results from Sec. III, and taking into account that matter effects are small for 7 Be solar neutrino energies, the electron neutrino survival probability for 7 Be neutrinos, integrating over the daughter neutrino energy, is well approximated by ‡ where f ( r, sin 2 θ 12 ) is a function of r and sin 2 θ 12 . The first two terms correspond to the contribution of the surviving parents while the last term comes from the visible daughter component. The function f ( r, sin 2 θ 12 ), while relatively cumbersome, has the following simple limit: f ( r → 0 , sin 2 θ 12 ) = 0 for all sin 2 θ 12 . This limit follows from the fact that, as the daughter mass m 1 → 0, all daughters have right-handed helicity and are hence invisible. f ( r, sin 2 θ 12 ) also has an approximate upper limit, which we will discuss momentarily. When m 2 Γ 2 L/E h is small, the decay effects are not significant. As discussed earlier, for fixed ∆ m 2 21 , m 2 Γ 2 depends exclusively, for all practical purposes, on g . For the Earth-Sun distance and 7 Be neutrino energies, m 2 Γ 2 L/E h ≪ 1 for g ≲ 0 . 001. In this region, the electron neutrino survival probability is This limiting case is apparent in the left panel in Fig. 3 where all values of g ≲ 0 . 001 are allowed, mostly independent from r . In the opposite regime m 2 Γ 2 L/E h ≫ 1 - Eq. (IV.8) simplifies to again keeping in mind that matter effects are small for 7 Be neutrino energies. In this region of the parameter space, the electron neutrino survival probability depends on r but does not significantly depend on g . This behavior is apparent in Fig. 3(left) where the contours become horizontal lines. The behavior of ∆ χ 2 is governed by two effects: the 'missing' ν 2 -component of the parent population and the behavior of the visible daughter contribution. The effect of the missing ν 2 -component can be seen when r ≪ 1 and f is very small. The fact that P ee < P NoDecay ee allows one to disfavor that region of the parameter space. For larger values of r , f is finite and the daughter contribution can make up for the missing ν 2 component of the flux, as is apparent in Fig. 3(left). More quantitatively, when the decays are prompt relative to the Earth-Sun distance, the daughter contribution is of order sin 2 θ 12 cos 2 θ 12 × Br(visible), where Br(visible) is the probability that the daughter from the decay has left-handed helicity and is therefore visible. Numerically, the combination sin 2 θ 12 cos 2 θ 12 ∼ 0 . 2 and, for Br(visible) ∼ 0 . 5, it turns out that sin 4 θ 12 ∼ sin 2 θ 12 cos 2 θ 12 × Br(visible). Note that for r → 1, Br(visible) → 1 and the decay solution 'overshoots' the no-decay electron-neutrino survival probability, a behavior that is also reflected in the left panel in Fig. 3. Finally, we highlight that values of r ∼ 0 . 8 are slightly more disfavored relative to other values of r in the limit where the decay is prompt. The reason is partially related to the distortion of the daughter neutrino energy spectrum relative to the parent one concurrent with a significant fraction of visible decays.", "pages": [ 4, 5, 6 ] }, { "title": "1. Dark Side", "content": "If one ignores solar neutrino data, all detailed information on sin 2 θ 12 comes from reactor antineutrino experiments. In fact, until the JUNO experiment [47] starts collecting and analyzing data, all detailed information comes from the KamLAND experiment. The experimental conditions are such that, to an excellent approximation, KamLAND is only sensitive to sin 2 2 θ 12 = 4sin 2 θ 12 cos 2 θ 12 and cannot distinguish θ 12 from π/ 2 -θ 12 . § 8 B solar neutrino data break the degeneracy and rule out the so-called dark side of the parameter space, sin 2 θ 12 > 0 . 5. Since we are introducing a hypothesis that modifies the flavor evolution of solar neutrinos, we investigate the constraints on g and r restricting θ 12 to the dark side. In the absence of oscillations, because matter effects are negligible for 7 Be solar neutrino energies, Borexino cannot rule out the dark side of the parameter space: for both sin 2 θ 12 = 0 . 3 and sin 2 θ 12 = 0 . 7, P NoDecay ee = 0 . 58. This does not hold when the parent neutrino is allowed to decay. We repeat the exercise discussed earlier in this section with the prior sin 2 θ 12 = 0 . 7 ± 0 . 05 and depict our results in the right panel in Fig. 3. Comparing to the left panel in Fig. 3, it is clear that the exchange symmetry is broken. Dark and light side priors on sin 2 θ 12 lead to different results for both the invisible and visible contribution, leading to significant differences between the hypotheses. As before, when g ≲ 0 . 001 decay effects are irrelevant - the lifetime is too long - and the Borexino data are not sensitive to g or r . When g ≳ 0 . 001, the results obtained with the two different priors differ considerably. These differences are simplest to analyze qualitatively when r is small. In this limiting case, the daughter neutrinos are effectively invisible and the ν 2 component of the flux has enough time to completely disappear. We are left with In the light side, as discussed earlier, sin 2 θ 12 = 0 . 3 translates into P ee ∼ 0 . 5, not too far from the central value preferred by the no-decay scenario, P ee = 0 . 58. Instead, in the dark side, cos 2 θ 12 = 0 . 3 and P ee ∼ 0 . 1, markedly smaller than the the preferred value in the no decay scenario. This is apparent when comparing the two panels in Fig. 3; the dark-side constraints are stronger than the light-side ones and the large g , small r region is excluded, in the dark side, at more than the three-sigma level. ¶ For larger values of r , the daughter contribution improves the quality of the fit in the dark side when the ν 2 decay is prompt. Following the discussion below Eq. (IV.10), the daughter contribution cannot exceed approximately 0.2 and, in the dark side assuming the decay hypothesis, when g ≳ 0 . 001, P ee < 0 . 3, always less than the central value preferred by the no-decay scenario ( P NoDecay ee = 0 . 58). Nonetheless, the region of parameter space for r → 1 and large values of g is disfavored at less than the three-sigma level.", "pages": [ 6, 7 ] }, { "title": "B. Super-Kamiokande", "content": "Super-Kamiokande (SK) is a 50 kton water Cherenkov detector running in Japan. Solar neutrinos interact inside the water mainly through neutrino-electron scattering, which is sensitive to neutrinos of all flavors. The scattered electron produces Cherenkov radiation in the water, which can be detected. To simulate events in SK, we compute the neutrino-electron cross-section and consider 504 days of data taking [39], consistent with phase I of SK solar neutrino data. SK has collected over 5,000 days of solar neutrino data but we restrict our analyses to phase I for a couple of reasons. First, the detector changed qualitatively after phase I and our simulations below are best matched to the results presented in [39]. Second, we are most interested in exploring the differences between Borexino, SK, and SNO (discussed in the next subsection) and how these different data sets complement one another, as opposed to how strict are the solar constraints on our neutrino decay hypothesis. For this reason, we benefit most from combining data sets that are roughly of the same size. The number of targets is taken to be N tar = 3 × 10 33 . The resolution function is taken to be S ( E tr ) = -0 . 084 + 0 . 376 √ E tr +0 . 040 E tr . SK is sensitive to 8 B neutrinos. The neutrino oscillation parameters are such that 8 B neutrinos are predominantly ν 2 : P 2 ≳ 0 . 9 for the energy range of interest [48] and, in the absence of neutrino decays, the electron neutrino survival probability is P ee ∼ | U e 2 | 2 . SK data are consistent with | U e 2 | 2 ≡ sin 2 θ 12 cos 2 θ 13 ∼ 0 . 3. Since cos 2 θ 13 ∼ 0 . 98, this translates into sin 2 θ 12 ∼ 0 . 3. As discussed in detail earlier, invisible ν 2 decays lead to a suppression of the measured flux. This suppression can, in principle, be compensated by increasing the value of sin 2 θ 12 . Similar to the Borexino discussion, here it is also easy to estimate the impact of the invisible decays. If a fraction ϵ of the ν 2 population survives, P ee ∼ (1 -P 2 ) cos 2 θ 12 + P 2 ϵ sin 2 θ 12 , ignoring sin 2 θ 13 effects. If epsilon is not zero, one could tolerate a population drop of up to almost 70% by jacking up sin 2 θ 12 all the way to one. On the other hand, when ϵ = 0, because P 2 ≳ 0 . 9, it is impossible to lower sin 2 θ 12 enough to reach P ee ∼ 0 . 3 even in the limit cos 2 θ 12 → 1. Of course, KamLAND data constrain sin 2 θ 12 ∼ 0 . 3. Hence, the combination of reactor and solar data prevent one from varying sin 2 θ 12 with complete impunity. KamLAND data are, however, also completely consistent with sin 2 θ 12 ∼ 0 . 7 - the dark side solution discussed in the last subsection - allowing some extra flexibility. The situation is different when the daughters of the decay are visible, something we expect when r ≡ m 1 /m 2 is large. In this case, the daughters contribute to the number of events in SK, contributing an amount of order P 2 cos 2 θ 12 (1 -ϵ ). Including the parent contribution, P ee ∼ cos 2 θ 12 -ϵP 2 cos 2 θ 12 . Now, if all ν 2 decay ( ϵ → 0), P ee ∼ cos 2 θ 12 which may provide a good fit to SK data if θ 12 is in the dark side. The discussion in the preceding paragraphs is meant to be qualitative and we now turn to a more quantitative estimate of SK's sensitivity. Fig. 4 shows the expected number of events as a function of the recoil-electron kinetic energy for different values of g and r . Here, ∆ m 2 21 = 7 . 54 × 10 -5 eV 2 , sin 2 θ 12 = 0 . 307, and sin 2 θ 13 = 0 . 0218. We find that for r = 0 . 1 (left-hand panel), as the value of the coupling g increases, the number of events decreases. In this regime, most of the decays are invisible and the impact of the nonzero daughter mass is negligible. However, for larger values of r , the visible contribution is significant. For r = 0 . 9 (right-panel) the impact of the visible daughters is strong enough that one observes an increase of the expected number of events as the coupling g increases. One should also note that the recoil-electron kinetic energy spectrum is distorted relative to the no-decay hypothesis. We first discuss the results presented in the left-hand panel of Fig. 5. As expected, there is no sensitivity to g ≲ 10 -3 ; in this region of the parameter space, the lifetime is too long relative to the Earth-Sun distance. There is no allowed scenario in the region of parameter space where all ν 2 decay into ν 1 ( g ≫ 0 . 001). The reasoning is, rather qualitatively, as follows. When the decays are invisible (small r ) one expects too few events in SK. Instead, when the decay is 100% visible, the survival probability is too large (of order cos 2 θ 12 ∼ 0 . 7). This implies there is a value of r where the visible branching ratio is optimal. For such a value, however, the energy spectrum is distorted enough that a fit comparable to that of the stable ν 2 case does not exist. When sin 2 θ 12 is constrained to the dark side, the situation is qualitatively different. In this case, when the lifetime is too long ( g ≪ 0 . 001), sin 2 θ 12 ∼ 0 . 7 is safely ruled out since it leads to too many events associated to 8 B neutrinos. For small r , large values of g are also excluded. In this case, the ν 2 decays invisibly and there are too few events associated to 8 B neutrinos. There is a range of values of g that lead to a reasonable fit when r is small. When r is large and the decays are mostly visible, the situation is again qualitatively different. In the limit r → 1 and large g , the ν 2 population decays into left-handed ν 1 with the same energy and, roughly, P ee ∼ cos 2 θ 12 , In the dark side, cos 2 θ 12 ∼ 0 . 3 and we expect a fit that is just as good as the standard, no-decay fit when sin 2 θ 12 = 0 . 3. This corresponds to the region of parameter space allowed at one-sigma when g is large and r → 1.", "pages": [ 7, 8 ] }, { "title": "C. Sudbury Neutrino Observatory", "content": "The Sudbury Neutrino Observatory (SNO) was an underground neutrino detector in Canada, consisting of 1kT of heavy water (D 2 O), designed to observe solar neutrinos. With heavy water, SNO could measure neutrinos through three channels: The CC interaction provided a direct measurement of the ν e flux from 8 B neutrinos coming from the Sun, while the NC processes could measure the net 8 B neutrino flux, irrespective of the flavor conversions. The impact of neutrino decay in SNO is similar to that in SK, with a few extra ingredients. The NC measurement is insensitive to the neutrino flavor but is sensitive to the total number of left-handed helicity neutrinos. Hence, invisible neutrino decays are constrained in a way that cannot be compensated by modifying the mixing parameters. On the other hand, visible neutrino decays allow one to accommodate the NC sample as long as the energy distribution of the daughter neutrinos is not very different from that of the parents. The CC measurement, instead, is only sensitive to electrontype neutrinos and hence it is impacted by neutrino decay in a way that is slightly different from the ES sample. Quantitatively, there is significantly more statistical power in the CC sample so we concentrate on it henceforth. To simulate events in SNO, we consider 1 year of data taking, corresponding to roughly the first two phases of SNO, and consider the CC channel only. We considered fourteen bins of equivalent-recoil-electron kinetic energy, 0.5 MeV wide [41]. The number of targets is taken to be N tar = 3 × 10 31 . The resolution function is considered to be S ( E tr ) = -0 . 462 + 0 . 5470 √ E tr +0 . 00872 E tr , following [41] and the energy threshold of the detector is taken to be E thr = 1 . 446 MeV. SNO reports the recoil kinetic energy T eff = E e -m e and information is extracted from T eff > 6 MeV. For lower energies, the NC sample dominates - see Fig. 2 of [49]. Fig. 6 shows the effect of visible decay on the CC event spectrum in SNO for different values of g and r . We find features very similar to those at SK. Fig. 7 shows the sensitivity of the SNO CC sample to neutrino decays as a function of the coupling g and the ratio of the daughter-parent masses r ≡ m 1 /m 2 . We simulate data consistent with no decay, and test for the impact of visible neutrino decays, as we vary g and r . We find results similar to those of SK, Fig. 5.", "pages": [ 9 ] }, { "title": "D. Combined Results", "content": "In this section we describe our combined analysis for the three experiments. The light-side result is presented in Fig. 8 and shows the combined ∆ χ 2 contour plot for SK, SNO, and Borexino for decay relative to no decay with a prior of sin 2 θ 12 = 0 . 30 ± 0 . 05. Contours are for ∆ χ 2 = 2 . 30 (solid), ∆ χ 2 = 6 . 18 (big dashed), and ∆ χ 2 = 11 . 83 (small dashed). We repeated the same exercise with a dark-side prior, sin 2 θ 12 = 0 . 70 ± 0 . 05, where, similar to Figs. 5 and 7, ∆ χ 2 is computed relative to the minimum value of χ 2 , χ 2 min , obtained in the light side. In this case, we find a very small allowed region - too small to capture in a useful figure - characterized by g ∼ 0 . 005 and r ∼ 0 . 98. There are, however, points in the parameter space allowed at close to the one-sigma level. Independent from the assumption on sin 2 θ 12 - light side versus dark side - the complementarity between the low energy solar neutrino data (Borexino) and the high energy solar neutrino data (SK and SNO) is clear. The reason is the mass-eigenstate composition of the 7 Be solar neutrino flux is quite different from that of the 8 B: the 7 Be flux is almost 70% ν 1 while the 8 B flux is more than 90% ν 2 . In the light side, Borexino data allow for short lifetimes as long as the ratio of m 1 and m 2 is around 0.9 but this possibility is safely excluded by SK and SNO data. In the dark side, instead, SK and SNO data allow for short lifetimes as long as the ratio of m 1 and m 2 is close to 1. This is disfavored by Borexino data. In the end, the combined data sets allow sin 2 θ 12 around 0.7 as long as m 1 and m 2 are close in mass and ν 2 decays to ν 1 between the Earth and the Sun. This hypothesis is only slightly disfavored and is certainly not excluded by our rather simplified analyses.", "pages": [ 10, 11 ] }, { "title": "V. CONCLUDING REMARKS", "content": "We explored how active neutrino decays impact the interpretation of solar neutrino data and how well solar neutrino data can test the hypothesis that the active neutrinos are unstable. We were especially interested in understanding the nontrivial impact of nonzero daughter neutrino masses in the different analyses. For the scenario of interest, we found that the value of the daughter neutrino mass can significantly modify the impact of active solar neutrino decay, sometimes qualitatively. We also explored the possibility that the solar angle resides in the so-called dark side of the parameter space, a hypothesis that, for stable neutrinos, is only excluded by solar neutrino data. It is important to revisit this possibility whenever one modifies the physics of solar neutrinos. It is well known, for example, that the addition of large non-standard neutrino-matter interactions allow a fit to the solar neutrino data in the dark side [50]. We found that measurements of 7 Be and 8 B neutrinos are quite complementary. 'Blind spots' in one type of experiment are often covered by the other type. Nonetheless, we find that the hypothesis that the solar angle resides on the dark side is allowed as long as ν 1 and ν 2 are close in mass and ν 2 decays relatively quickly relative to the Earth-Sun distance. A more sophisticated analysis, outside the aspirations of this manuscript, is necessary in order to reveal whether a dark-side solution with unstable ν 2 is indeed allowed by the combined SK, SNO, and Borexino data. We highlight the importance of Borexino when it comes to addressing this particular issue. We restricted our discussions to the hypothesis that neutrinos are Dirac fermions and that the decay is governed by Eq. (II.1), for a few reasons. We chose a chiral interaction such that, when the daughter neutrino is massless, it has right-handed helicity and is hence 'sterile.' The situation is very different when the mass of the daughter neutrino approaches that of the parent. This way, we can change the nature of the decay by dialing up or down the daughter neutrino mass. We also chose a very simple model - a two body decay - in order to render our results and discussions as transparent as possible. Finally, with Dirac neutrinos, we did not have to worry about the possibility of 'neutrinos' converting into 'antineutrinos.' There is a price to be paid by our choice of decay Lagrangian. Eq. (II.1) is not SU (2) L × U (1) Y invariant, for example. A more complete version of this model would include new interactions involving charged leptons, gauge bosons, or other hypothetical new particles. Furthermore, the same physics that mediates neutrino decay will also mediate other phenomena that will provide more nontrivial constraints on the new-physics coupling g . These include relatively long-range neutrino-neutrino interactions - see [51] for a recent thorough overview - the presence of new light degrees of freedom in the early universe - see, for example, [52-55] for recent analyses - and low-energy laboratory processes - see, for example, [56-58]. We did not take any of these constraints into account here. On the other hand, the results discussed here can be generalized to other interesting decay scenarios. As introduced and discussed in [38], for Dirac neutrinos, neutrino decay can be mediated by a four-fermion interaction that involves only right-chiral neutrino fields (see Eq. (II.5) in [38]). New interactions that involve only gauge-singlet fermions are virtually unconstrained by experiments and observations, and are probably best constrained by searches for active neutrino decays. When the neutrino mass ordering is inverted, the decay ν 2 → ν 1 ν 3 ¯ ν 3 , in the limit where m 2 and m 1 are quasi-degenerate and the mass m 3 of the lightest neutrino is very small relative to m 1 , ∗∗ is kinematically quite similar to the decay ν 2 → ν 1 φ . The results obtained here can be adapted to this other decay scenario without too much difficulty.", "pages": [ 11, 12 ] }, { "title": "Acknowledgements", "content": "We thank Pedro de Holanda, Orlando Peres, Renan Picoreti, and Dipyaman Pramanik for helpful discussions and collaboration in the initial stages of the project. This work was supported in part by the US Department of Energy (DOE) grant #de-sc0010143 and in part by the NSF grant PHY-1630782. AdG also acknowledges the warm hospitality of the Department of Theoretical Physics and Cosmology at the University of Granada, where some of this work was carried out. [40] SNO Collaboration, B. Aharmim et al. , 'Low Energy Threshold Analysis of the Phase I and Phase II Data Sets of the Sudbury Neutrino Observatory', Phys. Rev. C 81 (2010) 055504, arXiv:0910.2984 . [51] J. M. Berryman et al. , 'Neutrino self-interactions: A white paper', Phys. Dark Univ. 42 (2023) 101267, arXiv:2203.01955 . seen by Planck', Phys. Rev. D 106 (2022), no. 6, 063539, arXiv:2207.04062 .", "pages": [ 12, 13, 14 ] } ]

[ { "title": "Self-Dual Fields on Self-Dual Backgrounds and the Double Copy", "content": "Graham R. Brown, Joshua Gowdy, Bill Spence Centre for Theoretical Physics Department of Physics and Astronomy Queen Mary University of London Mile End Road, London E1 4NS, United Kingdom", "pages": [ 1 ] }, { "title": "Abstract", "content": "We explore the double copy for self-dual gauge and gravitational fields on self-dual background spacetimes. We consider backgrounds associated to solutions of the second Plebanski equation and describe results with different gauge-fixing conditions. Finally we discuss the kinematic and w -algebras and the double copy, identifying modified Poisson structures and kinematic structure constants in the presence of the self-dual background. The self-dual plane wave and Eguchi-Hanson spacetimes are studied as examples and their respective w -algebras derived.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The study of self-dual gauge and gravitational fields has provided a fertile source of ideas and results in physics and mathematics. Work some time ago showed that self-dual theories in a light-cone gauge could be described by equations for scalar fields [1-9] corresponding to the positive helicity sectors of Yang-Mills and gravity. The self-dual sector allows for a simplified study of many features of the full theories. An area of recent interest is the investigation of self-dual fields in order to learn more about the structure of the double copy 1 and kinematic algebras (see [12-33] for recent more general work on the double copy and CK duality). In [34] it was shown that self-dual Yang-Mills and gravity have manifest colour-kinematics duality, and the kinematic algebra was identified as that of area-preserving diffeomorphisms of the plane. In the context of celestial holography [35-41] this algebra was shown appear through the soft and collinear limit of positive helicity gravitons as (the wedge subalgebra of) w 1+ ∞ [42]. This link between kinematic algebras and OPEs in celestial holography [43-56] was recently discussed in detail in [57]. A natural generalisation of this is to study self-duality conditions on non-flat spacetimes. Some progress has been made in understanding how the double copy can be applied to curved backgrounds [58-66], with the case of AdS receiving particular attention [67-82]. In this letter we would like to study self-duality for the case of self-dual spacetime backgrounds (see [65,66,83-85] for some recent work on this topic). We will work with the spacetime metrics defined by solutions of the second Plebanski equation, and study the conditions for the existence of self-dual Yang-Mills fields, and self-dual metric variations, on these backgrounds. In the flat space case, dealt with in section 2, there are two formulations of the self-duality conditions which are related by a simple relabelling of coordinates. A general self-dual background (YangMills backgrounds are dealt with in section 3 and gravity backgrounds in section 4) does not have this symmetry, and we find that these two formulations generalise quite differently. The first class of solutions in gravity backgrounds, which we call a 'matched' gauge, can be seen as generalising the flat space solution to curved self-dual backgrounds by linearly perturbing the Plebanski scalar. We also find a second class of solutions, which we call a 'flipped' gauge, which requires a Kerr-Schild condition on the background and leads to a modified Poisson structure coming from the Plebanski equation in this gauge. We then describe aspects of the double copy, and kinematic and w -algebras revealed by these results. This general formulation is discussed in detail in two examples - the self-dual plane wave spacetime in section 5 and the Eguchi-Hanson (EH) metric in section 6. These case studies connect with some of the results developed recently in twistor space in [84] and [85]. For the self-dual plane wave background we find that a natural definition of a 'plane wave'-like solution to the wave/Plebanski equation in that spacetime leads to a kinematic algebra with modified structure constants when compared to the flat background. Nevertheless these structure constants match the flat-space case in the holomorphic collinear limit of the two 'plane wave' solutions and so generate the standard flat-space w -algebra. The soft generators generating the algebra are however altered and correspond to the expansion of the particular 'plane wave' solutions adapted to the self-dual plane wave background. The double copy in our formulation replaces the Lie algebra commutators with Poisson brackets and leads to a so-called double bracket in the Plebanski equation. When acting on 'plane wave' solutions, in flipped gauge, we show this procedure replaces colour structure constants with those from the kinematic algebra X PW ( k 1 , k 2 ) and so gives the expected squaring relation of the single copy, i.e. X PW ( k 1 , k 2 ) 2 . In the Eguchi-Hanson background, we express the more complicated solutions of the wave equation discussed in [85] in spacetime coordinates. We then show that the Poisson bracket of two of these 'plane waves' gives an expression for the deformed kinematic structure constants X EH ( k 1 , k 2 ), which we define in the holomorphic collinear limit. The double bracket of two 'plane waves' is then shown to give the square of this expression, but with additional terms, demonstrating that in the Eguchi-Hanson background the kinematic algebra squaring relations are modified by curvature terms. Since even in the holomorphic collinear limit X EH ( k 1 , k 2 ) differs from the flat-space and self-dual plane wave cases, we then expect a completely different ' w -algebra' of soft generators. We derive this algebra of soft generators following the same method as in the previous cases, by expanding the 'plane wave' solutions, giving a spacetime realisation of the results of [85] coming from twistor space.", "pages": [ 2, 3 ] }, { "title": "2 Flat background", "content": "We start by setting notation and briefly recalling the standard results for self-dual Yang-Mills (YM) and self-dual gravity in a flat background. In this section, we will generally follow the discussion in [34]. The spacetime coordinates are taken to be ( u, v, X, Y ), with the metric For real coordinates, this implies we are using (2 , 2) signature. The coordinates ( u, v, X, Y ) are related to the usual ( t, x, y, z ) as follows and in terms of the coordinates ( t, x, y, z ) the metric signature is (+ , -, + , -).", "pages": [ 3, 4 ] }, { "title": "2.1 Self-dual Yang-Mills", "content": "A gauge field A µ = ( A u , A v , A X , A Y ) on flat space with metric (2.1) is self-dual if its field strength 2 satisfies √ where g is the determinant of the metric (2.1). Imposing the gauge-fixing condition A v = 0, the self-duality condition above can be satisfied by setting A X = 0 and for a (Lie algebra valued) function φ ( u, v, X, Y ) satisfying the self-dual Yang-Mills equation where the scalar Laplacian is □ = 2( ∂ u ∂ v -∂ X ∂ Y ). In what follows, we will be using a notation where subscripts on scalar fields such as φ signify partial derivatives. For example, φ v = ∂ v φ, φ Xv = ∂ X ∂ v φ - this should not be confused with the use of subscripts to denote components of covectors, for example k µ = ( k u , k v , k X , k Y ). If we introduce the Poisson bracket then the self-dual Yang-Mills equation becomes where we have used a notation suggestive of colour-kinematics duality, as used in [80], For the covector k µ = ( k u , k v , k X , k Y ) and coordinate vector x µ = ( u, v, X, Y ), with k · x := k µ x µ , the plane wave e ik · x satisfies if k µ is a null vector. In momentum space, the cubic coupling arising from the self-dual YM equation (2.7) involves the kinematic structure constants along with the Lie algebra constants f abc . Explicitly, the bracket (2.8) of two plane waves and Lie algebra generators satisfies [34] There is also an alternative gauge-fixing condition A u = 0, for which the self-duality condition can be satisfied by setting A Y = 0 and for a function φ ( u, v, X, Y ) satisfying the self-dual YM equation (2.5) but with the coordinates u ↔ v and Y ↔ X exchanged, that is The flat metric is invariant under this exchange and so results obtained in this new gauge are trivially related to the previous gauge by a simple interchange of coordinates. This is not the case when we consider self-dual backgrounds in the sections below, since these backgrounds have no such symmetry, and we will describe the two different gauges separately.", "pages": [ 4, 5 ] }, { "title": "2.2 Self-dual gravity", "content": "Now we recall the analogous construction for self-dual gravity, where the metric is taken to be the following variation from the flat metric: with some function Ψ( u, v, X, Y ). Define the expression where the subscript 0 indicates the flat background, and define the operator ∆ 0 (Ψ) by the variation of this expression as Explicitly Then the anti-self-dual part of the Weyl tensor is zero except for the component C -uY uY (and components related to this by the symmetries of the tensor), and we find The non-vanishing components of the Ricci tensor are given by where a, b = ( u, Y ) and ¯ u = X, ¯ Y = v . Thus the metric g µν (Ψ) given by (2.14) is Ricci-flat and has self-dual Weyl tensor if the scalar field Ψ satisfies the gravitational Plebanski equation Defining the following gravitational bracket {{· , ·}} using (2.6) the Plebanski equation (2.20) can be written revealing the double copy relation [34] φ → Ψ , [ {· , ·} ] →{{· , ·}} compared with (2.7). Furthermore, we can consider the gravitational bracket acting on a pair of plane wave solutions in flat space and we find the following double copy structure where the colour structure constants in (2.11) have been replaced by additional kinematic ones. Alternatively, from (2.11) one may strip off the colour structure and isolate the kinematic algebra as the Poisson bracket of two plane waves As explained in [57], in the context of celestial holography the appearance of the 'left structure constants' X ( k 1 , k 2 ) in both YM and gravity implies the chirality of the operator product expansion in both cases. The second 'right structure constants' , f abc in the YM case and the second copy of X ( k 1 , k 2 ) in the gravity case, correspond to the structure constants of the OPEs. The soft expansion of the latter may be explored by noting that the null vector condition k 2 = 2( k u k v -k X k Y ) = 0 implies that we may set for some ρ . Thus we can write The soft limit of the momentum k then corresponds to ( k v , k x ) → 0 at fixed ρ . Expanding e ik · x in this limit gives where the 'soft mode generators' are given by e ab = ( ρY + v ) a ( ρu + X ) b . To make contact with the algebras appearing in celestial holography, we now need to take the collinear limit of the two momentum k 1 , k 2 appearing in the algebra (2.24). To do this we use the holomorphic collinear limit where ( ρ 1 -ρ 2 ) → 0. This makes k 1 and k 2 collinear since At leading order in the holomorphic collinear limit (corresponding to the first term in the OPE expansion in the celestial holography context) we may set ρ 1 = ρ 2 = ρ and substitute the expansion (2.27) into the kinematic algebra (2.24) to obtain Defining the conventional generators w p m = 1 2 e p -1+ m,p -1 -m we then find the wedge subalgebra of the w 1+ ∞ algebra The conditions that a and b are integers greater than or equal to zero translates to the conditions that p, m are half-integers and satisfy 1 -p ≤ m ≤ p -1 and p ≥ 1 (similarly for q, n ). This algebra has been studied in the celestial holography context [50,42,86] where it is generated by the commutation relations of operators inserting soft gravitons. As in the YM case, there is another gauge-fixing condition related to the above by interchanging u ↔ v and Y ↔ X , with (trivially equivalent) consequent equations. We reiterate that these different types of gauge will not be as trivially related once we consider self-dual backgrounds in the next section.", "pages": [ 5, 6, 7 ] }, { "title": "3 General self-dual YM backgrounds", "content": "Our first exploration of self-dual perturbations of self-dual backgrounds starts with YM backgrounds in flat space. To begin with we consider a background self-dual gauge field A ( χ ) in the gauge (2.4) Since the gauge field is linear in the scalar χ we can write a self-dual perturbation on this background as where A ( ψ ) is the perturbation. The total gauge field A ( χ + ψ ) must then satisfy the self-dual YM equation (2.5) where we have used (3.1) and defined a 'deformed' scalar Laplacian which is simply the scalar Laplacian in the background gauge field A ( χ ). The covariant derivative is given by 3 when acting on an adjoint valued field. Equation (3.3) is the analogue of (2.5) in a self-dual background YM field. We will see shortly that the discussion above can be double copied in two ways. First, we can just double copy the background gauge field to obtain equations of motion for self-dual YM on a self-dual gravitational background (Sec. 4.1). Second, we can double copy both the background and the perturbation, to obtain self-dual gravity perturbations on a self-dual background (Sec. 4.2). In either case, the scalar Laplacian (3.4) will double copy to a familiar object.", "pages": [ 7, 8 ] }, { "title": "4 General self-dual background spacetimes", "content": "We now turn to generalising the above results in section 2 valid for flat backgrounds to the case of self-dual background metrics. This leads us to the two possible double copies of the case considered in section 3 of self-dual YM fields on self-dual YM backgrounds, these are summarised in the diagram in (4.27). We consider self-dual metrics of the form for a scalar function Φ( u, v, X, Y ) satisfying the Plebanksi equation Pleb 0 (Φ) = 0. Given a co-vector k µ = ( k u , k v , k X , k Y ) we have where it proves useful to define The hatted momenta are those in the tangent space - if we write the vierbein satisfying e a µ e νa = g µν (Φ), with g µν (Φ) the metric in (4.1), then ˆ k a = e µ a k µ .", "pages": [ 8 ] }, { "title": "4.1 Self-dual Yang-Mills", "content": "With a gauge field A µ = ( A u , A v , A X , A Y ) on this spacetime we can choose the gauge-fixing condition n µ A µ = 0, with null vector n µ = (0 , 1 , 0 , 0), thus setting Now we require that the field strength F µν is self-dual, i.e. that the anti-self-dual components F -µν vanish. This imposes three independent conditions. Two of these are satisfied if we set for a scalar field φ ( u, v, X, Y ), and the final self-duality condition imposes the equation where □ Φ is the Laplacian in the metric (4.1). This gauge matches the choice made for the selfdual background metric, wherein the components of the metric g µν satisfy g vv = g vX = g XX = 0; we call this gauge the ' matched gauge '. Equation (4.7) is thus the generalisation of the self-dual Yang-Mills equation (2.5) to the background (4.1). If we define the Poisson bracket as in the flat space case then the Plebanski equation (4.7) can be written as with [ { φ, φ } ] defined in (2.8). The equation above can be viewed as the double copy of (3.3) where we only double copy the background gauge field. Explicitly, performing the double copy on only the background χ using [ { χ, ·} ] → {{ Φ , ·}} , the scalar Laplacian in gauge theory ˜ □ χ (3.4) becomes where in the last equality we combined -2 {{ Φ , ·}} with the flat scalar Laplacian to give us the curved Laplacian on the background Φ. As we have noted earlier, there is also a different gauge choice which reduces to the gaugefixing condition A u = 0 in the flat space case, and which has a quite different structure. We can find self-dual Yang-Mills fields in this gauge which satisfy a generalised Plebanski equation, for background metrics which are of course self-dual themselves, namely but also are of the Kerr-Schild form and so satisfy The above conditions imply that the flat Laplacian acting on Φ vanishes i.e. Φ uv = Φ XY . The self-dual gauge field in this case is given by where the previously defined ˆ k u and ˆ k Y are now regarded as differential operators defined by replacing the unhatted k 's in their expression by the corresponding derivatives. The perturbation field φ then satisfies a generalised Plebanski equation in the background Φ given by The gauge field above is not adapted to the background metric in the same fashion as the previous gauge, instead it features the non-trivial components of the background metric. Since in the flat space limit it is related to the previous gauge by the coordinate exchange u ↔ v and Y ↔ X we call it the ' flipped gauge '. The commutator term in (4.14) in this 'flipped gauge' reveals a different algebraic structure connected with the fact that one can define a curved space Poisson bracket for this spacetime [87,88]. We can define this by considering the expression ˆ k 1 u ˆ k 2 Y -ˆ k 2 u ˆ k 1 Y and as before replacing the unhatted k 's in this expression by coordinate derivatives with respect to the two functions in the Poisson bracket, i.e. or The Jacobi identity for the Poisson bracket { , } Φ is satisfied since the self-dual background Φ satisfies the Plebanski equation Pleb 0 (Φ) = 0. Furthermore, since the Kerr-Schild condition Φ 2 Xv -Φ XX Φ vv = 0 is satisfied the bracket { v, X } vanishes. The symplectic form connected with the Poisson bracket (4.15) is We observe that ω 2 = 0 and is closed, dω = 0 (c.f. [88]) when the Kerr-Schild condition and background Plebanski equation are satisfied. Using the notation the condition (4.14) on the field φ may then be written", "pages": [ 8, 9, 10 ] }, { "title": "4.2 Self-dual gravity", "content": "We now consider self-dual gravity perturbations on the background metric in (4.1). That is we simply consider the shifted metric g µν (Φ+Ψ) given by (4.1) with Φ replaced by Φ+Ψ. We take the metric g µν (Φ) to be the background self-dual spacetime, with Pleb 0 (Φ) = 0. This setup corresponds to the so called 'matched gauge' for the gravity perturbation. We can then define the gravitational Plebanski function in the background metric g µν (Φ) by with □ Φ the scalar Laplacian in the background metric. Once again this can be written in terms of the double bracket notation (2.21) illustrating the double copy structure compared with eqn. (4.9). Alternatively, (4.21) can be viewed as the double copy of (3.3), where we double copy both the YM background and the perturbation. Now one can check that the Plebanski equation satisfies the following identity This immediately gives the gravitational Plebanski equation in the background metric as simply This follows since the identity (4.22) shows that if Φ leads to a self-dual metric then Φ+Ψ does as well if the Plebanski equation for Ψ in a Φ metric background (4.23) is satisfied. The above conclusions can be confirmed explicitly. The relevant non-trivial component of the anti-self-dual part of the Weyl tensor for the metric g µν (Φ + Ψ) is given by where we have used the self-duality of the background metric, with Pleb 0 (Φ) = 0, and imposed the condition (4.23). It is also immediate that the variations of the Plebanski function (4.20) are related to variations of the flat Plebanski function - if we define the variation then as differential operators as expected. A similar argument, based on (2.19), shows Ricci-flatness of the shifted metric. In summary, we have shown the following commuting triangle of double copy relations for equations of motion in the matched gauge: The diagonal arrow above is just the usual self-dual flat space double copy applied to the sum of the background and perturbation fields χ + ψ in (3.2). The double copy properties of backgrounds and perturbations have been studied beyond the self dual context in [60]. We can also consider the 'flipped gauge' for which the natural double copy of the bracket in (4.18) replaces the YM commutator with the Poisson brackets { , } Φ of (4.15) 4 These double brackets have a related curved space Plebanski equation of the form which may be regarded as the double copy of (4.19). We discuss these brackets further in the examples below. It would be interesting to know if these equations are related to the conditions required for the self-duality of the curvature of metrics on self-dual backgrounds. One might also study self-dual backgrounds satisfying the Kerr-Schild condition Φ 2 Xv = Φ XX Φ vv more generally. Whilst we have not found answers to these questions in the general case, the study of interesting examples reveals more structure, as we will see in the following.", "pages": [ 10, 11, 12 ] }, { "title": "5 The self-dual plane wave spacetime", "content": "Plane wave backgrounds have been the object of some interest recently in the area of amplitudes, kinematic algebras and the double copy (see, for example, [58, 59, 83, 84, 89-91] and references therein). Here we study the self-dual plane wave metric where F ( v ) is a function related to the wave profile. This metric is an example of the general form (4.1) considered earlier and is also Kerr-Schild, we simply set Φ = Φ( v ) with Φ vv = 2 F ( v ). The self-dual plane wave metric is Ricci-flat and has self-dual Weyl tensor; the only non-vanishing components of the self-dual part of the Weyl tensor being C vY vY = -2 F '' [ v ] and those related to this by the symmetries of this tensor.", "pages": [ 12 ] }, { "title": "5.1 Self-dual Yang-Mills", "content": "A self-dual gauge field in the 'matched gauge' on this spacetime is given by where, in order to solve the self-duality conditions, the scalar field φ ( u, v, X, Y ) must satisfy the plane wave background Plebanski equation with □ PW the Laplacian in the metric (5.1). Using the Poisson bracket { f, g } = f v g X -f X g v which is the same as the flat space case, we can write (5.3) as where the double bracket notation (2.8) is defined as usual. There is also the 'flipped' self-dual gauge field solution in this background, from (4.13) which can be used to elucidate the algebraic structure of self-dual perturbations on the self-dual plane wave background. We find where the field φ satisfies This leads us to the modified Poisson bracket in the plane wave background and the re-writing of (5.6) as We are now tasked with finding the analogue of plane wave solutions to the wave equation in flat space, but for solutions to the wave equation in the background (5.1). Such solutions then act as generators of our kinematic Poisson algebra. We begin by constructing a null vector in flat space k µ satisfying k u k v -k X k Y = 0 so that (as before) k u = ρk X , k Y = ρk v for some ρ . Then for the function G ( v ) given by the indefinite integral of F ( v ), ie G ' = F , we may define the quantity Then one can show that the vector K µ = ∇ µ Q k is null, K µ K µ = 0, divergence free, ∇ µ K µ = 0 (which is just the wave equation on Q k ), and geodesic, K ν ∇ ν K µ = 0, where ∇ µ is the covariant derivative in the plane wave metric. One consequence is that any function of Q k is annihilated by the Laplacian, in particular Whence the function e iQ k ( u,v,X,Y ) satisfies the wave equation in the plane wave background and furthermore reduces to the usual plane wave e ik · x in the flat space limit. The Poisson bracket of two of these solutions is leading to a modification of the structure constants defining the kinematic algebra compared to the flat space case. This modification is however sub-leading in the holomorphic collinear limit so we expect it to not alter the w -algebra, which we confirm in the next section. This result, and hence also the w -algebra in (5.20), also holds if one uses the flipped gauge Poisson bracket (5.7), although in that case it is more natural to write the function (5.9) in terms of k u and k Y as follows where ˜ ρ := 1 /ρ . The flipped Poisson bracket of two plane waves is then", "pages": [ 12, 13, 14 ] }, { "title": "5.2 Self-dual gravity", "content": "The gravitational analogue of the discussion above in the 'matched gauge' is based on the metric This metric has vanishing Ricci tensor and self-dual Weyl tensor if the following Plebanski equation is satisfied: where □ PW is the Laplacian in the self-dual background. Employing the double bracket notation (2.21) as usual we have revealing the double copy structure compared with (5.3). If we apply the double bracket to two of the 'plane waves' (5.9) we find which is not just the simple square of the relation (5.11). Despite this, we can still derive a w -algebra as follows. Similarly to the flat space case, we may expand the above solutions to the wave equation in powers of soft momenta variables k v , k X to find where we have defined e ab = ( ρY + v ) a ( ρu + X + 1 ρ G ( v )) b in the self-dual plane wave background. Defining the modified w generators in analogy with the flat space case. We recover the standard w 1+ ∞ -algebra for these modified generators, working to leading order in the holomorphic collinear limit We may also consider the 'flipped gauge' with its modified Poisson bracket (5.7) which satisfies a double copy relation analogous to (2.23) acting on two solutions e iQ k ( u,v,X,Y ) . First we define a modified double bracket then we find the expected double copy of (5.13), that is Interestingly, in contrast to the matched double bracket (5.17) of plane waves, the above does exhibit a simple squaring relation when compared to the single bracket (5.13). As mentioned before, we can also define analogous soft generators ˜ w p m in the flipped gauge, now as coefficients of k a u and k b Y . One can then show that these generators also satisfy the w 1+ ∞ algebra (5.20), but now with the bracket (5.7).", "pages": [ 14, 15 ] }, { "title": "6 The Eguchi-Hanson spacetime", "content": "We now move on to consider a more complicated example, the Eguchi-Hanson space-time. This is self-dual, and in the form (4.11) has the scalar function with m a constant, satisfying the Plebanski equation in flat space The full metric is then and satisfies the Kerr-Schild condition. We now repeat the methods laid out for the general case and the plane wave example but now with the function Φ EH . We will encounter a much richer algebraic structure than was found in the self-dual plane wave background, reproducing in spacetime some of the results recently described via twistor space in [85].", "pages": [ 15 ] }, { "title": "6.1 Self-dual Yang-Mills", "content": "Consider firstly self-dual Yang-Mills in an Eguchi-Hanson background. From the results earlier, a gauge field A µ in the 'matched gauge' A v = 0 has self-dual field strength if in addition A X = 0 , A u = φ X and A Y = φ v , with φ satisfying the Plebanski equation in the EH background where □ EH is the Laplacian in the metric (6.3). In the case at hand, the EH Plebanski equation can be written in terms of the flat space Poisson bracket (2.6) as The alternative 'flipped gauge' (4.13) in the Eguchi-Hanson case comes from the null vector m µ = (1 , 0 , 0 , v/X ) and gauge-fixing condition m µ A µ = 0 and sets This gauge field has self-dual field strength if the scalar field φ satisfies Using the notation equation (6.7) may be written (c.f. (4.9)) The deformed Poisson bracket (4.15) in the Eguchi-Hanson metric is then and we note that the terms quadratic in m in the above Poisson bracket in fact drop out. To find the equivalent of plane wave solutions in the EH background we introduce a null co-vector k µ = ( k u , k v , k X , k Y ) whose components satisfy k u k v = k X k Y so that as before we may write for some parameter ρ . As was the case for the self-dual plane wave, we look for solutions to the EH wave equation which are of an exponential form and return the usual e ik · x plane wave in the flat space limit m → 0. Following [85], we define the function Then the vector K µ = ∇ µ R k is null, K µ K µ = 0, divergence free, ∇ µ K µ = 0 (which is just the wave equation on R k ), and geodesic, K ν ∇ ν K µ = 0, where ∇ µ is the covariant derivative in the EH metric. One consequence is that any function of R k is annihilated by the Laplacian, in particular √ where e i √ R k ( u,v,X,Y ) gives the standard plane wave e ik · x in the flat space limit m → 0. Note the qualitative difference between the Eguchi-Hanson function R k , which is quadratic in the null momenta k µ , versus Q k in the self-dual plane wave background which is linear in k µ . √ We can now perform the Poisson bracket of two of the solutions e i R k to the wave equation with momenta k 1 , k 2 , using the form of R k on the second line of (6.12). We work in the holomorphic collinear limit ρ 1 = ρ 2 = ρ which is all that is needed to recover a w -algebra. This gives √ √ √ √ where here and dot products k · x here mean ( ρY + v ) k v +( ρu + X ) k X . The final factor may be compared to the right-hand side of (6.12). Eqn (6.14) may be viewed as the Eguchi-Hanson background version of the expression in equation (2.11). We note that the kinematic algebra has modified kinematic structure 'constants' compared to the flat-space case and the modification survives in the holomorphic collinear limit so we expect the w -algebra to also be modified. As in the plane wave case, the Poisson bracket relation (6.14) also holds if we use the Eguchi-Hanson flipped bracket (6.10), up to an overall factor which also appeared in (5.13).", "pages": [ 15, 16, 17 ] }, { "title": "6.2 Self-dual gravity", "content": "For the case of self-dual gravity, a perturbation of the EH metric in the matched gauge is given by g µν (Φ EH + Ψ) = g µν (Φ EH ) + g µν (Ψ) has vanishing anti-self-dual components of the Weyl tensor except for Thus, the perturbed EH metric has self-dual Weyl tensor if the EH Plebanski equation is satisfied. The EH Plebanski equation for self-dual gravity in this case is given by using the double bracket (2.21). Similarly for the Ricci tensor one finds that its components vanish except for R EH ab with a, b ∈ ( u, Y ) and for these components where ¯ u = X, ¯ Y = v . The non-trivial form of the single bracket (6.14) suggests that the double copy, realised by using a double bracket, may involve more than just the square of X ( k 1 , k 2 ). This proves to be the case - the double brackets of two plane wave solutions e i √ R k in the EH background in the holomorphic collinear limit give a double copy-type formula (c.f (2.23) in the flat space case) where X EH ( k 1 , k 2 ) is given in (6.15) and the terms indicated by dots are more complicated expressions which multiply ( R 1 ) -1 / 2 , ( R 2 ) -1 / 2 and ( R 1 R 2 ) -1 / 2 and are of order m,m 2 or m 3 and hence vanish in the flat space limit m → 0. These results suggest that in general the double copy and related kinematic algebra on curved space backgrounds are not just given by a simple squaring operation of the relevant curved space term, as seen in the first term on the right-hand side of eqn. (6.19), but can involve other curvature corrections. We now consider the soft expansion of the solution e i √ R k ( u,v,X,Y ) in powers of the soft momentum variables k Y , k u and once again work in the holomorphic collinear limit where ρ 1 = ρ 2 = ρ . We define functions X g , Y g , Z g which give the coefficients of k 2 v , k 2 X and k v k X in the function R k which satisfy and the discriminant constraint The quantities X g , Y g , Z g correspond to the X,Y,Z of [85]. The parameter c 2 ( λ ) in that reference is related to ours by c 2 ( λ ) = m 2 ρ 2 2 . One can then expand the 'plane wave' e i √ R k in powers of the variables k v , k X 5 and the Poisson bracket of the coefficients in this expansion generates a w -type algebra. Due to the constraint (6.22) one can define a new basis of generators V 2 p, 2 q := X p g Y q g , V 2 p +1 , 2 q +1 := X p g Y q g Z g , and the Poisson brackets of these generates the underlying algebra The full celestial chiral algebra of self-dual gravity on an Eguchi-Hanson background can then be written in terms of sums of these generators (see [85]). We can also consider the double brackets of the flipped gauge (4.28) in the EH background which are given by and using these in the holomorphic collinear limit we find a double copy-type formula like (6.19) with the same leading term, but with different sub-leading terms. As in the plane wave case, we could expand the solution (6.12) in terms of k 2 u , k 2 Y and k u k Y instead to define analogous soft generators ˜ X g , ˜ Y g and ˜ Z g . These then satisfy the same algebra as (6.23) but with the flipped Poisson bracket (6.10).", "pages": [ 17, 18, 19 ] }, { "title": "7 Conclusions", "content": "We have studied the self-duality of gauge and gravitational fields on the self-dual background spacetimes defined by solutions of Plebanski's second equation. In light-cone gauges we showed that the conditions for self-duality could be reduced to second order scalar equations generalising the flat space equations. We found two classes of general solutions. One, which we called a 'matched' gauge, was a direct generalisation of the flat space solutions to the curved self-dual backgrounds under consideration. The other involves a Kerr-Schild condition on the gravitational background, which we called the 'flipped' gauge, and can be seen as the curved space versions of 'flipped' flat space solutions. We discussed the double copy and kinematic algebra in these two cases. Finally, we studied two examples in more detail - the self-dual plane wave spacetime and the Eguchi-Hanson (EH) metric - connecting with some recent results from [84] and [85], and noting that in the EH background the kinematic algebra squaring relations are modified by curvature terms. There are a number of avenues of research which follow from this. It would be interesting to explore more examples in detail, and investigate perturbative solutions to the equations where direct solutions prove difficult. Gravitational analogues of the 'flipped' gauge self-dual YM solution, eqn. (4.13) could be studied further, in general and in particular examples. Plebanskitype conditions of the generic form □ φ -{{ φ, φ }} = 0 for the different double brackets given above would be expected to feature. In radiative spacetimes this should connect with the very recent analysis of self-dual deformations in [84], which relates these to twistor sigma models and MHV generating functionals. It would also be interesting to explore applications to known deformations of the Plebanski equations such as those involving Moyal brackets (c.f. [12, 85] and references therein). The application of the formalism used recently for self-dual YM in [92] could also be explored in self-dual backgrounds. Acknowledgements: We would like to thank Ricardo Monteiro, Chris White and Sam Wikeley for helpful comments. This work was supported by the Science and Technology Facilities Council (STFC) Consolidated Grants ST/P000754/1 'String theory, gauge theory and duality' and ST/T000686/1 'Amplitudes, strings and duality'. The work of GRB and JG is supported by STFC quota studentships.", "pages": [ 19 ] }, { "title": "References", "content": "JHEP 08 (2022) 083, arXiv:2204.09313 [hep-th] . theories and cosmological scattering equations,' JHEP 08 (2022) 054, arXiv:2204.08931 [hep-th] .", "pages": [ 21, 24 ] } ]

[ { "title": "Imprints of Einstein-Maxwell dilaton-axion gravity in the observed shadows of Sgr A* and M87*", "content": "Siddharth Kumar Sahoo ∗ , Neeraj Yadav † and Indrani Banerjee ‡ Department of Physics and Astronomy, National Institute of Technology, Rourkela, Odisha-769008 India", "pages": [ 1 ] }, { "title": "Abstract", "content": "Einstein-Maxwell dilaton-axion (EMDA) gravity provides a simple framework to investigate the signatures of string theory. The axion and the dilaton fields arising in EMDA gravity have important implications in inflationary cosmology and in addressing the late time acceleration of the universe. It is therefore instructive to explore the implications of such a model in explaining the astrophysical observations. In this work we explore the role of EMDA gravity in explaining the observed shadows of black holes (M87* and Sgr A*) released by the Event Horizon Telescope (EHT) collaboration. The Kerr-Sen metric represents the exact, stationary and axisymmetric black hole solution of EMDA gravity. Such a black hole is characterized by the angular momentum a acquired from the axionic field and the dilatonic charge r 2 arising from string compactifications. We study the role of spin and the dilaton charge in modifying the shape and size of the black hole shadow. We note that black holes with larger dilaton charge cast a smaller shadow. We investigate the consequences of such a result in addressing the EHT observations of M87* and Sgr A*. Our analysis reveals that the shadow of M87* exhibits a preference towards the Kerr scenario. However, when 10% offset in the shadow diameter is considered, 0 . 1 ≲ r 2 ≲ 0 . 3 is observationally favored within 1σ . The shadow of Sgr A* on the other hand shows a preference towards the Kerr-Sen scenario since the central value of its shadow can be better explained by a non-zero dilaton charge 0 . 1 ≲ r 2 ≲ 0 . 4. However, when the 1σ interval is considered the Kerr scenario is included. We discuss the implications of our results.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "General relativity (GR), the successor of Newtonian theory of gravity has radically changed our understanding pertaining to gravitational interaction. In GR the mass of a body produces curvature in the spacetime, which changes the metric of the spacetime from Minkowski metric [1]. The particles in curved spacetime move along geodesics which are obtained by solving the geodesic equation associated with the metric describing the spacetime. The metric itself is obtained by solving the Einstein field equations and depends crucially on the matter distribution. GR has many interesting predictions [1], namely, the perihelion precession of mercury, the bending of light, the gravitational redshift of radiation from distant stars, to name a few, which have been experimentally verified [2,3]. The detection of gravitational waves by the LIGO-VIRGO collaboration [4-6] and the release of black hole images of M87* and Sgr A* by the Event Horizon Telescope collaboration [7-18] have further demonstrated GR as a successful theory of gravity. Despite being a very successful theory, GR also has certain limitations. The theory allows formation of singularities [19-21] namely, the black hole and the big bang singularities, where the theory loses its predictive power [19,20,22,23]. This indicates that GR is not a complete theory of gravity [2] and that at very small length scales it must receive considerable corrections from a more complete theory that incorporates its quantum nature [24-27] [28]. In the observational front GR falls short in explaining the nature of dark matter [29-31] and dark energy [32-34], which are invoked to explain the flat rotation curves of galaxies and the accelerated expansion of the universe, respectively. These inadequacies have lead to the development of many alternate theories of gravity which address the limitations of GR [35-41] and deviate from GR in the strong field regime. Therefore, to test the effectiveness of alternative theories of gravity, it is necessary to study how effectively they explain observations related to strong field tests of gravity [42,43]. The alternatives to GR include higher curvature gravity, e.g., f ( R ) gravity, [44-46] and Lanczos Lovelock models [47-50], extra dimensional models [51-57] and scalar-tensor/scalar-vector-tensor theories of gravity [58-62]. Many of these models are string inspired which provides a framework for force unification [63-66] . In this work we intend to discern the signatures of the string inspired model, namely, the Einstein-Maxwell dilaton-axion (EMDA) gravity, from observations related to black hole shadows. EMDA, a scalar-vector-tensor theory of gravity, arises in the low energy effective action of superstring theories [67] on compactifying the ten dimensional heterotic string theory on a six dimensional torus. In the EMDA theory the scalar field dilaton and the pseudo scalar axion are coupled to the Maxwell field and the metric. The axion and dilaton fields which originate from string compactifications have interesting implications in inflationary cosmology and the late-time acceleration of the universe [68, 69]. It is, therefore, important to explore the footprints of EMDA gravity in astrophysical observations which is the goal of the present work. In particular, we aim to decipher the imprints of the dilaton charge in black holes from observations related to black hole shadows. Black holes(BH) are compact objects with extremely strong gravity. Among the various systems that possess strong gravitational field, black holes are the most interesting and the simplest ones. Different black hole solutions have been constructed in the context of string inspired low-energy effective theories [70-73]. Interestingly, the charge neutral axisymmetric black hole solution in string theory resembles the Kerr solution in GR [74,75]. In EMDA gravity the stationary and axisymmetric black hole solution is represented by the Kerr-Sen metric which is similar to the Kerr-Newman spacetime in GR. Despite the similarities, the intrinsic geometry of the two black holes vary considerably which have been explored extensively in the past [76-79]. Investigating the observational signatures of the Kerr-Sen black hole is important as it can provide an indirect testbed for string theory. Astrophysical signatures of Kerr-Sen black hole have been studied previously in the context of photon motion, null geodesics, strong gravitational lensing and black hole shadows [76,80-84]. In [84], the authors have worked out the shadow of the Kerr-Sen black hole but there they have not compared their result with the observed shadows and hence no constrain on the dilaton charge was reported. Recently, the shadow of dyonic Kerr-Sen black holes have been studied [85] and an upper bound on the magnetic monopole charge of Sgr A* has been mentioned. We explore the role of the dilaton charge in modifying the structure of the black hole shadow from that of the Kerr scenario. We compare the theoretically derived black hole shadow (which depends on the dilaton charge and the spin) with that of the observed images of M87* and Sgr A* (released by the EHT collaboration). Such a study enables us to establish constrains on the dilaton parameter of the Kerr-Sen black hole and allows us to comment on the possible feasibility of string theory in explaining the observed black hole shadows. The structure of the paper is as follows: In Section 2 we give a brief overview of the Kerr-Sen BH. In Section 3 we derive the shadow outline of the Kerr-Sen BH. In Section 4 we discuss our results related to constrains on the dilaton parameter r 2 from EHT observations of M87* and SgrA*. We give summary of our results and concluding remarks in Section Section 5. In our paper we have chosen the metric signature ( -, + , + , +) and used geometrized units G = c = 1.", "pages": [ 1, 2, 3 ] }, { "title": "2 Black hole in Einstein-Maxwell dilaton axion gravity", "content": "The Einstein-Maxwell dilaton-axion (EMDA) gravity [67, 86] results from the compactification of ten dimensional heterotic string theory on a six dimensional torus T 6 . In EMDA gravity, N = 4, d = 4 supergravity is coupled to N = 4 super Yang-Mills theory which can be suitably truncated to a pure supergravity theory exhibiting S and T dualities. The bosonic sector of this supergravity theory when coupled to the U (1) gauge field is known as the Einstein-Maxwell dilaton-axion (EMDA) gravity [86] which provides a simple framework to study classical solutions. The four dimensional effective action for EMDA gravity consists of a generalization of the Einstein-Maxwell action such that the metric g µν is coupled to the dilaton field χ , the U (1) gauge field A µ and the Kalb-Ramond field strength tensor H αβγ . The action corresponding to EMDA gravity assumes the form, In Eq. (1) g is the determinant and R the Ricci scalar associated with the 4-dimensional metric g µν , χ represents the dilatonic field, F µν = ∇ µ A ν -∇ ν A µ is Maxwell field strength tensor and H ρσδ is given by where A µ is the vector potential and B µν is the second rank antisymmetric tensor field called the KalbRamond field while its cyclic permutation with A µ denotes the Chern-Simons term. In four dimensions the Kalb-Ramond field strength tensor H ρσδ can be written in terms of the pseudo-scalar axion field ψ , such that, The action in Eq. (1) written in terms of the axion field assumes the form, Variation of the action with respect to the dilaton, axion and Maxwell fields give their corresponding equations of motion. The equation of motion associated with the axion field is given by, while that of the dilaton field assumes the form, The Maxwell's equations with couplings to the dilaton and the axion fields are given by, Solving the aforesaid equations one obtains solutions for the dilaton, axion and the Maxwell field, respectively [67, 86, 87], where M is the mass, a is the spin and q is the charge of the black hole. In Eq. (9) r 2 is associated with the dilaton parameter and is given by r 2 = q 2 M e 2 χ 0 where χ 0 represents the asymptotic value of the dilatonic field. The dilaton parameter also depends on the electric charge of the black hole, which owes its origin from the axion-photon coupling and not from the in-falling charged particles. This is because the axion and dilaton field strengths vanish if the electric charge q = 0 (see Eq. (10) and Eq. (9)). It is further important to note that the axion field renders a non-zero spin to the black hole since the field strength corresponding to the axion field vanishes if the black hole is non-rotating (Eq. (10)). Varying the action with respect to the metric gives the Einstein field equations, where, G µν is the Einstein tensor and T µν the energy-momentum tensor which is given by, The Kerr-Sen metric [67] is obtained when one looks for the stationary and axisymmetric solution of the aforesaid Einstein's equations [88-90]. In Boyer-Lindquist coordinates the Kerr-Sen metric takes the form, where, The non-rotating counterpart of the Kerr-Sen metric corresponds to a pure dilaton black hole characterized by its mass, electric charge and asymptotic value of the dilaton field [71,91]. In order to obtain the event horizon r h of the Kerr-Sen black hole one solves for g rr = ∆ = 0, which gives, Since r 2 = q 2 M e 2 χ 0 > 0, the presence of real, positive event horizon requires 0 ≤ r 2 M ≤ 2 (see Eq. (15)). Since we are interested in black hole solutions we will be interested in this regime of r 2 in this work.", "pages": [ 3, 4, 5 ] }, { "title": "3 Shadow of Kerr-Sen black holes", "content": "When photons from a distant astrophysical object or the accretion disk surrounding the black hole come close to the black hole horizon, a few of them get trapped inside the horizon while others escape to infinity. Since some photons get trapped inside the horizon, the observer sees a dark patch in the image of the black hole, known as the black hole shadow. The outline of the black hole shadow is associated with the motion of photons near the event horizon and hence we expect to extract valuable information regarding the nature of strong gravity from the shape and size of the black hole shadow [92-96]. It may be noted that the shape of the shadow depends on the background spacetime while the size of the shadow is related to the mass and distance as well as the background metric. Thus, a non-rotating black hole gives rise to a circular shadow in which case the size is the only parameter based on which one can study deviations from the Schwarzschild geometry in GR [97,98]. Rotating black holes cast a non-circular shadow provided the black hole is viewed at a high inclination angle. In such a scenario, both the size and the shape of the shadow can be used to study deviation from GR [83,92-96,99,100]. In this section we investigate the motion of photons in the Kerr-Sen background. This enables us to compute the outline of the black hole shadow in EMDA gravity which in turn can be compared with the Kerr scenario in general relativity. For a stationary, axisymmetric metric, the Lagrangian L for the motion of any test particle is given by, The action S representing the motion of test particles satisfying the Hamilton-Jacobi equation is given by, where H is the Hamiltonian, λ is a curve parameter, and p µ , the conjugate momentum corresponding to the coordinate x µ is The Hamiltonian is given by, where k denotes the rest mass of the test particle which is zero for photons. Since the Kerr-Sen metric does not explicitly depend on t and ϕ , the first term in the the Euler-Lagrange equation is zero. Therefore, the energy E and the angular momentum L z of the photon are conserved. Using Eq. (18) these constants are given by We further note from Eq. (18) that, Integrating Eq. (22) the action S can be written as It turns out that ¯ S ( r, θ ) can be separated in r and θ giving us From Eq. (19) we have g µν p µ p ν = 0 giving us Using Eq. (22), Eq. (23) and Eq. (25), Eq. (26) can be written as which on substitution of the metric components g µν (see Eq. (14)) gives The above equation can be separated in r and θ such that, where Q is called the Carter's constant. From Eq. (29) the angular part is given by where The radial equation is given by We also note that where while Therefore the first order geodesic equations for r and θ can be respectively written as, where ξ = L z /E and η = Q/E 2 represent the two impact parameters. While ξ denotes the distance from the axis of rotation, η signifies the distance from the equatorial plane. The first order geodesic equations for t and ϕ are obtained from Eq. (22) and are given by,", "pages": [ 5, 6, 7 ] }, { "title": "· Analysis of the θ equation", "content": "In this section we simplify the angular equation of motion by defining a new variable u = cos θ . Then the angular equation Eq. (37) is given by, Note that the left hand side of Eq. (40) is positive which implies that the right hand side also needs to be positive. Since G (1) = -ξ 2 is negative, the photon cannot access θ = 0. To obtain the maximum accessible value of θ denoted by θ max we solve for G ( u ) = 0 which gives, If η > 0 one can only consider the positive root of Eq. (41) since the left hand side of Eq. (41) is positive. Such orbits cross the equatorial plane reaching a maximum height of θ max given by the solution of Eq. (41). For negative η , we define η = -| η | such that Eq. (41) can be rewritten as From Eq. (42) it is easy to note that for its right hand side to be positive, which is the condition to be satisfied by the impact parameters. Finally, we note that η = 0 has two solutions, namely, If ξ 2 > a 2 only u 2 1 is valid else both u 2 1 and u 2 2 are valid solutions.", "pages": [ 7, 8 ] }, { "title": "· Analysis of the radial equation", "content": "In this section we consider the geodesic equation associated with the radial coordinate given by Eq. (36), We will be interested in spherical photon orbits of constant radius which yields ˜ V ( r ) = ˜ V ' ( r ) = 0. Thus, we have to solve the following two equations for η and ξ : From Eq. (46) we obtain two classes of solutions for η and ξ . 1. The first solution has η < 0 which requires a 2 + | η | -ξ 2 to be positive (see previous discussion). Substituting η and ξ from the first solution we note that a 2 + | η | -ξ 2 = -2 r ( r + r 2 ) < 0 which makes the first solution unphysical and hence unacceptable. In the case of the second solution η may assume any sign depending on the value of r and it can be shown that the suitable conditions as discussed earlier are satisfied. We will therefore work with the second solution. 2.", "pages": [ 8 ] }, { "title": "3.1 Equation of the shadow outline", "content": "In this section we use the derived impact parameters from the last section to evaluate the celestial coordinates x and y of the black hole shadow as viewed by an observer at infinity. The position of the distant observer is taken to be ( r 0 , θ 0 ) where we take r 0 →∞ and θ 0 is the inclination angle of the observer. In order to obtain the outline of the black hole shadow in the observer's sky we consider the projection of the photon sphere onto the image plane. In order to obtain the celestial coordinates we write the metric in terms of Bardeen tetrads [98,101,102], which are associated with observers to whom the black hole appears static. From the tetrads we can compute the components of four momentum p ( i ) = e j ( i ) p j of a locally inertial observer. The contravariant components of the four momentum p ( k ) = η ( k )( l ) p ( l ) of the locally inertial observer are given as, Distant observer located at ( r 0 , θ 0 ) will find the local apparent velocities of a photon to be v ( θ ) = p ( θ ) p ( r ) and v ( ϕ ) = p ( ϕ ) p ( r ) in which case the apparent perpendicular distance from the axis of rotation and the equatorial plane are respectively given by d ϕ = r 0 v ( ϕ ) and d θ = r 0 v ( θ ) . These are associated with the celestial coordinates x and y such that Fig. 1 illustrates the variation of the shape and size of the black hole shadow with the dilaton parameter r 2 , inclination angle θ , and the black hole spin a . The figure reveals that the shadow size decreases with an increase in the magnitude of the dilaton parameter r 2 . We further note that when a and θ are enhanced the shadow becomes increasingly non-circular [103-106].", "pages": [ 9 ] }, { "title": "4 Comparison with observations and constrains on the dilaton parameter", "content": "In this section we aim to constrain the Kerr-Sen parameter r 2 using observations of M87* and SgrA* by the EHT collaboration. In order to obtain constraints on the Kerr-Sen parameter r 2 we theoretically calculate the observables, namely, the angular diameter ∆ θ , the axis ratio ∆ A and the deviation from circularity ∆ C [93] for the black hole shadow, assuming the spacetime around the black hole to be described by the Kerr-Sen metric. In our approach we use measurements for distance D , mass M and the inclination angle θ 0 (angle between the line of sight and the jet axis) of the black hole determined from previous observations. The observables related to black hole shadow which will be used to find best estimate on the Kerr-Sen parameter r 2 are discussed below: Angular diameter of shadow ∆ θ : It is a measure of the angular width of the shadow. If the maximum width of the shadow is ∆ y (also called the major axis length), mass of the black hole is M and distance of the black hole from the observer is D then the angular diameter of the shadow ∆ θ [93] is defined as: The value of ∆ y is calculated from the equation of the shadow which contains the impact parameters ξ and η . The impact parameters in turn depend on the metric components r 2 , a and the inclination angle θ 0 . Therefore, the angular diameter also depends on the three aforesaid parameters and thus r 2 can be constrained using experimental observations of ∆ θ for predetermined inclination angle θ 0 . Axis ratio ∆ A of the black hole shadow: As the shadow of the black hole is in general not circular the major axis ∆ y and the minor axis ∆ x may not be equal. From Fig. 2, the axis ratio ∆ A is defined as [93]: where the minor axis ∆ x is also calculated from the equation of the shadow and hence, ∆ A also depends on r 2 , a and θ 0 . Deviation from Circularity ∆ C : Deviation from circularity ∆ C measures the amount of deviation from the circular shape of the shadow [93]. It is defined as follows: In the above expression R avg is the average radius of the shadow. l ( ϕ ) is the length of the line joining the point ( x ( ϕ ) , y ( ϕ )) on the shadow and the geometric centre ( x c , 0) (see Fig. 2). It must be noted that due to reflection symmetry of the Kerr-Sen metric, the shape of the shadow is symmetric about the x -axis, hence the y coordinate of the geometric center is 0. The x coordinate of the geometric centre is calculated using the formula, EHT observations of M87*: The EHT collaboration measured the angular diameter ∆ θ , the axis ratio ∆ A and the deviation from circularity ∆ C for the image of M87*, the supermassive black hole candidate at the center of the galaxy M87 [7-9]. The values reported are given below: In order to determine the observationally favored Kerr-Sen parameter r 2 , we need to theoretically derive the above three observables as functions of the metric parameters r 2 and a . As evident from Eq. (53) a theoretical computation of the angular diameter ∆ θ requires independent measurements of the black hole mass, distance and inclination (required to derive ∆ y ). We use previously estimated masses and distance of this source to compute the theoretical angular diameter. The distance of M87* as reported from stellar population measurements turns out to be D = (16 . 8 ± 0 . 8) Mpc [107-109]. The angle of inclination which is the angle between the line of sight and the jet axis (the jet axis is believed to coincide with the spin axis of the black hole) is 17 · [110]. The mass of M87* has been measured using different methods. The mass measurement by modelling surface brightness and dispersion in stellar velocity was found to be M = 6 . 2 +1 . 1 -0 . 6 × 10 9 M ⊙ [8, 111, 112]. Mass measurements from kinematic study of gas disk gives M = 3 . 5 +0 . 9 -0 . 3 × 10 9 M ⊙ [8, 113]. Mass measured from the image of M87* by the EHT collaboration assuming general relativity turns out to be M = (6 . 5 ± 0 . 7) × 10 9 M ⊙ [7-9]. EHT observations of SgrA*: In May 2022 the EHT collaboration released the image of the black hole SgrA* present at the galactic center of the Milky Way galaxy. The angular diameter of the image is found to be ∆ θ = (51 . 8 ± 2 . 3) µas [13-18]. The angular diameter of the shadow is ∆ θ = (48 . 7 ± 7) µas [13-18]. The mass and distance of SgrA* reported by the Keck collaboration keeping the redshift parameter free, are M = (3 . 975 ± 0 . 058 ± 0 . 026) × 10 6 M ⊙ [114] and D = (7959 ± 59 ± 32)pc [114] respectively. Fixing the value of redshift parameter to unity the mass and distance reported by the Keck team are M = (3 . 951 ± 0 . 047) × 10 6 M ⊙ and D = (7935 ± 50)pc. The mass and distance of Sgr A* reported by the GRAVITY collaboration are M = (4 . 261 ± 0 . 012) × 10 6 M ⊙ and D = (8246 . 7 ± 9 . 3)pc [115, 116] respectively. When systematics due to optical aberrations are taken into account the GRAVITY collaboration constrains the mass and distance of Sgr A* to M = 4 . 297 ± 0 . 012 ± 0 . 040 × 10 6 M ⊙ and D = 8277 ± 9 ± 33 pc respectively. Apart from mass and distance we also need to provide independent measurements of the inclination angle to establish observational constrains on r 2 . From [117] we take θ ≃ 134 · (or equivalently 46 · ). When models based on extensive numerical simulations are compared with the the observed image of Sgr A*, one concludes that the inclination angle of the source is θ < 50 · . The estimates for axes ratio ∆ A and the deviation from circularity ∆ C for image of SgrA* by EHT collaboration are yet to be released, hence, for SgrA* the observable ∆ θ will only be used for estimating r 2 . To constrain the Kerr-Sen/dilaton parameter r 2 using EHT observations, we proceed with the following approach: Constrains on the dilaton parameter r 2 from EHT observations of M87* : Here we discuss the observationally favored magnitude of the dilaton parameter r 2 derived from the shadow of M87* released by the Event Horizon Telescope collaboration in April 2019. In order to get an understanding of the observationally preferred value of r 2 we theoretically compute the observables, namely, ∆ θ (angular diameter), ∆ A (axis ratio) and ∆ C (deviation from circularity) related to the black hole shadow, which have been discussed towards the beginning of this section. It is important to recall that these observables depend on the metric parameters r 2 , a and the inclination angle θ 0 . In addition, the theoretically derived ∆ θ requires independent estimates of the mass M and the distance D of the black hole (see Eq. (53)). The inclination angle is assumed to be 17 · and the distance D is taken to be D = 16 . 8 Mpc (obtained from stellar population measurements) throughout this discussion. In Fig. 3a we plot the theoretical angular diameter ∆ θ of M87* as functions of r 2 and a assuming mass M ≃ 3 . 5 × 10 9 M ⊙ (obtained from gas dynamics studies). We note from Fig. 3a that there is no suitable r 2 in the range 0 to 2 (obtained from the considerations of a real, positive event horizon) which can reproduce the observed angular diameter of M87* denoted by Φ = 42 ± 3 µas . Even when the maximum offset of 10% in the angular diameter is considered i.e Φ = 37 . 8 ± 2 . 7 µas is taken as the observed value, the mass M ≃ 3 . 5 × 10 9 M ⊙ falls short in addressing the observations. Since the theoretical angular diameter is directly proportional to the mass (Eq. (53)) therefore a larger mass of the source is required to reproduce the observations. Hence, it seems that the mass of M87* measured from gas dynamics studies needs to be revisited. We next consider calculating the theoretical angular diameter of shadow of M87* keeping the distance fixed to D = 16 . 8 Mpc but using the mass M ≃ 6 . 2 × 10 9 M ⊙ obtained from stellar dynamics measurements. With these values of mass and distance we evaluate the theoretical angular diameter ∆ θ for M87* which is plotted in Fig. 3b. From the figure it is evident that no value of r 2 can reproduce the observed image diameter Φ = 42 ± 3 µas . However, when maximum offset of 10% in the angular diameter is considered, 0 . 1 ≲ r 2 ≲ 0 . 3 is required to explain the observed angular diameter within 1σ (35 . 1 µas = (37 . 8 -2 . 7) µas , denoted by the red dashed line in Fig. 3b). Thus, when angular diameter is calculated with mass M ≃ 6 . 2 × 10 9 M ⊙ a non-zero r 2 can only explain the observations within 1σ if maximum offset of 10% in the image diameter is allowed. For completeness we also calculate the theoretical angular diameter with mass M ≃ 6 . 5 × 10 9 M ⊙ which is the mass derived by the EHT collaboration from the observed shadow of M87* assuming general relativity. Since this is the largest among all the three masses, it can explain the the observed image diameter of Φ = 42 ± 3 µas within 1σ (39 µas denoted by the blue dashed line in Fig. 4). If the maximum offset of 10% is allowed then a non-zero dilaton charge 0 ≲ r 2 ≲ 0 . 2 can explain the observed shadow diameter (Φ = 37 . 8 µas , denoted by the red solid line in Fig. 4). However, M ≃ 6 . 5 × 10 9 M ⊙ should not be used to infer the observationally favored magnitude of r 2 since this mass is obtained from shadow measurements assuming GR. Therefore, using this mass estimate we cannot constrain another alternative gravity theory. We now discuss the constrains on r 2 from the other two observables ∆ C and ∆ A . The theoretical computation of these two observables does not require information about the mass and distance of the source. One however needs to provide the inclination angle of the source (which in the present case in 17 · ) to obtain ∆ C and ∆ A as functions of r 2 and a . According to the EHT results, the deviation from circularity ∆ C ≲ 10% [7-9] or 0.1 for M87*. The density plot of ∆ C for M87* is shown in Fig. 5a. From the density plot we observe that for all values of spin and r 2 ∆ C < 10% or 0.1 is realized. Thus, ∆ C estimate for M87* does not give any additional bound on the Kerr-Sen parameter r 2 . The EHT collaboration estimates an upper bound on the axis ratio ∆ A for the image of M87*, i.e, ∆ A < 4 3 [7-9]. The density plot for axes ratio ∆ A in Fig. 5b indicate that for all values of r 2 and a the axes ratio ∆ A < 4 3 . Thus, EHT estimate of axes ratio ∆ A for M87* does not provide any additional constrain on the dilaton parameter r 2 . It can be said that the axis ratio estimate allows non-zero values of Kerr-Sen parameter, although it does not constrain it. Constrains on the dilaton parameter r 2 from EHT observations of Sgr A*: The EHT collaboration measured the angular diameter for image of SgrA* to be ∆ θ = (51 . 8 ± 2 . 3) µas while the shadow diameter is estimated to be ∆ θ = (48 . 7 ± 7) µas [13-18]. The theoretical angular diameter depends on the mass M , the distance D , the inclination angle θ 0 and the metric parameters r 2 and a (see Eq. (53)). As before, we use previously determined masses and distances of the source to compute the theoretical angular diameter which is then compared with the observations to establish constrains on r 2 . The angle of inclination has an estimated upper bound θ 0 < 50 · obtained by comparing the image of Sgr A* with extensive numerical simulations [13]. Following [117] we fix the inclination angle to be θ 0 = 46 · for the present work. In Fig. 6 the contours of theoretical angular diameter ∆ θ of the shadow of Sgr A* are plotted for different estimates of mass and distance. The mass and distance of the source have been well constrained by the Keck team and the GRAVITY collaboration. We first discuss the constrains on r 2 assuming distance and mass measurements by the Keck team [114]. Keeping the red-shift parameter free the mass and distance of Sgr A* turns out to be M = (3 . 975 ± 0 . 058 ± 0 . 026) × 10 6 M ⊙ and D = (7959 ± 59 ± 32) pc, respectively. When the red-shift parameter is fixed to unity the distance and mass estimates by the Keck team yield D = (7935 ± 50) pc and M = (3 . 951 ± 0 . 047) × 10 6 M ⊙ . In Fig. 6a and Fig. 6b we plot the contours of angular diameter of the shadow using the masses and distances estimated by the Keck team. From the figures it is evident that the observed shadow diameter of Φ = 48 . 7 µas can be reproduced by 0 . 1 ≲ r 2 ≲ 0 . 2 (red solid line in Fig. 6a and Fig. 6b). When the lower 1σ interval is considered, i.e., Φ = (48 . 7 -7 = 41 . 7) µas , then, r 2 can be as high as unity (red dashed line in Fig. 6a and Fig. 6b). Since the error bar associated with the shadow diameter is quite high ( ± 7 µas ) we do not assign much importance to this result but emphasize that a small but non-trivial value of r 2 ≃ 0 . 1 -0 . 2 is required to reproduce the central value of the observed shadow diameter. We further note that 0 ≲ r 2 ≲ 0 . 1 can explain the observed image diameter within 1σ , (Φ = 51 . 8 -2 . 3 = 49 . 5 µas , blue dashed line in Fig. 6a and Fig. 6b). We now discuss the constrains on the dilaton parameter r 2 from the distance and mass measurements by the GRAVITY collaboration [115, 116]. According to the results of the GRAVITY collaboration the mass and distance of Sgr A* turn out to be M = (4 . 261 ± 0 . 012) × 10 6 M ⊙ and D = (8246 . 7 ± 9 . 3)pc [115,116] respectively. However, when one takes into account the systematics due to optical aberrations, the GRAVITY collaboration constrains the mass and distance of Sgr A* to M = (4 . 297 ± 0 . 012 ± 0 . 040) × 10 6 M ⊙ and D = (8277 ± 9 ± 33) pc respectively. In Fig. 6c and Fig. 6d we plot contours of theoretical angular diameter of shadow of Sgr A* assuming masses and distances reported by the GRAVITY collaboration. From the figures we note that once again a non-zero dilaton parameter 0 . 3 ≲ r 2 ≲ 0 . 4 is required to explain the central value of the observed shadow diameter of Φ = 48 . 7 µas (red solid line in Fig. 6c and Fig. 6d). To reproduce the central value of the image diameter Φ = 51 . 8 µas (blue solid line in Fig. 6c and Fig. 6d), a smaller but non-zero r 2 (0 ≲ r 2 ≲ 0 . 1) is required. The upper 1σ interval of the image diameter Φ = 51 . 8 -2 . 3 = 49 . 5 µas , (denoted by the blue dashed line in Fig. 6c and Fig. 6d) can be reproduced by 0 . 2 ≲ r 2 ≲ 0 . 3. Therefore, for all mass and distance estimates of Sgr A*, a small positive dilaton charge is required to explain the observed shadow/image diameter. This implies that the observed shadow of Sgr A* can be better explained by the Kerr-Sen scenario. For completeness we plot in Fig. 7 the dependence of the the deviation from circularity ∆ C and the axis ratio ∆ A on the metric parameters r 2 and a . This is a theoretical plot which only requires independent estimate of the inclination of the source, which is taken to be θ 0 = 46 · as discussed earlier. EHT has not provided any data related to ∆ A and ∆ C for Sgr A*. These results therefore cannot impose additional constrains on the dilaton parameter r 2 at present. These plots can however be useful in future when EHT releases data pertaining to ∆ A and ∆ C for Sgr A*.", "pages": [ 9, 10, 11, 12, 13, 14, 15, 16, 18 ] }, { "title": "5 Concluding Remarks", "content": "In this work we investigate the signatures of Einstein-Maxwell dilaton-axion (EMDA) gravity in the shadows of Sgr A* and M87* observed by the Event Horizon Telescope collaboration. EMDA gravity arises in the low energy effective action of superstring theories and is associated with the dilaton and the axion fields coupled to the Maxwell field and the metric. Exploring the astrophysical implications of such a theory is important as it can potentially provide a possibility to test string inspired models. Moreover, the axion and dilaton fields are often invoked to address the inflationary paradigm or the present accelerated expansion of the universe [68,69]. EMDA gravity is a scalar-vector-tensor theory of gravity that differs substantially from the standard general relativistic scenario. The stationary and axisymmetric black hole solution of EMDA gravity corresponds to the Kerr-Sen spacetime associated with the dilatonic charge and angular momentum acquired from the axionic field. The Maxwell field imparts electric charge to the Kerr-Sen black hole and in the absence of the Maxwell field the field strengths corresponding to dilaton and axion vanish. In that event the metric reduces to the Kerr background in general relativity. It is important to note that the charge of Kerr-Sen black hole originates from the axion-photon coupling and not from the charged particles falling onto the black hole. In the present work we aim to constrain the charge of the Kerr-Sen black hole from observations related to black hole shadow. For this purpose we examine the motion of photons in the Kerr-Sen background and analyze the nature of the light rings and spherical photon orbits. These light rings when projected onto the observer's sky give rise to the black hole shadow which depends sensitively on the background metric. The x and y coordinates of the shadow are dependent on the inclination angle θ 0 and the two impact parameters ξ and η which denote the distances of the photon from the axis of rotation and the equatorial plane, respectively. These impact parameters in turn depend on the charge r 2 , the spin a of the black hole and the radius of the spherical photon orbit r . Thus, we have x and y as functions of θ 0 , a , r 2 and r . The outline of the shadow y ( x ) is obtained by eliminating r between x and y which is often achieved numerically. Since x and y also depend on θ 0 , a and r 2 , the shape and size of the shadow depend sensitively on these three parameters. For example, a rapidly rotating black hole viewed at a high inclination angle casts a non-circular shadow. In fact to observe a deviation from circularity in the shape of the shadow, one needs to have non-zero spin and inclination angle, both. The dilaton parameter r 2 on the other hand mainly has an impact on the shadow size. An increase in r 2 leads to a decrease in the shadow diameter. Once the role of the metric on the shadow structure is derived we next compute the various observables associated with the black hole shadow. These include the angular diameter ∆ θ , the axis ratio ∆ A and the deviation from circularity ∆ C . Since the shadow is non-circular in general we can define a major axis and a minor axis associated with the shadow and a ratio of the two gives us the axis ratio ∆ A . In order to theoretically compute these observables one need to provide estimates of the inclination angle of the source. Computation of ∆ θ further requires independent estimates of mass and distance of the compact object. For M87* and Sgr A* the mass, distance and inclination angle have been previously determined. We use these data to theoretically compute the observables pertaining to the shadow of M87* and Sgr A* which are eventually compared with observations to establish constrains on the dilaton parameter r 2 . When the shadow of M87* is computed with predetermined mass M = 3 . 5 × 10 9 M ⊙ , obtained from gas dynamics studies, the observed angular diameter cannot be reproduced for any value of r 2 , including r 2 = 0 which corresponds to the Kerr scenario. This mass measurement therefore possibly needs to be revisited. When mass determined from stellar dynamics measurements ( M = 6 . 2 × 10 9 M ⊙ ) is used, the observed shadow diameter can be reproduced by 0 . 1 ≲ r 2 ≲ 0 . 3 within 1σ only when the maximum offset of 10% is considered. In both cases the distance is taken to be D = 16 . 8 Mpc (obtained from stellar population measurements) to compute the theoretical shadow diameter. For completeness we also compute the shadow diameter with M = 6 . 5 × 10 9 M ⊙ which is the mass reported by the EHT collaboration from the shadow measurements assuming GR. Since this mass is larger than the previous measurements, it can explain the observations better. However, theoretical shadow calculated using this mass should not be used to constrain r 2 as such a mass is derived from the observed shadow angular diameter assuming GR. With this mass therefore we cannot constrain another alternate gravity model. We note that with previously estimated masses the theoretical shadow is smaller than the observed one and since an increase in r 2 further shrinks the shadow, therefore the Kerr scenario can better explain the image of M87* compared to the Kerr-Sen scenario. It may be worthwhile to mention here that the Kerr solution is not unique to GR but arises even in several other alternative gravity scenarios. For example, the stationary, axisymmetric uncharged black hole solution in string theory resembles the Kerr solution in GR [74, 75]. We further mention that the Kerr scenario also fails to reproduce the observed shadow diameter of (42 ± 3) µas . Therefore, it seems that if in an alternate gravity model the shadow is larger than the GR scenario, it will better explain the observations, e.g. the braneworld scenario [96]. This may also be the reason why the mass of M87* obtained by the EHT collaboration is larger than the previous two measurements. For Sgr A*, the mass and distance have been estimated independently by the Keck team and the GRAVITY collaboration. When the theoretical shadow is computed assuming mass and distance estimates by the Keck team, 0 . 1 ≲ r 2 ≲ 0 . 2 is required to explain the central value of the observed shadow diameter of Φ = 48 . 7 µas . Further, 0 ≲ r 2 ≲ 0 . 1 is required to address the observed image diameter (51 . 8 -2 . 3 = 49 . 5 µas ) within 1σ . On considering mass and distance estimates by the GRAVITY collaboration, 0 . 3 ≲ r 2 ≲ 0 . 4 is required to reproduce the central value of the observed shadow diameter while 0 ≲ r 2 ≲ 0 . 1 is necessary to explain the central value of the observed image diameter. Thus, a small positive dilaton charge is required to address the observed shadow of Sgr A* for both the mass and distance estimates by the Keck team and the GRAVITY collaboration. The general relativistic scenario however can explain the observations when 1σ interval is considered in the observed shadow diameter. Thus, the Kerr-Sen scenario better explains the observed shadow of Sgr A* compared to the Kerr scenario. The charge of Kerr-Sen black hole has been constrained previously from different astrophysical observations, e.g. black hole continuum spectrum [118] and relativistic jets [119]. Comparison of the theoretical spectrum of eighty Palomar Green quasars with their optical observations reveal that r 2 ∼ 0 . 2 best explains the observations. The general relativistic scenario with r 2 = 0 is however included when 1σ interval is considered [118]. When the jet power associated with ballistic jets in microquasars is used to constrain the dilaton parameter, r 2 ≃ 0 seem to be favored by observations [119]. Thus, we note that astrophysical observations, e.g, shadows, continuum spectra or jets either indicate a small dilatonic charge in black holes or exhibit a preference towards the Kerr scenario. All the observations rule out black holes with large dilaton charge. This is interesting as different astrophysical observations on completely different observational samples consistently indicate the same result. The scope to verify this finding will further increase as EHT releases more black hole images with greater resolution. The present astrophysical observations like quasi-periodic oscillations observed in the power spectrum of black holes or the Fe-line observed in the black hole spectrum can be further used to verify our result. This will be addressed in a future work.", "pages": [ 18, 19, 20 ] }, { "title": "Acknowledgements", "content": "Research of I.B. is funded by the Start-Up Research Grant from SERB, DST, Government of India (Reg. No. SRG/2021/000418).", "pages": [ 20 ] } ]

[ { "title": "Potentials for general-relativistic geodesy", "content": "Claus Lammerzahl 1 , 2 , 3 and Volker Perlick 1 1 Center for Applied Space Technology and Microgravity (ZARM), University of Bremen, 28359 Bremen, Germany 2 Institute of Physics, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany 3 Gauss-Olbers Space Technology Transfer Centre, c/o University of Bremen, Am Fallturm, 28359 Bremen, Germany Geodesy in a Newtonian framework is based on the Newtonian gravitational potential. The general-relativistic gravitational field, however, is not fully determined by a single potential. The vacuum field around a stationary source can be decomposed into two scalar potentials and a tensorial spatial metric, which together serve as the basis for general-relativistic geodesy. One of the scalar potentials is a generalization of the Newtonian potential while the second one describes the influence of the rotation of the source on the gravitational field for which no non-relativistic counterpart exists. In this paper the operational realizations of these two potentials, and also of the spatial metric, are discussed. For some analytically given spacetimes the two potentials are exemplified and their relevance for practical geodesy on Earth is outlined.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Beside astronomy, geodesy is one of the oldest science. It is about the shape of the Earth, its orientation, and its gravitational field. While its shape, the topography, can be observed directly e.g. from space with satellites equipped with Lidar systems, its orientation is inferred e.g. from VLBI observations and from direct measurements of the rotation of the Earth, e.g. [1], and today provided by the International Earth Rotation Service IERS [2]. The gravitational field can be determined from the measurements of gravimeters like falling corner cubes [3], superconducting gravimeters with a resolution better than 1 nm / s 2 [4] and gradiometers. These devices measure the vector of the gravitational acceleration g which is the gradient of the gravity potential W = U + Ω 2 r 2 where U is the Newtonian gravitational potential and Ω 2 r 2 is the centrifugal potential. Equipotential surfaces, and in particular the geoid, are constructed from g -measurements and a procedure called geodetic leveling [5] which possesses an inconsistency within Europe of approx. 1 m. New developments on the experimental side open up new possibilities and improved precision to measure the gravity field of the Earth: (i) atom interferometers serve as a new class of gravi- and gradiometers [6] and they are sensitive to differences of the gravity potential, (ii) clocks through the general-relativistic redshift are also sensitive to differences of the gravity potential [7], and (iii) the new laser ranging interferometer LRI on board of the GRACE Follow-On satellites, which were launched in 2018, yield improved data for determining the gravity field of the Earth on a global scale. The precision of the LRI of 1 nm [8] and in particular the use of the gravitational redshift of clocks now make it mandatory to describe these measurements within the formalism of General Relativity. The main task of geodesy is to determine and to characterize the gravitational field of a compact gravitating body like the Earth. On the Newtonian level a characteristic quantity is the geoid which is a certain surface of a constant Newtonian gravity potential possessing the topology of a sphere. Also in General Relativity it was possible to define a fully general-relativistic geoid which can be determined through clocks or through gravimetric measurements [9]. However, since the gravitational field within General Relativity possesses more degrees of freedom than in Newtonian gravity (we have 10 metrical components compared with one Newtonian gravitational potential) one may wonder whether there might also be more than one kind of geoid within the framework of Gdeneral Relativity. In fact, in this paper we define with the help of a second potential a second geoid which is related to the gravitational field of a stationary rigid body. While the first geoid is mainly related to the mass density of the gravitating source, the second geoid is related to the mass current density of the source, in particular its rotation, i.e., to the gravitomagnetic part of the general-relativistic gravitational field. Both geoids are related to the stationarity of the gravitational field and the second one requires, in addition, that Einstein's vacuum field equation is satisfied, i.e., it is defined only outside of the source. The other degrees of freedom of the gravitational field are included in the remaining metrical components in a 3-dimensional rest frame. In developing the notions we first state the model of the Earth which we assume to rotate rigidly and to be isolated from all other gravitating bodies. As a consequence, the 4-dimensional general-relativistic spacetime around the Earth possesses a timelike Killing vector field. The (pseudo-)norm of this vector field is related to the first geoid which describes the gravitational redshift and, at the same time, the acceleration of falling corner cubes. The curl of this vector field gives a twist covector field. Outside of the Earth, where Einstein's vacuum field equation is assumed to hold, this covector field admits a potential which is known as the twist potential , so we have a second potential related to the gravitational field of the Earth. The first potential is analogous to an electrostatic potential while the second one is analogous to a magnetostatic potential. Accordingly, our two gravita- tional potentials can be regarded as a gravitoelectric and a gravitomagnetic potential, respectively.", "pages": [ 1, 2 ] }, { "title": "II. THE MODEL OF THE EARTH", "content": "The Earth is an extended gravitating body. Therefore, it is most efficiently modeled in terms of a congruence of non-intersecting timelike worldlines [10] describing the constituents of the Earth. As a first approximation, it is reasonable to assume that the Earth is rigidly rotating with a constant angular velocity. In this section we want to recollect the well-known fact that then the (appropriately parametrized) worldlines of the constituents are the integral curves of a timelike Killing vector field. As most of the gravimetric measurements are taking place in the vacuum region outside of the Earth, what is important for us is the fact that the timelike Killing vector field that describes the motion of the constituents of the Earth can be extended, as a timelike Killing vector field, to the exterior region. The integral curves of this extended Killing vector field may be interpreted as the worldlines of geostationary satellites (or, if on the surface of the Earth, as the worldlines of observers that are at rest there with respect to the rotating Earth). For a congruence of timelike curves with four-velocity u µ one defines acceleration a µ , rotation ω µν , expansion θ and shear σ µν by the equations [10] where h µ ν = δ µ ν -u µ u ν is the projection onto the local rest space and D µ is the covariant derivative defined by the Levi-Civita connection of the metric. Round brackets and square brackets denote symmetrization and antisymmetrization, respectively. In the following we discuss the operational meaning of these quantities for the special congruence associated with the rigidly rotating Earth. To that end we need the notion of a standard clock and of the radar distance between neighboring observers. Within General Relativity it is possible to uniquely characterize a particular parameter along the worldline of an observer which is called proper time . An (idealized) clock that shows proper time is called a standard clock . To give an operational characterization of this notion, we first define the radar distance ∆ x of an event p from the worldline of a fixed observer. In our units with c = 1, we simply have ∆ x = ∆ t were ∆ t is half the time span, measured with a clock of the observer, a light ray needs to propagate from the observer's worldline to p and back to the observer. The observer's clock is a standard clock, i.e., the parameter t is proper time, if and only if ( 1 -( dx/dt ) 2 ) -1 d 2 x/dt 2 takes the same value for all freely falling particles emitted in the same spatial direction, see [11]. It has been shown that the energy levels of atoms are influenced by the spacetime curvature according to δE ∼ Ra 2 B where R is a typical component of the curvature tensor and a B is the Bohr radius [12]. On Earth, this will amount to a relative frequency change of the order δν/ν ∼ 10 -42 which is more than 20 orders of magnitude beyond the present uncertainty of atomic clocks. Accordingly, with very high precision atomic clocks on Earth are standard clocks. With standard clocks it is also possible to uniquely define a standard distance. Furthermore, using light rays it is also possible to operationally define whether a congruence of timelike curves is rotating. This is Pirani's bouncing photon construction [13]. Fix any two infinitesimally neighboring curves A and B in the congruence and send a light ray from A to B . Reflect the light ray at B in such a way that the tangent vectors to the incoming light ray, to the reflected light ray and to the worldline of B are linearly dependent. We say that the congruence is irrotational if, in any such situation, the reflected light ray arrives back at A , i.e., if the light rays bouncing back and forth between A and B form a two-dimensional timelike worldsheet. It is well known that a congruence is irrotational if and only if it is hypersurface-orthogonal. In this case for any pair of infinitesimally neighboring worldlines A and B in the congruence the following is true: The normalized connecting vector r σ from A to B , which is assumed to be orthogonal to the four-velocity vector u µ tangent to the wordline of A , satisfies the Fermi(-Walker)-transport law h µ ν u σ D σ r ν = 0. Any deviation from that describes a rotation. This notion of rotation is often discussed in the relativistic theory of continua, see e.g. [10], but it is well-defined also for congruences of worldlines in vacuum. ̸ ̸ As already mentioned, we want to assume, as a reasonable first approximation, that the Earth is rigid. In relativity a congruence is called (Born-)rigid if the (radar) distance between any two infinitesimally close worldlines of the congruence is time-independent. As a consequence, also angles between directions to neighboring worldlines remain constant in time. This is possible only if the congruence has vanishing shear and expansion [10]. The rigidity condition still allows the Earth to rotate with ω µν = 0 and accelerate with a µ = 0. We assume now in addition that an observer co-moving with a constituent of the Earth always experiences the same situation. This means, in particular, that the acceleration of this comoving observer is co-rotating, h µ ν u ρ D ρ a ν = ω µ ν a ν . Furthermore, if we assume that the angular velocity of the rigidly rotating Earth is time-independent, the rotation of the rigid Earth is assumed to be Fermi-constant, that is, h κ ρ h σ λ u µ D µ ω ρ σ = 0. These three conditions which are fulfilled by the Earth to high precision then imply that the congruence describing the Earth is a Killing congruence, that is, u ∼ ξ with ξ being a timelike Killing vector field [10]. In reality, the Earth experiences small deformations, tides, winds, ocean whirls, snow falls and ice melting, and further time-dependent processes. All this happens with very low velocities and small masses so that the time-dependency of the gravitational field can be treated adiabatically to very high precision. To sum up, with high precision the Earth is described adiabatically by means of a Killing congruence. This Killing congruence can then be extended to the exterior of the Earth. In this approximation, the analysis of the relativistic gravitational field of the Earth is thus tantamount to the analysis of a Killing congruence. In spherical polar coordinates ( t, r, ϑ, φ ) the Killing vector field is represented as ξ = ∂ t + Ω ∂ φ with a constant Ω that gives the angular velocity of the rotating Earth. Note that neither ∂ t nor ∂ φ are Killing vector fields, unless we assume that the Earth is axisymmetric. (The axisymmetric case, where we have an entire family of Killing vector fields parametrized by Ω, will be treated in Section VI below.) Strictly speaking, such an irregularly shaped rotating body would emit gravitational waves which would cause the angular velocity to decrease. However, for the Earth and all other planets and moons this energy loss by gravitational waves is totally negligible.", "pages": [ 2, 3 ] }, { "title": "III. THE GEOMETRY OF KILLING CONGRUENCES", "content": "A Killing congruence is given by a timelike Killing vector field ξ proportional to the 4-velocity of a family of observers, ξ ∼ u . A Killing vector field possesses a (pseudo-)norm, e 2 ϕ := g µν ξ µ ξ ν as well as a curl ∂ [ µ ξ ν ] which is equivalent to the twist vector field ϖ µ = ϵ µνρσ ξ ν ∂ ρ ξ σ . Here ϵ µνρσ denotes the totally antisymmetric Levi-Civita tensor field (or volume form) associated with the spacetime metric where we choose the orientation such that in the spherical polar coordinates used below ϵ trϑφ > 0. The twist vector is Fermi(-Walker) propagated, h µ ν u σ D σ ϖ ν = 0. Using the Killing property of ξ , it can be shown that the (negative) gradient of the scalar function ϕ is equal to the acceleration of the Killing congruence, a µ = -∂ µ ϕ [10], which fulfills the co-moving condition h µ ν u σ D σ a ν = ω µ ν a ν . If the metric satisfies Einstein's vacuum field equation, it can be shown that the twist covector field ϖ µ = g µν ϖ ν possesses a potential, ϖ µ = ∂ µ ϖ [14]. Therefore, in vacuum, outside the Earth, we have two gravitational potentials, ϕ and ϖ . Owing to their properties, see also below, these two potentials may be called gravitoelectric and gravitomagnetic potentials. It was already mentioned that they are the gravitational analogues of the electrostatic and magnetostatic potentials which are well-known in standard electromagnetism [15]. Up to an additive constant, each of the two gravitational potentials is directly related to measurements, either through a measurement of potential differences or through a measurement of the gradient. This will be outlined in the following. In our case of a stationary spacetime the line element can be 3+1 decomposed according to where i, j = 1 , 2 , 3 and ϵ ijk is the volume form associated with the spatial metric γ ij [16]. It is the g 00 -component and the g 0 i -component which can be expressed through the gravitoelectric and -magnetic potentials.", "pages": [ 3 ] }, { "title": "IV. THE PHYSICS OF THE GENERAL-RELATIVISTIC 'GRAVITOELECTRIC' POTENTIAL OF THE EARTH", "content": "The gravitoelectric potential ϕ is obtained from the equation e 2 ϕ = g µν ξ µ ξ ν , where ξ is assumed to be a timelike Killing vector field. As outlined above, we model the Earth as a rigid body that rotates with constant angular velocity; this allows us to choose spherical polar coordinates ( t, r, ϑ, φ ) in the vacuum region outside of the Earth such that ξ = ∂ t + Ω ∂ φ , with a constant Ω that is to be identified with the angular velocity of the Earth. The gravitoelectric potential foliates the spacetime into 3-dimensional hypersurfaces ϕ = const . . Because of the Killing property of ξ , these hypersurfaces project onto 2-dimensional surfaces in the 3-dimensional space of integral curves of ξ which are known as isochronometric surfaces . The general-relativistic geoid can be defined as one of these surfaces, see Philipp et al. [9]. In the case of the Earth it is natural to choose the isochronometric surface that is closest to the mean sea level. This surface is also used as a height reference, i.e., it has by definition zero height. Note that this definition of the geoid in terms of isochronometric surfaces makes sense not only for the Earth but also for all other planets and moons and even for neutron stars and black holes. In the latter cases the choice of a particular isochronometric surface is purely conventional. The application to compact and ultracompact objects is made possible by the fact that exact relativistic equations are used, rather than post-Newtonian approximations. The idea of defining the geoid in terms of isochronometric surfaces was brought forward already in 1985 by Bjerhammar [17] who, however, did not work out any mathematical details. Also inspired by Bjerhammar, Kopeikin et al. [18] discussed a relativistic geoid based on a particular fluid model of the Earth. For an alternative fully relativistic definition of a geoid, not in general related to an operational realization with clocks, we refer to Oltean et al. [19]. The gravitoelectric potential can be most easily measured and, thus, operationally defined through the redshift in clock-comparison experiments [9]: For any two stationary clocks (i.e., clocks whose worldlines are integral curves of the Killing vector field ξ ) the redshift z is given by the equation ln(1 + z ) = ϕ 2 -ϕ 1 where ϕ 1 and ϕ 2 are the values of ϕ at the positions of the two clocks. (Note that ϕ is constant along each worldline of ξ .) As outlined in [9], for clocks on the surface of the Earth the comparison may be done with the help of optical fibers. Such fiber links already exist and they may have a length of more than thousand kilometers [20]. As an alternative to redshift measurements, the same potential difference ϕ 2 -ϕ 1 can also be obtained with atom interferometry [21-23]. From measurements of the acceleration a µ = -∂ µ ϕ of a falling corner cube one can also calculate the equipotential surfaces. Therefore, within full General Relativity, all three types of measurements yield the same potential ϕ ( x ) which makes data fusion and improved geoid determination possible [24].", "pages": [ 3, 4 ] }, { "title": "V. THE PHYSICS OF THE GENERAL-RELATIVISTIC 'GRAVITOMAGNETIC' POTENTIAL OF THE EARTH", "content": "There are many ways to determine the gravitomagnetic potential through measurements of the twist potential. The Sagnac experiment runs with a ring laser interferometer with counter-propagating laser beams. The two interfering beams give a proper time difference given by [25] where dx i is the spatial differential within the interferometer's rest frame. Using Stokes's theorem and the twist vector this can be rewritten as for a small interferometer with area ⃗ Σ and ϕ 0 as gravitoelectric potential at the position of the beam splitter. If the Einstein vacuum field equations are fulfilled, then we introduce Φ J = e -2 ϕ ϖ which is the angular momentum potential and obtain Similar characterizations of the gravitomagnetic potential result from atom interferometry [22]. (ii) Also the propagation of classical objects with spin S µ couple to the gravitomagnetic field. This is known as the Schiff effect first derived in [26]. The relation to the twist has been shown in [27] see also [28]. This effect has experimentally been confirmed by the space mission Gravity Probe B [29]. (iii) Finally, atomic spectroscopy is also sensitive to the Sagnac effect [30]. All these methods measure the same twist and, thus, the same gravitomagnetic potential.", "pages": [ 4 ] }, { "title": "VI. STATIONARY AND AXISYMMETRIC SPACETIMES", "content": "In order to get a better physical and intuitive understanding of these potentials we are now calculating and visualizing the gravitoelectric and the gravitomagnetic potential for certain examples of spacetimes. For doing so, we now specialize to the case that the spacetime is stationary and axisymmetric. In this case we can use spherical polar coordinates ( t, r, ϑ, φ ) such that ∂ t and ∂ φ are Killing vector fields. We can then consider the gravitoelectric and gravitomagnetic potentials with respect to the Killing vector field ∂ t + Ω ∂ φ , with any constant Ω. The potentials are well-defined on the domain where ∂ t +Ω ∂ φ is timelike. Because of the symmetry they are functions of r and ϑ only. For modeling the gravitational field around the Earth the assumption of axisymmetry is of course a strong idealization. The realistic Earth can be modeled using a relativistic multipole expansion. If the stationary and axisymmetric spacetime is asymptotically flat (i.e., if g tt → 1 and g φφ / ( r 2 sin 2 ϑ ) → -1 for r → ∞ ) , the Killing vector field ∂ t is distinguished among the Killing vector fields ∂ t +Ω ∂ φ by the property that it is timelike near infinity. The integral curves of ∂ t can then be interpreted as the worldlines of observers who co-rotate with the source. The gravitoelectric and gravitomagnetic potentials related to this Killing vector field are commonly combined to give the complex Ernst potential Einstein's vacuum field equation then reduces to a partial differential equation for E known as the Ernst equation , see e.g. Griffiths and Podolsk'y [31] for details. The Ernst potential determines all components of the metric. In other words, a stationary axisymmetric vacuum spacetime is completely determined by the two real potentials ϕ and ϖ which depend only on the two coordinates r and ϑ . This should be contrasted with an arbitrary spacetime, where the metric has ten independent components which depend on all four coordinates, and with a stationary vacuum spacetime, where the metric is determined by eight scalar-valued functions that depend on the three spatial coordinates. The idea of using the real and the imaginary part of the Ernst potential as coordinates was brought forward already in the 1980s by Perj'es [32]. These two real potentials coordinatize the planes ( t, φ ) = const . , i.e., together with t and φ they can be used as coordinates on the spacetime. Of course, this is true only where dϕ and dϖ are linearly independent. Whereas the potentials corresponding to ∂ t + Ω ∂ φ with Ω = 0 are the ones most naturally related to the spacetime geometry, these potentials can be defined with respect to any Ω. This is of particular relevance in the case that the source is nonrotating, i.e., that ∂ t is hypersurface-orthogonal. Then the gravitomagnetic potential associated with Ω = 0 is constant and cannot be used as a coordinate. If the metric is given as with the g µν depending only on r and ϑ , the gravitoelectric potential associated with the Killing vector field ∂ t +Ω ∂ φ is given by the equation The twist covector field reads with g = ( g tt g φφ -g 2 tφ ) g rr g ϑϑ . If the metric satisfies Einstein's vacuum field equation, it is guaranteed that the twist covector field admits a potential, ϖ µ = ∂ µ ϖ . We now discuss the gravitoelectric and gravitomagnetic potentials for a few specific stationary and axisymmetric vacuum solutions to Einstein's field equation, thereby illustrating that they can be applied also to the spacetime around a black hole or another (ultra-)compact object. In all cases, we consider the metric in spherical polar coordinates ( t, r, φ, ϑ ), where ∂ t and ∂ φ are Killing vector fields. We determine the gravitoelectric and gravitomagnetic potentials with respect to the Killing vector field ∂ t +Ω ∂ φ and we plot the potentials ϕ ( r, ϑ ) and ϖ ( r, ϑ ) in diagrams with r sin ϑ on the horizontal axis and r cos ϑ on the vertical axis. The examples demonstrate how the two families of equipotential lines give a coordinatization of the r -ϑ -plane. For sufficiently small | Ω | the gravitoelectric equipotential lines close to the central body are circles, i.e., the gravitoelectric geoid in 3-space has the topology of a sphere, as usually associated with the term 'geoid'. By contrast, the gravitomagnetic equipotential lines are not usually closed, i.e., the gravitomagnetic geoid in 3-space has the topology of R 2 . To put this another way, the gravitoelectric potential may be interpreted as a height coordinate whereas the gravitomagnetic potential may be viewed as a latitude coordinate.", "pages": [ 4, 5 ] }, { "title": "A. Schwarzschild, Kottler and Reissner-Nordstrom metrics", "content": "For a spherically symmetric and static metric of the form (10) gives us the gravitoelectric potential as and (11) gives us the twist covector field as For the Scharzschild spacetime, this specifies to and Integration of the latter equation gives us the gravitomagnetic potential More generally, we can consider the Kottler spacetime, also known as the Schwarzschild-(anti)deSitter spacetime, In this case the cosmological constant gives a contribution to the gravitoelectric potential, whereas it drops out from the equation for the twist covector field, which is still given by (17). So also in this case we have a gravitomagnetic potential given by (18). However, in the case of the Reissner-Nordstrom spacetime the twist covector field is not integrable (unless Ω = 0 or Q = 0 ), i.e., in this case the gravitomagnetic potential does not exist. This is in line with the theorem in [14]. The equipotential surfaces for Schwarzschild are shown in Fig. 1 (a) and (b) for two different values of the angular velocity Ω of the observer field. (For Ω = 0 we have of course ϖ = 0, so in this case the potentials cannot be used as coordinates.) There is a forbidden region, shown in gray shading, where the potentials are not defined because the Killing vector field ∂ t + Ω ∂ φ is spacelike. Outside of this region, the differentials dϕ and dϖ are linearly independent, for all non-zero values of Ω, with the exception of the axis. If | Ω | is smaller than the critical value Ω crit = ( √ 27 M ) -1 ≈ 0 . 19245 M -1 , the forbidden region consists of two connected components: One is adjacent to the horizon, the other one is the region outside of the so-called light-cylinder. The inner equipotential surfaces of the gravitoelectric potential are topological spheres in 3-space, the outer ones are topological cylinders; the borderline case is an equipotential surface with a self-crossing along a circle in the equatorial plane. The equipotential surfaces of the gravitomagnetic potential all have the topology of R 2 , with the exception of one which consists of the sphere r = 3 M and part of the equatorial plane. If | Ω | = Ω crit the two connected components of the forbidden region touch at the photon circle r = 3 M in the equatorial plane. If | Ω | > Ω crit , the forbidden region is connected. Now all the equipotential surfaces have the topology of R 2 in 3-space. For the Earth the critical angular velocity is Ω Earth , crit ∼ 800 GHz which is 16 orders of magnitude larger than the actual value. The outer part of the forbidden region begins at approx. 5 × 10 9 km which is 10 times the distance to Pluto. And the inner part of the forbidden region does not exist for the real Earth.", "pages": [ 5, 6 ] }, { "title": "B. Kerr metric", "content": "We now consider the Kerr metric where, for a Kerr black hole, a 2 ≤ M 2 . By (10) the gravitoelectric potential reads From (11) we find the twist covector field with ϖ (1) µ dx = 2cos ϑ ( r + a ) - 2 a ( r + a - 2 Mr )sin ϑ + a sin ϑ dr This gives us the gravitomagnetic potential In Fig. 2 (a) we show the equipotential surfaces for non-rotating observers, Ω = 0, where the potentials simplify to [33] µ ( 2 2 2 2 2 2 2 4 4 ) The surfaces ϕ = const . are topological spheres in 3-space; the innermost one is the boundary of the ergoregion. The surfaces ϖ = const . all have the topology of R 2 . For rotating observers we have to distinguish the case a Ω > 0 (Fig. 2 (b) and (c)) and the case a Ω < 0 (Fig. 2 (d) and (g)). If | Ω | is sufficiently small, the region where the potentials are not defined (gray shaded in the figure) is connected; beyond a critical value this region decomposes into two connected components which are separated from the equatorial plane.", "pages": [ 6, 7 ] }, { "title": "C. The NUT metric", "content": "The NUT metric [34] is a solution to Einstein's vacuum field equation that reads where M is the mass parameter and n is the socalled NUT parameter, also known as a gravitomagnetic charge . From (10) we find the gravitoelectric potential and from (11) we find the twist covector field The twist potential reads Fig. 3 (a) and (b) show the potentials for the NUT metric with two different values of Ω. Qualitatively, the equipotential surfaces are similar to the Schwarzschild case. In particular there is a critical value for Ω beyond which the region where the potentials are not defined decomposes into two connected components. Note, however, that in contrast to the Schwarzschild spacetime the potentials are no more symmetric with respect to the equatorial plane.", "pages": [ 7 ] }, { "title": "D. The Kerr-NUT metric", "content": "The Kerr-NUT metric is a solution to Einstein's vacuum field equation that depends on a mass parameter M , a NUT parameter (gravitomagnetic charge) n and a spin parameter a , see e.g. Griffiths and Podolsk'y [31], p.312. The metric reads For simplicity, we restrict here to Ω = 0. Then the gravitoelectric potential is and the twist covector field is so the gravitomagnetic potential equals The potentials for the Kerr-NUT metric with Ω = 0 are shown in Fig. 3 (c) for the case an > 0 and in Fig. 3 (d) for the case an < 0. In either case, the surfaces ϕ = const . are topological spheres in 3-space and the surfaces ϖ = const . are diffeomorphic to R 2 . Again, it is clearly seen that the NUT parameter breaks the symmetry with respect to the equatorial plane.", "pages": [ 8, 9 ] }, { "title": "E. The rotating q -metric", "content": "The rotating q -metric is a solution to Einstein's vacuum field equation that was found by Toktarbay and Quevedo [35], also see [36]. It depends on 3 parameters, a mass parameter M , a quadrupole parameter q and a spin parameter a . It features a naked singularity which is considered as unphysical by most authors. Therefore, when working with this metric one usually assumes that this vacuum solution is valid only outside of a certain sphere which covers the naked singularity and that inside this sphere the metric has to be matched to a regular interior solution. If interpreted in this sense the rotating q -metric describes the spacetime around a spinning body with a non-zero quadrupole moment that is very compact but not compact enough to have undergone gravitational collapse. As the surface where the matching is done can be chosen arbitrarily close to the naked singularity, in the following we consider the metric down to the naked singularity. As the q -metric has a non-zero quadrupole moment, it is of geodetic relevance since it might be used to relativistically model the flattened Earth. Higher-order multipole moments can be introduced via a series expansion of the metric, see e.g., [16]. The Ernst potential E for the rotating q metric can be found in [35]. From that we can find for non-rotating observers, Ω = 0, the gravitoelectric and gravitomagnetic potentials via (8). In prolate spheroidal coordinates ( t, x, y, φ ), the Ernst potential reads where The prolate spheroidal coordinates are related to spherical polar coordinates ( t, r, ϑ, φ ) by the transformation ̸ Here σ is a constant parameter. For rotating observers (Ω = 0) the potentials are given by very involved equations which will not be written out here but can be easily evaluated numerically. ̸ Fig. 3 (e) and (f) show the potentials for the nonspinning q -metric ( a = 0) for two different values of Ω. Roughly speaking, the pictures look like squashed versions of the Schwarzschild case, which is of course an effect of the non-zero quadrupole moment. Fig. 3 (g) and (h) show the potentials for the spinning q -metric ( a = 0) where the observers are non-rotating (Ω = 0).", "pages": [ 9 ] }, { "title": "VII. DISCUSSION", "content": "This paper is based on the observation that for a stationary solution to Einstein's vacuum field equation there are two scalar gravitational potentials. While the gravitoelectric potential is a general-relativistic generalization of the Newtonian potential the gravitomagnetic one has no non-relativistic analogue. There are many classical and quantum methods to operationally realize these potentials. The gravitoelectric potential can be interpreted as a height and the gravitomagnetic potential as a measure of the latitude. This means that these two potentials might be used as a physically given reference system in the vicinity of the rotating Earth. More precisely, if some degenerate cases are excluded they can be used as two of the three coordinates one needs for parametrizing 3-dimensional space. As the general-relativistic geoid is a particular level surface of the gravitoelectric potential, the intersection lines of the gravitomagnetic equipotential surfaces with the geoid give a latitudinal coordinatization of the geoid. Unfortunately, the gravitomagnetic effects on Earth are very small though they have been measured by LAGEOS and by Gravity Probe B via the Lense-Thirring and the Schiff effect. As we have seen, the Kerr parameter of the Earth also influences the gravitoelectric potential. However, its influence is just one order below the current accuracy of gravimeters. So, the next generation of instruments measuring the gravitoelectric and gravitomagnetic effects will be sensitive to the influence of the Earth's rotation on its gravitational field. The latter may also become observable with the help of the gravitomagnetic clock effect [37, 38]. The full gravitational field of the Earth can be given by a multipole expansion of the two potentials and the spatial metric γ ij , see (2). These components of the full metric can be measured with stationary and moving clocks, interferometers and gyroscopes. Within this framework, any adiabatic change of the gravitational field can be described through time-dependent multipole parameters. It has to be worked out how the potentials as well as the spatial metric can be measured from space, that is, with moving clocks or with GRACE-like constellations. A particular question is the following: Whereas for the gravitoelectric potential it is possible to measure poten- tial differences, in particular in terms of the redshift of clocks, for the gravitomagnetic potential the measurement methods discussed above only yield the gradient. It would be interesting to find out if there is a method to measure differences of the gravitomagnetic potential.", "pages": [ 9, 10 ] }, { "title": "ACKNOWLEDGMENT", "content": "We thank Eva Hackmann and Dennis Philipp for fruitful discussions. We acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2123 'QuantumFrontiers' - Grant No. 390837967 and the CRC 1464 'Relativistic and", "pages": [ 10 ] } ]

[ { "title": "Gravitational Spin Hall Effect of Dirac Particle and the Weak Equivalence Principle", "content": "Zhen-Lai Wang ∗ Center for Fundamental Physics and School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China (Dated: October 27, 2023) We present a spin-induced none-geodesic effect of Dirac wave packets in a static uniform gravitational field. Our approach is based on the Foldy-Wouthuysen transformation of Dirac equation in a curved spacetime, which predicts the gravitational spin-orbit coupling. Due to this coupling, we find that the dynamics of the free-fall Dirac wave packets with opposite spin polarization will yield the transverse splitting in the direction perpendicular to spin orientation and gravity, which is known as the gravitational spin Hall effect. Even in a static uniform gravitational field, such effect suggests that the weak equivalence principle is violated for quantum particles.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The question whether the weak equivalence principle (WEP) also holds for quantum particles has been received an theoretical and experimental interest for a long time [1-10]. The WEP is one of the foundational assumption of classical gravitational theory, which states that free-fall point particles will follow mass-independent trajectories. In classical physics the WEP is well-defined in terms of spacetime trajectories, but in quantum physics it is ill-defined because trajectories and even point particles are vague concepts. Quantum particles bring in many new properties distinct from classical point-like particles, such as matter/anti-matter [11-13], spin [14-16] and internal structures[5, 7, 17], which might raise objections to the validity of the WEP and add new physics contents to the WEP. Furthermore, WEP-test based on quantum particle can offer vital clue to understanding the connection between the quantum and gravitational theories. For example, almost all theories trying to unify gravitational theory and the standard model of particle physics predict the violations of WEP [18, 19]. The investigations on the WEP of quantum system have a vast spectrum, from the effects described by the simplest Schrodinger equation with gravitational potential to the effects originated from the quantization of gravitational field [20-25]. However, theoretical literature frequently offers conflicting views on whether the WEP of quantum system is violated or not. The major reason for this unpleasant situation is that there is no consensus about quantum version of the WEP's notion, which is still an open issue [26-28]. Quantum physic is formulated on the distinct concepts (such as quantum states, measurements and probabilities) from the classical ones. Naturally, the notions of quantum WEP can not be transferred directly from the classical statements of WEP and ought to be explained in the language of these quantum concepts [29, 30]. In this paper, we take the notions of quantum WEP introduced recently in Ref. [29] and [30]. In Ref. [29], the notion of quantum WEP is reconstructed as the statement that the Fisher information about the mass of quantum probe in free fall is the same as the free case without gravitational field . To extract information of mass through measurements of quantum probe's position, the Fisher information can be defined as with the wave function ψ ( x , t ) of the quantum probe. The notion of quantum WEP based on Fisher information is F g x ( m ) = F f x ( m ), where F g x ( m ) is the Fisher information in free fall and F f x ( m ) in the free case. In other words, the gravitational field can not create more information about the mass of free-fall quantum probe. In Ref. [30], the notion of quantum WEP is put into the following statement: The probability distribution of position for a free-falling particle is the same as the probability distribution of a free particle, modulo a massindependent shift of its mean . For a static uniform gravitational field, both the above the notions of quantum WEP are valid with taking no account of the internal degrees of freedom of quantum probe. However, both notions are invalid in the complex gravitational fields such as gravitational gradient field [29] and gravitational wave [31]. One of the interesting and important questions is that how much the spin of quantum particle is important on these two notions of quantum WEP. For this reason, we consider the dynamics of a Dirac particle freely falling in a static uniform gravitational field. An interesting effect is revealed that the free-fall Dirac wave packets with opposite spin polarization are split transversely in the direction perpendicular to spin orientation and the gravitational acceleration. Such effect is known as the gravitational spin Hall effect [32]. Like the spin Hall effects in other realms of physics, the gravitational spin Hall effect is also originated from the spin-orbit interaction [33]. The gravitational spin Hall effects or similar effects have been reported in various physical systems such as the Dirac field [37-41], the electromagnetic/light waves [42-47] and gravitational waves [48, 49]. Most of the studies present the polarizationor helicitydependent ray trajectories of motion which suggest the gravitational spin Hall effect. However, many results of these studies are inconsistent with each other due to the dependence upon different methods or models. Besides, the similar effects of Dirac field discussed in the literature, the gravitational spin-orbit coupling are mainly in the charge of interaction between spin and gravity through the gradient or torsion of the gravitational field. In this paper, with the help of Foldy-Wouthuysen transformation [50], a gravitational spin-orbit coupling is derived from the Dirac equation in a static uniform gravitational field (without gravitational gradient and torsion), which also yields the gravitational spin Hall effect even the gravitational field is so simple. Additionally, unlike the semi-classical approach where the external (position) degrees of freedom treated as a classical variable, we analyze the dynamical evolution of the wave packets of Dirac particle and show the entanglement between the internal (spin) and external (position) degrees of freedom in its full quantum-mechanical form. More importantly, we find that both the notions of quantum WEP mentioned earlier are unacceptable to the gravitational spin Hall effect presented in this paper, and so the gravitational spin Hall effect of Dirac particle can be treated as a new probe of quantum WEP's notion. The content of the paper is organized as follows: First, using the standard Foldy-Wouthuysen (F-W) transformation leads to a gravitational spin-orbit coupling from the Dirac equation in a static uniform gravitational field. Then, from the Schrodinger equation in the F-W picture, we consider the dynamical evolution of the Dirac wave packets, and find that the gravitational spin-orbit coupling plays a key role in the gravitational spin Hall effect going against the notions of quantum WEP. Finally, discussion and summary are presented.", "pages": [ 1, 2 ] }, { "title": "II. DIRAC EQUATION IN A STATIC GRAVITATIONAL FIELD AND FOLDY-WOUTHUYSEN PICTURE", "content": "Let us start with Dirac equation in a curved spacetime [51], which describes the dynamics of a spin-1/2 particle in a gravitational field ( ℏ = c = 1), Hereafter Latin indices, ( a, b, c . . . ), denote local flatspacetime indices and run from 0 to 3. Greek indices, ( µ, ν, . . . ) denote curved-spacetime indices and run from 0 to 3. Three-vectors are denoted by bold letters and their components are labeled by Latin indices from the middle of the alphabet, ( i, j, k, . . . ). Einstein summation convention is used. The spacetime metric g µν and the local flat metric η ab are connected by the tetrad field g µν = e a µ e b ν η ab , which satisfies the orthogonality conditions e µ a e a ν = δ µ ν , e a µ e µ b = δ a b . We adopt the flat metric η ab = diag(+ , -, -, -). The covariant derivative D a for the spinor field is given by Here σ ab = i [ γ a , γ b ] / 2 with [ ... , ... ] denoting the commutator, and the spin connection obey ω ab µ = -ω ba µ which can be expressed as with the affine connection Γ ν µλ and the tetrad field. The affine connection is constructed by the so-called Christoffel symbol Γ λ µν = g λσ ( ∂ µ g νσ + ∂ ν g σµ -∂ σ g µν ) / 2 from the spacetime metric g µν . Throughout this paper, we choose the Dirac representation of gamma matrices: where σ i is the Pauli matrices. In this paper, we discuss only a Dirac particle traveling in a static uniform gravitational field. Following the notion of [34, 35], we consider the static uniform spacetime where Here the gravitational acceleration g along the positive direction of z -axis. Under such metric, Eq.(2) can be rewritten in a Schrodinger-like form i∂ t ψ = Hψ . The Hamiltonian has the form Here α i = γ 0 γ i , β = γ 0 , p = -i ℏ ∇ and { ... , ... } denotes the anticommutator. Strictly speaking, the Dirac equation is self-consistent only in the context of quantum field theory due to creation and annihilation of the particle-antiparticle pairs. Fortunately, the effect of particle-antiparticle pair creation or annihilation is negligible in the non-relativistic limit of the Dirac Hamiltonian. In order to obtain the non-relativistic physical interpretation of the Dirac Hamiltonian, there is a need of carrying out the F-W transformation, which is a systematic way of separating the positive and the negative energy states. In the FWrepresentation, the Hamiltonian and all operators are block-diagonal and the operators are exactly analogous to the ones in the non-relativistic quantum theory. We start from the Hamiltonian (5) in the form with where E denotes the even operator commuting with β , E β = β E , and O denotes the odd operator, O β = -β O . After two successive standard F-W transformations, the Dirac Hamiltonian can be put into the form [52] Substituting Eq.(7) explicitly and performing the operator products to the first order of g and 1 /m , we obtain where Σ = ( σ 23 , σ 31 , σ 12 ) is the spin operator (or Σ k = ε ijk σ ij / 2). This F-W Hamiltonian is in agreement with [36], and confirms the weak-field and non-relativistic approximation of [35]. The last term on the right-hand side of (9) describe a gravitational (inertial) spin-orbit coupling, first introduced in [36]. This coupling plays a crucial role in determining gravitational spin Hall effect of Dirac wave packets in a static uniform gravitational field, as we will discuss in the following section.", "pages": [ 2, 3 ] }, { "title": "III. GRAVITATIONAL SPIN HALL EFFECT OF DIRAC PARTICLE AND THE WEAK EQUIVALENCE PRINCIPLE", "content": "In this core section of this paper, we analyze the full quantum dynamics of the gravitational spin Hall effect of Dirac particle in terms of quantum-mechanical postulates. We start from the Schrodinger equation in the F-W representation (9) Since H FW independent of time, the general solution to Eq.(10) can be represented as as the initial wavefunction ψ ( x , 0) is known. The wavefunction at some later time t is governed by the timeevolution operator U ( t, 0) = U = e -i H FW t on the initial wavefunction. Our motivation is to look for whether the WEP is violated if considering spin effect of Dirac particles in a gravitational filed. For this purpose, let us consider Dirac particles of opposite spin-polarization with the same initial mean position and momentum falling freely in our static uniform gravitational field and compare their evolution. For concreteness, the initial states of particles are represented by the following normalized gaussian wave packets, respectively, with the normalized gaussian wave packet and the spinors Here the letter T represents the standard transposition operation. ψ + ( x , 0) and ψ -( x , 0) describe the particle of spin along the positive and negative y -direction initially well localized in position and momentum with the expected values ⟨ x ⟩ = 0 and ⟨ p ⟩ = 0, respectively, as depicted in Fig.(1). Weturn now to the calculation of the expected position of the wave packets of opposite spin-polarization in a uniform gravitational field: There is a need for calculating the operator U -1 x U , but it is difficult to solve the exact solution because of the non-commutation of operators. If considering the weakfield and non-relativistic approximation, U -1 x U can be written via Baker-Hausdorff lemma [53] as Using (9) and keeping operator products to the first order of g and 1 /m , we can deduce the following commutation rules: Substituting these commutation rules back into Eq.(16), and performing the integration of (15) with the initial wave packets (12), we can get Such results can be rewritten in a compact form of 3vectors as where σ = +1 for particle's spin along the positive ydirection and σ = -1 for particle's spin along the negative y-direction, respectively. Without considering the quantum fluctuations, the classical trajectory x ( t ) = g t 2 / 2 is recovered in the semi-classical limit ℏ → 0. From Eq.(25), we can find that the wave packets of opposite spin-polarized Dirac particles in free fall can be split transversely in the direction perpendicular to spin and gravity, as illustrated in Fig.(1), which is known as the gravitational spin Hall effect. Free-fall Dirac particles with different spin orientations follow 'different paths'. Besides, unlike the classical case, the expected trajectory of the quantum particle in free fall is dependent on its mass, and the quantum particle with spin polarization is able to fall freely in a different 'path structure'. In these senses, this effect implies a kind of violations of quantum WEP induced by the spin of quantum particle. Restoring explicit factors ℏ and c , we can get the transverse shift from the origin point along x -axis by an amount Here, λ is the reduced Compton wavelength. Although this effect originates from the wave feature of quantum particle, the split is due to the geodesic deviation out of gravitational (inertial) spin-orbit coupling, existing even in a uniform gravitational field. In addition, this effect implies the entanglement between the internal (spin) and external (position) degrees of freedom of the wave packets. In order to analyze the detail dynamical evolution of wavefunction and the detail entanglement of the internal and external degrees of freedom, we can try to look for the solution (11) of the wavefunction at time t . The F-W Hamiltonian can be written as with defining the free Hamiltonian and the interaction Hamiltonian We can deduce the following commutation rules to the first order in term of 1 /m and g : Using these commutation rules and the Baker-CampbellHausdorff (BCH) formula [54], we can factorize the time evolution operator as Thus, the evolution of the wavefunction will become Here ψ f ± ( x , t ) = e -i H f t ψ ± ( x , 0) = e -imt e -it p 2 2 m ψ ± ( x , 0) describes the free evolution of the initial wavefunction ψ ± ( x , 0) without gravitational interaction, which can be written explicitly as [55] where Again, e -i H g t can also be further factorized by the BCH formula as Inserting Eq.(35) into Eq.(32) by elementary calculation, we can obtain the approximation of the wavefunction ψ ± ( x , t ) as follows with Equation (36) confirms the evolution towards an entangled state between the internal (spin) and external (position) degrees of freedom. In fact, with the wavefunction for general time t (36), straightforward calculation can verify the expected position as The probability distribution of position In our case, since the mass dependence of u and v and the spin-position entanglement, the probability distribution is mass-dependent, and the functional mass-dependence of probability distribution is different from that of the free case in the absence of gravity. From this point, we can conclude that the notion of quantum WEP in Ref. [30] is invalid, even in a static uniform gravitational field. ̸ In addition, from the view of the Fisher information framework, one can extract mass information of Dirac particle in free fall with spin polarization, even in a static uniform gravitational field, namely F g x ( m ) = F f x ( m ). Thus it also imply the violation of the notion of quantum WEP in terms of Fisher information. In this sense, the gravitational spin Hall effect can be as a new probe to test the quantum version of WEP's notion.", "pages": [ 3, 4, 5 ] }, { "title": "IV. DISCUSSION AND SUMMARY", "content": "In conclusion, we have revealed the gravitational spin Hall effect holding simultaneously quantum and gravitational effects, even in a static uniform gravitational field. For the free-fall Dirac wave packets with opposite spin polarization, such remarkable effect suggests the transverse splitting in the direction perpendicular to spin orientation and gravity. In the F-W picture, we analyze the dynamical evolution of the free-fall Dirac wave packets and show that the gravitational spin Hall effect produces the entanglement between the internal and external degrees of freedom due to the gravitational spin-orbit coupling. Interestingly, the gravitational spin Hall effect manifest the violation of the quantum WEP's notion presented recently, even in the very simple case of a static uniform gravitational field. To test the gravitational spin Hall effect will be of great interest and importance for the possible observation of the violation of WEP in the quantum realm. However, as showed previously, the shift is an order of magnitude smaller than the Compton wavelength of the Dirac particle. It does not seem feasible to detect this effect by the current detectors. If the future detectors realizing high spatial resolution and the free-fall particles traveling enough time, possible future observations of this effect could be as a new probe of WEP of quantum systems, so as to clarify the notion of quantum WEP.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGES", "content": "This work is supported by the Scientific Research Project of Hubei Polytechnic University (Project No. 20xjz02R).", "pages": [ 5 ] } ]

[ { "title": "Gravitational Lensing by Born-Infeld Naked Singularities", "content": "Yiqian Chen a , ∗ Peng Wang a , † Houwen Wu a,b , ‡ and Haitang Yang a § a Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610064, China and b Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK We examine the gravitational lensing phenomenon caused by photon spheres in the BornInfeld naked singularity spacetime, where gravity is coupled with Born-Infeld electrodynamics. Specifically, our focus lies on relativistic images originating from a point-like light source generated by strong gravitational lensing near photon spheres, as well as images of a luminous celestial sphere. It shows that Born-Infeld naked singularities consistently exhibit one or two photon spheres, which project onto one or two critical curves on the image plane. Interestingly, we discover that the nonlinearity nature of the Born-Infeld electrodynamics enables photons to traverse the singularity, leading to the emergence of new relativistic images within the innermost critical curve. Furthermore, the presence of two photon spheres doubles the number of relativistic images compared to the scenario with only a single photon sphere. Additionally, the transparency inherent to Born-Infeld naked singularities results in the absence of a central shadow in the images of celestial spheres.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Gravitational lensing, the phenomenon of light bending in curved space, is a captivating and fundamental effect predicted by general relativity [1-3]. Due to its pivotal role in astrophysics and cosmology, extensive research has been conducted on gravitational lensing in the past decades. It has contributed significantly to addressing crucial topics such as the distribution of structures [4-6], dark matter [7-9], dark energy [10-13], quasars [14-17], and gravitational waves [18-20]. In an idealized lens model involving a distant source in a Schwarzschild black hole, the slight deflection of light in a weak gravitational field gives rise to the observation of a primary and a secondary image. Moreover, strong gravitational lensing near the photon sphere generates an infinite sequence of higher-order images, known as relativistic images, on both sides of the optic axis [21]. Remarkably, relativistic images are minimally affected by the characteristics of the astronomical source, making them valuable for investigating the nature of the black hole spacetime. Recently, the remarkable achievement of high angular resolution by the Event Horizon Telescope collaboration [22-35], has facilitated the study of gravitational lensing in the strong gravity regime, reigniting interest in the shadow of black hole images and the associated phenomenon of strong gravitational lensing [36-62]. It has been demonstrated that strong gravitational lensing exhibits a close connection to bound photon orbits, which give rise to photon spheres in spherically symmetric black holes. Intriguingly, certain horizonless ultra-compact objects have been discovered to harbor photon spheres, effectively mimicking black holes in numerous observational simulations [63-72]. Among these objects, naked singularities have garnered significant attention. Although the cosmic censorship conjecture forbids the formation of naked singularities, it is possible for these entities to arise through the gravitational collapse of massive objects under specific initial conditions [7379]. Given that the presence of photon spheres allows naked singularities to emulate the optical appearance of black holes, the gravitational lensing phenomena associated with naked singularities have been extensively investigated [80-87]. In the context of Reissner-Nordstrom (RN) naked singularities characterized by a mass M and charge Q , it is noteworthy that a photon sphere exists only if 1 < Q/M ≤ √ 9 / 8, whereas no photon sphere is present when Q/M > √ 9 / 8. The phenomenon of strong gravitational lensing by the photon sphere has been investigated within the spacetime of RN naked singularities [85, 87]. Analogous to the case of black holes, two sets of relativistic images can be observed beyond the critical curve, which arises from photons originating from the photon sphere. Remarkably, two additional sets of brighter relativistic images have been identified within the critical curve due to the absence of an event horizon and the existence of a potential barrier near the anti-photon sphere. Furthermore, as demonstrated below, when a celestial sphere illuminates an RN naked singularity, the absence of a shadow at the center of the image is observed since light rays entering the photon sphere are reflected by the potential barrier at the singularity. These distinctive observational characteristics can serve as means to differentiate between RN singularities and RN black holes. The Born-Infeld electrodynamics was initially proposed to regulate the divergences arising from the electrostatic self-energy of point charges, achieved through the introduction of an electric field cutoff [88]. Subsequently, it was recognized that Born-Infeld electrodynamics can emerge from the low-energy limit of string theory, which describes the dynamics of D-branes at low energies. Coupling the Born-Infeld electrodynamics field to gravity, the Born-Infeld black hole solution was obtained in [89, 90]. Since then, a multitude of properties pertaining to Born-Infeld black holes have been extensively examined [91-103]. More recently, it has been reported that Born-Infeld naked singularity solutions can exhibit two photon spheres within specific parameter ranges [104]. The primary objective of this paper is to investigate gravitational lensing phenomena exhibited by Born-Infeld naked singularities. Remarkably, our findings unveil the ability of photons to traverse these singularities, which, in conjunction with the presence of double photon spheres, gives rise to distinct observational signatures. The subsequent sections of this paper are structured as follows: In Section II, we provide a concise overview of the Born-Infeld naked singularity solution and present their domain of existence. Section III focuses on the analysis of photon trajectories in an effective geometry and explores their behavior in proximity to the singularity. The discussion then proceeds to examine relativistic images of a distant light source in Section IV, followed by the analysis of images produced by a luminous celestial sphere in Section V. Finally, Section VI presents our conclusions. We adopt the convention 16 πG = c = 1 throughout the paper.", "pages": [ 2, 3, 4 ] }, { "title": "II. BORN-INFELD NAKED SINGULARITY", "content": "We consider a (3 + 1) dimensional gravity model coupled to a Born-Infeld electromagnetic field A µ . The action S is given by where Here, s and p are two independent nontrivial scalars constructed from the field strength tensor F µν = ∂ µ A ν -∂ ν A µ and none of its derivatives, i.e., where ϵ µνρσ ≡ -[ µ ν ρ σ ] / √ -g is a totally antisymmetric Lorentz tensor, and [ µ ν ρ σ ] denotes the permutation symbol. The coupling parameter a is related to the string tension α ' as a = (2 πα ' ) 2 . In the limit a → 0, the Born-Infeld Lagrangian L ( s, p ) reduces to the Lagrangian of the Maxwell field. The equations of motion can be obtained by varying the action (1) with respect to g µν and A µ , yielding where T µν is the energy-momentum tensor, The spherically symmetric ansatz yields a solution to the equations of motion (4) [89, 90, 104]. The metric is given by where The mass, electrical charge, and magnetic charge of the black hole are denoted by M , Q and P , respectively, and 2 F 1 ( a, b, c ; x ) is the hypergeometric function. Moreover, the solution appears to have a singularity at r = 0. The nature of the singularity is investigated using the Kretschmann scalar K = R µνρσ R µνρσ . Our calculation reveals that the origin is a physical singularity as Thus, the solution (6) describes a naked singularity at r = 0 or a black hole if an event horizon exists. To determine the separatrix between naked singularity and black hole solutions, we investigate extremal black holes with horizon radius r e and mass M e . The conditions f ( r e ) = 0 = d ( rf ( r )) /dr | r = r e yield the expressions for r e and mass M e as It is evident that extremal black holes do not exist for a > 4 ( Q 2 + P 2 ) . So when a < 4 ( Q 2 + P 2 ) and M < M e , the spacetime is a naked singularity. However, when a > 4 ( Q 2 + P 2 ) , the spacetime can have at most one horizon. The presence of the horizon can be determined by investigating rf ( r ), which vanishes at the horizon radius. In fact, one finds d ( rf ( r )) /dr > 0 and lim r → 0 rf ( r ) = 4 [ a ( Q 2 + P 2 )] 3 / 4 Γ(1 / 4) Γ (5 / 4) / (3 a √ π ) -2 M , indicating the appearance of a naked singularity when The left panel of FIG. 1 shows the domain of existence for Born-Infeld naked singularities in the a/M 2 -√ Q 2 + P 2 /M parameter space, with the dashed black line denoting the separatrix between the black hole and naked singularity solutions. Solutions above the dashed black line in the colored regions represent Born-Infeld naked singularities.", "pages": [ 4, 5, 6 ] }, { "title": "III. PHOTON TRAJECTORIES", "content": "Nonlinear electrodynamics theories allow for self-interaction of the electromagnetic field, leading to changes in the direction of photon propagation and deviation from null geodesics. Propagation equations describing photon trajectories can be obtained by analyzing the electromagnetic field's discontinuity at the characteristic surface of wave propagation. An effective metric is then introduced, in which photons travel on null geodesics [105]. In one-parameter theories with the Lagrangian as a function of s , a single effective geometry determines the photon trajectories. However, in two-parameter theories with the Lagrangian as a function of s and p , two possible solutions exist, leading to birefringence. The Born-Infeld theory, on the other hand, ensures the uniqueness of the photon path via its equations of motion, and the effective metric ˜ g µν is given by [105] Using the underlying Born-Infeld metric (6), the effective metric takes the form where ˜ g µρ ˜ g ρν = δ ν µ , and Although the effective metric appears to lack electric-magnetic duality, this symmetry is present when the metric is multiplied by a conformal factor ( aP 2 + r 4 ) -2 , which does not alter null geodesics. Therefore, the electric-magnetic duality of photon trajectories is expected. In the Hamiltonian canonical formalism, a photon with 4-momentum vector p µ = ( ˙ t, ˙ r, ˙ θ, ˙ φ ), where dots stand for derivative with respect to some affine parameter λ , has canonical momentum q µ = ˜ g µν p ν , which satisfies the null condition p µ q µ = 0. The null geodesic equations in the effective metric (12) are separable and can be fully characterized by three conserved quantities, which denote the total energy, the angular momentum parallel to the axis of symmetry, and the total angular momentum, respectively. Here, the tensor K µν is an symmetric Killing tensor Note that ˜ ∇ ( λ K µν ) = 0, where ˜ ∇ is the covariant derivative compatible with the effective metric. The canonical 4-momentum q = q µ dx µ can be expressed in terms of E , L z and L as where the two choices of sign ± r and ± θ depend on the radial and polar directions of travel, respectively. Then, null geodesic equations are given by p µ = ˜ g µν q ν , i.e., where b ≡ L/E is the impact parameter, and the effective potential of photons in the effective metric is defined as In the study of black holes, the near-singularity behavior of photons that have entered the horizon is usually not considered. However, since naked singularities lack horizons, it is crucial to investigate the behavior of photons in the vicinity of the singularities. Eqn. (17) gives the behavior of photons around the singularity at r = 0, yielding which implies that a photon can pass through the singularity within a finite coordinate time. Furthermore, we find that, near the singularity, solutions of the null geodesic equations (17) can be expanded as where x µ 0 are the constant of integration, and the coefficients c µ nm are calculated recursively order by order. Particularly, the leading coefficients are given by As a light ray traverses the singularity, it splits into two branches, namely the radially outgoing one associated with the upper sign of ± r and ∓ r , and the radially ingoing one with the lower sign. We adopt λ > 0 and λ < 0 for the ingoing and outgoing branches, respectively. It is worth emphasizing that the affine parameter approaches ±∞ when the light ray approaches the singularity, leading to x r 0 = 0. At the singularity, the ingoing and outgoing branches are connected by the conditions The effective potential V eff ( r ) determines the locations of circular light rays, with unstable and stable light rays corresponding to local maxima and minima, respectively. The unstable circular light rays form photon spheres, which are critical for observing black holes. From eqn. (12), it follows that V eff ( ∞ ) = 0 = V eff (0) when a > 0, indicating the existence of at least one photon sphere in the Born-Infeld naked singularity spacetime. The left panel of FIG. 1 illustrates the regions where one or two photon spheres exist in the a/M 2 -√ Q 2 + P 2 /M parameter space: The effective potential V eff ( r ) of RN naked singularities with a = 0 possesses a photon sphere provided that 1 < √ Q 2 + P 2 /M < √ 9 / 8 (e.g., indicated by the black line in the upper-right panel of FIG. 1), while no photon sphere exists when √ Q 2 + P 2 /M ≥ √ 9 / 8 (e.g., the black line in the lower-right panel of FIG. 1) [113]. In addition, V eff ( r ) diverges at r = 0 for RN naked singularities, preventing photons from reaching the singularity. Interestingly, when the effects of nonlinear electrodynamics are present, V eff ( r ) approaches zero instead of infinity as r → 0, enabling photons with a sufficiently small impact parameter to overcome the potential barrier and reach the singularity.", "pages": [ 6, 7, 8, 9 ] }, { "title": "IV. RELATIVISTIC IMAGES", "content": "In this section, we explore the phenomenon of gravitational lensing caused by Born-Infeld naked singularities in the context of the strong deflection limit. Our analysis starts with determining the deflection angle, which allows us to derive the angular positions of relativistic images. We employ an idealized thin lens model that assumes a high degree of alignment among the source, lens and observer. The lens equation, as presented in [21], is expressed as where β represents the angular separation between the source and the lens, ϑ denotes the angular separation between the lens and the image, and ∆ α represents the offset of the deflection angle after accounting for all the windings experienced by the photon. Here, the distances D OL , D LS and D OS correspond to the observer-lens, lens-source and observer-source distances, respectively. For the sake of simplicity, we confine our analysis to the equatorial plane, taking advantage of the spherical symmetry. In the idealized model, the deflection angle α ( b ) is described by the following expression from [21], where I ( b ) represents the change in φ , and b denotes the impact parameter related to ϑ through the equation b = D OL ϑ . When a photon approaches a turning point at r = r 0 and then gets deflected towards a distant observer, the integral I ( b ) is given by Alternatively, if the photon passes through the singularity at r = 0, the azimuthal angle φ increases by π , resulting in the expression, In the strong deflection limit, the integral I ( b ) diverges as the impact parameter b approaches the critical value b c , which represents the impact parameter for photon trajectories on the photon sphere at r = r c . By expanding I ( b ) around b = b c (or, equivalently r 0 = r c ), we can obtain α ( b ) in the strong deflection limit.", "pages": [ 10 ] }, { "title": "A. Single Photon Sphere", "content": "We first study strong gravitational lensing in a Born-Infeld naked singularity with a single photon sphere. When the impact parameter of photons approaches the critical value, whether from below or above, they undergo significant deflections. It is important to highlight that photons can traverse the naked singularity if their impact parameter is smaller than b c . Moreover, photons can orbit the photon sphere in either a clockwise or counterclockwise direction. Consequently, a distant source yields four relativistic images of n th -order, where n is a specified value. Photons with b > b c reach a turning point r 0 , located just outside the photon sphere r c , as depicted by the red lines in FIG. 2. In such cases, the deflection angle in the regime of strong lensing is described by the equation [114, 115] where Here, the term I R ( r c ) represents a regular integral that can be computed numerically. In the idealized lens model, the angular position of the image is related to the impact parameter by b = D OL ϑ . By utilizing the deflection angle formula (27) in conjunction with the lens equation (23), one can solve for the angular position ϑ > ± n for n th -order relativistic images produced by photons orbiting the photon sphere n times. It is noteworthy that -and + in the subscript of ϑ > ± n signify counterclockwise and clockwise orbits, respectively, while the superscript > indicates photons with b > b c . Specifically, the angular position ϑ > ± n is given by [114] where e n = e ¯ b -2 πn ¯ a , and ϑ > 0 ± n , satisfying α ( ϑ > 0 ± n ) = ± 2 nπ , is given by When b < b c , photons emitted from the source can pass through the singularity, resulting in the generation of relativistic images within the critical curve, as depicted by the green lines in FIG. 2. To facilitate the derivation of the integral (26), we introduce a variable z defined as The integral I ( b ) can then be expressed as where Note that A ( z ) is a regular function of z , whereas D ( z, b ) diverges at z = 0 as b → b c . Hence, we decompose the integral I ( b ) into a divergent part I D ( b ) and a regular part I R ( b ), as follows, Here, we employ a Taylor expansion within the square root in D ( z, b ) to obtain where γ and η are the expansion coefficients. Consequently, the divergent part I D ( b ) is given by Since the coefficient γ approaches zero as b → b c , the deflection angle in the strong limit is obtained by expanding I D ( b ) around b = b c , where Similarly, the angular position of n th -order relativistic images is given by where the angles ϑ < 0 n and ϑ < 0 -n are defined as To obtain numerical estimations of ϑ ≷ ± n in an astrophysical setting, we consider a Born-Infeld naked singularity with parameters corresponding to the supermassive black hole Sgr A* located at the center of our Galaxy. Specifically, we assume a mass of M = 4 . 31 × 10 6 M ⊙ and a lens-source distance of D OL = 7 . 86 kpc. Additionally, a source is positioned at D LS = 7 . 86 kpc with an angular separation of β = 2 · . For Born-Infeld naked singularities with √ P 2 + Q 2 /M = 1 . 2 and various values of a/M 2 , Table I presents ∆ ϑ ≷ ± n ≡ ϑ ≷ ± n -ϑ ±∞ , where ϑ ±∞ = lim n →∞ ϑ ≷ 0 ± n = ± b c /D OL is the angular position of the relativistic image formed at the photon sphere. Note that the corresponding effective potentials of the singularities are displayed in FIGs. 1 and 2. The results demonstrate that as the nonlinear parameter a increases, the potential peak becomes less pronounced, leading to larger values of ∆ ϑ ≷ ± n . This, in turn, facilitates the resolution of higher-order relativistic images. Moreover, the relativistic images with b < b c are more widely separated compared to those with b > b c due to the significant bending of light rays upon entering or exiting the photon sphere. Considering a resolution of 0 . 01 microarcseconds, which is capable of resolving the first-order relativistic image in a Schwarzschild black hole [85, 114, 116], it is observed that relativistic images associated with the singularity having a/M 2 = 0 . 2 are too closely spaced to be resolved. However, all n = 1 relativistic images of the singularity with a/M 2 = 1 . 5, as well as n ≤ 3 images of the singularity with a/M 2 = 2, can be distinguished.", "pages": [ 11, 12, 13, 14 ] }, { "title": "B. Double Photon Spheres", "content": "In the presence of a Born-Infeld naked singularity with a double-peak effective potential, two photon spheres are observed at distinct locations, namely, r = r in and r = r out , with r in FIG. 3. Left : The effective potential of a Born-Infeld naked singularity with √ P 2 + Q 2 /M = 1 . 2 and a/M 2 = 0 . 2. Notably, it displays two peaks corresponding to the inner photon sphere at r in = 0 . 2919 M and the outer one at r out = 1 . 7833 M . The horizontal lines denote b -2 of light rays in the right panel, which undergo significant lensing effects near the outer photon sphere with the impact parameter b out = 3 . 8021 M . Right : Light rays are depicted by red and green lines, indicating those with b > b out and b < b out , respectively. The blue dashed lines represent the photon spheres, while the solid and dashed lines demonstrate light rays orbiting once and twice around the outer photon sphere, respectively. FIG. 3. Left : The effective potential of a Born-Infeld naked singularity with √ P 2 + Q 2 /M = 1 . 2 and a/M 2 = 0 . 2. Notably, it displays two peaks corresponding to the inner photon sphere at r in = 0 . 2919 M and the outer one at r out = 1 . 7833 M . The horizontal lines denote b -2 of light rays in the right panel, which undergo significant lensing effects near the outer photon sphere with the impact parameter b out = 3 . 8021 M . Right : Light rays are depicted by red and green lines, indicating those with b > b out and b < b out , respectively. The blue dashed lines represent the photon spheres, while the solid and dashed lines demonstrate light rays orbiting once and twice around the outer photon sphere, respectively. similar to the single-peak scenario. However, when the height of the inner peak surpasses that of the outer peak, a distant source can generate a total of eight n th -order relativistic images due to strong gravitational lensing near the inner and outer photon spheres. The light rays responsible for these relativistic images are categorized based on their impact parameter b , · For b < b in , depicted by the green lines in FIG. 2, where the potential peak is treated as the inner one. The light rays emitted from the source pass through the singularity and produce two relativistic images at ϑ = ϑ in < ± n , with the minus ( -) and plus (+) signs representing the counterclockwise and clockwise directions, respectively. · For b > b in , illustrated by the red lines in FIG. 2. The light rays reach a turning point r 0 just outside the inner photon sphere before escaping towards the observer, generating two relativistic images at ϑ = ϑ in > ± n . · For b < b out , shown by the green lines in FIG. 3. The light rays are reflected at r = r 0 by the potential barrier between the two photon spheres, producing two relativistic images at ϑ = ϑ out < ± n . · For b > b out , demonstrated by the red lines in FIG. 3. The light rays reach a turning point r 0 slightly outside the outer photon sphere, resulting in two relativistic images at ϑ = ϑ out > ± n . Note that the angular position of the images, ϑ out >, in > ± n and ϑ in < ± n , can be computed using eqns. (29) and (39), respectively. Moreover, the deflection angle of light rays with b < b out has been previously shown to be [85] α ( b ) = -¯ a log ( b 2 out /b 2 -1 ) + ¯ b + O (( b out /b -1) ln ( b out /b -1)) , (41) where ¯ a = 2 √ 2 f ( r m ) h ( r out ) [ R '' ( r m ) f ( r m ) -R ( r m ) f '' ( r m )] , ¯ b = ¯ a log [ r 2 m ( r m r out -1 )( R '' ( r m ) R ( r m ) -f '' ( r m ) f ( r m ) )] + I R ( r out ) -π. (42) Here, r m is the critical turning point when b is very close to b out . Since the divergent parts of the deflection angles (37) and (41) share the same form, one can utilize eqn. (39) to calculate ϑ out < ± n with the values of ¯ a and ¯ b from eqn. (42). Similarly, in the aforementioned astrophysical scenario of Born-Infeld naked singularities with a single photon sphere, the angular positions of the relativistic images can be estimated numerically. Specifically, we present the values of ∆ ϑ in ≷ ± n ≡ ϑ in ≷ ± n -ϑ in ±∞ and ∆ ϑ out ≷ ± n ≡ ϑ out ≷ ± n -ϑ out ±∞ for a/M 2 = 0 . 2, 1 and 1 . 5 in Table II. Here, ϑ in ±∞ and ϑ out ±∞ are the angular positions of the relativistic images formed at the inner and outer photon spheres, respectively. We also assume a resolution of 0 . 01 microarcseconds, which enables the resolution of the first-order image in a Schwarzschild black hole. For a singularity with a/M 2 = 0 . 2, the inner potential peak is significantly sharper and higher than the outer peak. Consequently, the relativistic images at ϑ = ϑ in ≷ ± n are closely situated near the inner critical curve at ϑ = ϑ in ±∞ , making them indistinguishable on the image plane. However, the images near the outer critical curve at ϑ = ϑ out ±∞ are well-separated from the outer critical curve, allowing for the distinction of the first-order images with b > b out and the first three orders ( n ≤ 3) images with b < b out . When a/M 2 = 1, the inner potential peak becomes flatter, resulting in greater separation among the images near the inner critical curve. It should be emphasized that, due to the inner potential peak not being significantly higher than the outer peak, the formulas for ϑ in > ± n and ϑ out < ± n in the strong deflection limit may exhibit substantial errors. For the singularity with TABLE II. Angular separation of relativistic images near the inner and outer critical curves in Born-Infeld naked singularities with double photon spheres. The parameters M , D OL , D LS , β and √ P 2 + Q 2 /M are chosen to be consistent with those presented in Table I. Left : The angular separation ∆ ϑ in ≷ ± n = ϑ in ≷ ± n -ϑ in ±∞ between n th -order relativistic images near the inner critical curve and the relativistic image formed at the inner photon sphere. Right : The angular separation ∆ ϑ out ≷ ± n = ϑ out ≷ ± n -ϑ out ±∞ between n th -order relativistic images near the outer critical curve and the relativistic image formed at the outer photon sphere. TABLE II. Angular separation of relativistic images near the inner and outer critical curves in Born-Infeld naked singularities with double photon spheres. The parameters M , D OL , D LS , β and √ P 2 + Q 2 /M are chosen to be consistent with those presented in Table I. Left : The angular separation ∆ ϑ in ≷ ± n = ϑ in ≷ ± n -ϑ in ±∞ between n th -order relativistic images near the inner critical curve and the relativistic image formed at the inner photon sphere. Right : The angular separation ∆ ϑ out ≷ ± n = ϑ out ≷ ± n -ϑ out ±∞ between n th -order relativistic images near the outer critical curve and the relativistic image formed at the outer photon sphere. a/M 2 = 1 . 5, the inner peak is lower than the outer peak, leading to the existence of relativistic images solely near the outer critical curve. V. CELESTIAL SPHERE IMAGES In this section, we investigate gravitational lensing by a Born-Infeld naked singularity through the image of a luminous celestial sphere centered at the singularity and surrounding the observers. To obtain the image of the celestial sphere, we use the numerical backward ray-tracing method to calculate light rays from the observer to the celestial sphere. Light rays can be described either by numerically integrating eqn. (17) or eight first-order differential equations dx µ dλ = p µ , dp µ dλ = -˜ Γ µ ρσ p ρ p σ , (43) where λ represents the affine parameter, and ˜ Γ µ ρσ are the Christoffel symbols compatible with the effective metric. Numerically solving eqn. (43) for light rays enables us to avoid the need to account for turning points during the integration, resulting in improved numerical accuracy. Thus, we use eqn. (43) to calculate the light rays connecting the observer with the celestial sphere. For a static observer located at ( t o , r o , θ o , φ o ), we introduce a tetrad basis e ( t ) = ∂ t √ -g tt ( r o , θ o ) , e ( r ) = ∂ r √ g rr ( r o , θ o ) , e ( θ ) = ∂ θ √ g θθ ( r o , θ o ) , e ( φ ) = ∂ φ √ g φφ ( r o , θ o ) , (44) which span the tangent bundle at the observer. To obtain initial conditions for eqn. (43), a photon captured by the observer is considered, whose local 4-momentum ( p ( t ) , p ( r ) , p ( θ ) , p ( φ ) ) in the tetrad basis is related to the 4-momentum p µ o = dx µ /dλ | ( t o ,r o ,θ o ,φ o ) as p ( t ) = √ f BI ( r o ) p t o , p ( r ) = p r o / √ f BI ( r o ) , p ( θ ) = r o p θ o , p ( φ ) = r o | sin θ o | p φ o . (45) The observation angles Θ and Φ, as defined in [117], are given by sin Θ = p ( θ ) p , tan Φ = p ( φ ) p ( r ) , (46) which p = √ p ( r )2 + p ( θ )2 + p ( φ )2 . We express p ( r ) , p ( θ ) and p ( φ ) in terms of p , Θ and Φ as p ( r ) = p cos Θ cos Φ , p ( θ ) = p sin Θ , p ( ϕ ) = p cos Θ sin Φ . (47) Moreover, the condition ˜ g µν p µ o p ν o = 0 and eqn. (45) give p ( t ) = p √ f BI ( r o ) f ( r o ) √ f BI ( r o ) h ( r o ) cos 2 Θcos 2 Φ+ R ( r o ) r 2 o ( sin 2 Θ+sin 2 Φcos 2 Θ ) . (48) Using eqn. (45), we can rewrite p µ o in terms of p , Θ and Φ, which, together with the coordinates of the observer, provide initial conditions for eqn. (43). Without loss of generality, we set p = 1 in what follows. The Cartesian coordinates ( x, y ) of the image plane of the observer is defined by x ≡ -r o Φ , y ≡ r o Θ , (49) where the sign convention for Φ leads to the minus sign in the x definition. Note that the direction pointing to the singularity corresponds to the zero observation angles (0 , 0). As discussed previously, when a light ray travels through the singularity, its affine parameter λ becomes divergent at r = 0, posing a challenge for our numerical implementation. To circumvent this issue, we introduce a small sphere of radius Mϵ to enclose the singularity. Outside the sphere, we can solve eqn. (43) numerically, ensuring the accuracy and stability of the light ray calculation. FIG. 4. The relative error ( φ ϵ -φ 0 ) /φ 0 as a function of ϵ for a light ray emitted from ( r e , φ e ) = (25 M,φ ϵ ) and arriving at ( r o , φ o ) = (10 M,π ) with (Θ , Φ) = (0 , 3 / 20) on the equatorial plane. The light ray passes through the singularity, and the ingoing and outgoing branches are joined at r = Mϵ during numerical calculations. Here, φ 0 = φ ϵ =10 -3 , a/M 2 = 1 and √ Q 2 + P 2 /M = 1 . 05. FIG. 4. The relative error ( φ ϵ -φ 0 ) /φ 0 as a function of ϵ for a light ray emitted from ( r e , φ e ) = (25 M,φ ϵ ) and arriving at ( r o , φ o ) = (10 M,π ) with (Θ , Φ) = (0 , 3 / 20) on the equatorial plane. The light ray passes through the singularity, and the ingoing and outgoing branches are joined at r = Mϵ during numerical calculations. Here, φ 0 = φ ϵ =10 -3 , a/M 2 = 1 and √ Q 2 + P 2 /M = 1 . 05. Within the sphere, we can use the expansions in eqn. (20) to describe the light ray and provide a connection formula for its entry and exit points from the sphere. Specifically, we consider the light ray entering and leaving the sphere at ( t in , r in , θ in , φ in ) and ( t out , r out , θ out , φ out ), respectively. With eqns. (21) and (22), one has t out = t in + O ( ϵ 2 ) , r out = r in = Mϵ, θ out = π -θ in + O ( ϵ 3 ) , φ out = π + φ in + O ( ϵ 3 ) , p t out = p t in , p r out = -p r in , p θ out = -p θ in + O ( ϵ 5 log ϵ ) , p φ out = p φ in + O ( ϵ 5 log ϵ ) . (50) In this section, we employ the leading terms of eqn. (50) to connect the ingoing and outgoing branches. To explore the numerical error caused by the finite size of ϵ , we investigate a light ray on the equatorial plane of a Born-Infeld naked singularity with a/M 2 = 1 and √ Q 2 + P 2 /M = 1 . 05. The light ray originates from ( r e , φ e ) = (25 M,φ ϵ ), and an observer located at ( r o , φ o ) = (10 M,π ) captures it with observation angles (Θ , Φ) = (0 , 3 / 20). To obtain the coordinate φ ϵ , we trace the light ray backward from the observer to r e = 25 M while connecting the ingoing branch with the outgoing one at r = Mϵ . We present the relative error ( φ ϵ -φ 0 ) /φ 0 as a function of ϵ in FIG. 4, where φ 0 ≡ φ ϵ =10 -3 . To maintain numerical precision and efficiency, we set ϵ = 10 -1 in the following numerical simulations, for which the relative error is well below 10 -3 . To illustrate gravitational lensing by Born-Infeld naked singularities, we position a luminous celestial sphere at r CS = 25 M , while an observer is situated at x µ o = (0 , 10 M,π/ 2 , π ). The celestial sphere is divided into four quadrants, each distinguished by a different color, and a white dot FIG. 5. Observational image of the celestial sphere in the Minkowski spacetime. The observer is positioned at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of 2 π/ 3. FIG. 5. Observational image of the celestial sphere in the Minkowski spacetime. The observer is positioned at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of 2 π/ 3. is placed in front of the observer. Additionally, we overlay a grid of black lines representing constant longitude and latitude, where adjacent lines are separated by π/ 18. To generate an observational image, we vary the observer's viewing angle and numerically integrate 2000 × 2000 photon trajectories until they intersect with the celestial sphere. The resulting image of the celestial sphere in Minkowski spacetime is presented in FIG. 5. FIG. 6 displays images of the celestial sphere in both RN and Born-Infeld naked singularities, both of which possess a single photon sphere. The dashed circular lines in the images correspond to the critical curve formed by light rays originating from the photon sphere. Beyond this critical curve, the images of the celestial sphere in naked singularities bear resemblance to those observed in black hole spacetime. Notably, unlike shadows observed in black hole images, the celestial sphere images persist within the critical curve due to the absence of an event horizon. Additionally, higher-order images of the celestial sphere can be observed both inside and outside this critical curve. The left panel of FIG. 6 illustrates the image of the celestial sphere in a RN naked singularity characterized by √ Q 2 + P 2 /M = 1 . 05. Inside the critical curve, three distinct white rings can be observed. These rings correspond to the Einstein ring generated by the white dot positioned on the celestial sphere. Specifically, the innermost white ring originates from light rays emitted by the white dot, undergoes reflection at the potential barrier, and eventually reaches the observer after experiencing an angular coordinate change of ∆ φ = π . Within this innermost white ring, FIG. 7 showcases images of the celestial sphere in Born-Infeld naked singularities featuring two photon spheres. The inner and outer photon spheres give rise to corresponding inner and outer critical curves, as indicated by the dashed lines in the images. Higher-order celestial sphere images are observed on both sides of these critical curves. Similarly to the case of a single photon sphere, a central white spot appears in the images due to the transparency of the Born-Infeld naked singularity. In the left panel, characterized by √ Q 2 + P 2 /M = 1 . 05 and a/M 2 = 0 . 2, two FIG. 7 showcases images of the celestial sphere in Born-Infeld naked singularities featuring two photon spheres. The inner and outer photon spheres give rise to corresponding inner and outer critical curves, as indicated by the dashed lines in the images. Higher-order celestial sphere images are observed on both sides of these critical curves. Similarly to the case of a single photon sphere, a central white spot appears in the images due to the transparency of the Born-Infeld naked singularity. In the left panel, characterized by √ Q 2 + P 2 /M = 1 . 05 and a/M 2 = 0 . 2, two FIG. 6. Images of the celestial sphere in naked singularities featuring a single photon sphere. The observer is situated at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of π/ 4. The dashed lines depict the critical curve formed by photons escaping from the photon sphere. Left: The RN naked singularity with √ Q 2 + P 2 /M = 1 . 05. The image within the critical curve is generated by light rays that rebound off the infinitely high potential barrier at the singularity. Three white rings, representing the Einstein ring of the white dot on the celestial sphere, can be observed within the critical curve. Right: The Born-Infeld naked singularity with √ Q 2 + P 2 /M = 1 . 2 and a/M 2 = 2. The image within the critical curve is formed by light rays passing through the singularity. A central white dot is visible, surrounded by two white rings. the reflections from the infinitely high potential barrier at the singularity produce a mirror image of the celestial sphere. Moreover, the middle and outermost white rings arise from light rays with angular coordinate changes of ∆ φ = 3 π and 5 π , respectively. In the right panel of FIG. 6, the image captured in a Born-Infeld naked singularity with √ Q 2 + P 2 /M = 1 . 2 and a/M 2 = 2 is displayed. As expected, light rays passing through the singularity and undergoing an angular coordinate change of ∆ φ = π result in a white dot positioned at the center of the image. Additionally, two white rings emerge within the critical curve, representing photons that traverse the singularity with angular coordinate changes of ∆ φ = 3 π and 5 π , respectively. FIG. 7. Images of the celestial sphere in Born-Infeld naked singularities with √ Q 2 + P 2 /M = 1 . 05 for a/M 2 = 0 . 2 ( Left ) and a/M 2 = 1 ( Right ), featuring both inner and outer photon spheres. The observer is located at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of π/ 4, and the corresponding inner and outer critical curves are represented by dashed lines. The image within the inner critical curve is formed by light rays traversing the singularity, while the image between the inner and outer critical curves is a result of light rays reflecting off the potential barrier situated between the inner and outer potential peaks. FIG. 7. Images of the celestial sphere in Born-Infeld naked singularities with √ Q 2 + P 2 /M = 1 . 05 for a/M 2 = 0 . 2 ( Left ) and a/M 2 = 1 ( Right ), featuring both inner and outer photon spheres. The observer is located at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of π/ 4, and the corresponding inner and outer critical curves are represented by dashed lines. The image within the inner critical curve is formed by light rays traversing the singularity, while the image between the inner and outer critical curves is a result of light rays reflecting off the potential barrier situated between the inner and outer potential peaks. white rings can be observed positioned between the inner and outer critical curves. These rings originate from photons that are reflected by the potential barrier located between the inner and outer potential peaks with ∆ φ = 3 π and 5 π . In the right panel, with √ Q 2 + P 2 /M = 1 . 05 and a/M 2 = 1 . 5, a white ring emerges between the two critical curves. This ring arises from photons that undergo reflection at the potential barrier with an angular coordinate change of ∆ φ = 5 π . Additionally, a white ring is observed inside the inner critical curve, which occurs due to photons passing through the singularity and experiencing an angular coordinate change of ∆ φ = 3 π . VI. CONCLUSIONS This paper investigated the phenomenon of gravitational lensing by Born-Infeld naked singularities, which are solutions of a (3 + 1)-dimensional gravity model coupled to a Born-Infeld electromagnetic field. Owing to the nonlinearity inherent in Born-Infeld electrodynamics, photons follow null geodesics of an effective metric, deviating from the background metric, and remarkably, they are capable of traversing naked singularities. Additionally, we demonstrated that Born-Infeld naked singularities can exhibit the presence of two photon spheres with distinct sizes within specific parameter ranges. The existence of these double photon spheres, combined with the transparency of naked singularities, significantly impacts the gravitational lensing of light sources, leading to various effects such as the emergence of new relativistic images. Consequently, these findings provide a potent tool for detecting and studying Born-Infeld naked singularities through their distinctive gravitational lensing signatures. Naked singularities with double photon spheres have been infrequently reported; however, it has been discovered that asymptotically flat black holes can possess two photon spheres outside the event horizon [104, 107, 118]. A recent investigation focused on studying the relativistic images produced by point-like light sources and luminous celestial spheres in the presence of black holes with either a single or double photon spheres [108]. The key findings regarding strong gravitational lensing by black holes and naked singularities can be summarized as follows: · Black holes with a single photon sphere: In the celestial sphere image, a shadow is observed enclosed by the critical curve, which originates from light rays escaping the photon sphere. For a point-like source, two n th -order relativistic images are present just outside the critical curve, corresponding to clockwise and counterclockwise winding around the black hole. · Black holes with double photon spheres: The celestial sphere image exhibits both inner and outer critical curves, formed by the inner and outer photon spheres, respectively. Within the image, there exists a shadow enclosed by the inner critical curve. Two n th -order relativistic images of a point-like source appear just outside the outer critical curve, two images are found just inside the outer critical curve, and two additional images emerge just outside the inner critical curve. · Born-Infeld naked singularities with a single photon sphere: The celestial sphere image lacks a shadow, and the image within the critical curve is formed by light rays passing through the singularity. For a point-like source, there are four n th -order relativistic images, specifically, two images situated just inside the critical curve and two images positioned just outside the critical curve. · Born-Infeld naked singularities with double photon spheres: The celestial sphere image does not exhibit a shadow, and the image within the inner critical curve is produced by light rays that traverse the singularity. For a point-like source, there are eight n th -order relativistic images, two images on each side of the inner and outer critical curves. Although current observational facilities lack the capability to distinguish higher-order relativistic images within the Born-Infeld naked singularity spacetime, the next-generation Very Long Baseline Interferometry has emerged as a promising tool for this purpose [119-121]. Furthermore, it has been demonstrated that relativistic images located inside the critical curves are more readily detectable compared to those outside the critical curves. Hence, it would be highly intriguing if our analysis could be extended to encompass more astrophysically realistic models, such as the rotating Born-Infeld naked singularity solution and the imaging of accretion disks. ACKNOWLEDGMENTS We are grateful to Qingyu Gan and Xin Jiang for useful discussions and valuable comments. This work is supported in part by NSFC (Grant No. 12105191, 12275183, 12275184 and 11875196). Houwen Wu is supported by the International Visiting Program for Excellent Young Scholars of Sichuan University. [1] F. W. Dyson, A. S. Eddington, and C. Davidson. A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919. Phil. Trans. Roy. Soc. Lond. A , 220:291-333, 1920. doi:10.1098/rsta.1920.0009 . I [2] Albert Einstein. Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field. Science , 84:506-507, 1936. doi:10.1126/science.84.2188.506 . [3] A. Eddington. SPACE, TIME AND GRAVITATION. AN OUTLINE OF THE GENERAL RELATIVITY THEORY . 1987. I [4] Yannick Mellier. Probing the universe with weak lensing. Ann. Rev. Astron. Astrophys. , 37:127-189, 1999. arXiv:astro-ph/9812172 , doi:10.1146/annurev.astro.37.1.127 . I [5] Matthias Bartelmann and Peter Schneider. Weak gravitational lensing. Phys. Rept. , 340:291-472, 2001. arXiv:astro-ph/9912508 , doi:10.1016/S0370-1573(00)00082-X . [6] Catherine Heymans et al. CFHTLenS tomographic weak lensing cosmological parameter constraints: Mitigating the impact of intrinsic galaxy alignments. Mon. Not. Roy. Astron. Soc. , 432:2433, 2013. arXiv:1303.1808 , doi:10.1093/mnras/stt601 . I [7] Nick Kaiser and Gordon Squires. Mapping the dark matter with weak gravitational lensing. Astrophys. J. , 404:441-450, 1993. doi:10.1086/172297 . I [8] Douglas Clowe, Marusa Bradac, Anthony H. Gonzalez, Maxim Markevitch, Scott W. Randall, Christine Jones, and Dennis Zaritsky. A direct empirical proof of the existence of dark matter. Astrophys. J. Lett. , 648:L109-L113, 2006. arXiv:astro-ph/0608407 , doi:10.1086/508162 . [9] Farruh Atamurotov, Ahmadjon Abdujabbarov, and Wen-Biao Han. Effect of plasma on gravitational lensing by a Schwarzschild black hole immersed in perfect fluid dark matter. Phys. Rev. D , 104(8):084015, 2021. doi:10.1103/PhysRevD.104.084015 . I [10] Marek Biesiada. Strong lensing systems as a probe of dark energy in the universe. Phys. Rev. D , 73:023006, 2006. doi:10.1103/PhysRevD.73.023006 . I [11] Shuo Cao, Marek Biesiada, Rapha Gavazzi, Aleksandra Pi'orkowska, and Zong-Hong Zhu. Cosmology With Strong-lensing Systems. Astrophys. J. , 806:185, 2015. arXiv:1509.07649 , doi: 10.1088/0004-637X/806/2/185 . [12] T. M. C. Abbott et al. Dark Energy Survey Year 1 Results: Cosmological constraints from cluster abundances and weak lensing. Phys. Rev. D , 102(2):023509, 2020. arXiv:2002.11124 , doi:10.1103/ PhysRevD.102.023509 . [13] T. M. C. Abbott et al. Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing. Phys. Rev. D , 105(2):023520, 2022. arXiv:2105.13549 , doi:10.1103/ PhysRevD.105.023520 . I [14] Xiaohui Fan et al. The Discovery of a luminous z = 5.80 quasar from the Sloan Digital Sky Survey. Astron. J. , 120:1167-1174, 2000. arXiv:astro-ph/0005414 , doi:10.1086/301534 . I [15] Chien Y. Peng, Chris D. Impey, Hans-Walter Rix, Christopher S. Kochanek, Charles R. Keeton, Emilio E. Falco, Joseph Lehar, and Brian A. McLeod. Probing the coevolution of supermassive black holes and galaxies using gravitationally lensed quasar hosts. Astrophys. J. , 649:616-634, 2006. arXiv:astro-ph/0603248 , doi:10.1086/506266 . [16] Masamune Oguri and Philip J. Marshall. Gravitationally lensed quasars and supernovae in future widefield optical imaging surveys. Mon. Not. Roy. Astron. Soc. , 405:2579-2593, 2010. arXiv:1001.2037 , doi:10.1111/j.1365-2966.2010.16639.x . [17] Minghao Yue, Xiaohui Fan, Jinyi Yang, and Feige Wang. Revisiting the Lensed Fraction of Highredshift Quasars. Astrophys. J. , 925(2):169, 2022. arXiv:2112.02821 , doi:10.3847/1538-4357/ ac409b . I [18] Uros Seljak and Christopher M. Hirata. Gravitational lensing as a contaminant of the gravity wave signal in CMB. Phys. Rev. D , 69:043005, 2004. arXiv:astro-ph/0310163 , doi:10.1103/PhysRevD. 69.043005 . I [19] Jose M. Diego, Tom Broadhurst, and George Smoot. Evidence for lensing of gravitational waves from LIGO-Virgo data. Phys. Rev. D , 104(10):103529, 2021. arXiv:2106.06545 , doi:10.1103/PhysRevD. 104.103529 . [20] Andreas Finke, Stefano Foffa, Francesco Iacovelli, Michele Maggiore, and Michele Mancarella. Probing modified gravitational wave propagation with strongly lensed coalescing binaries. Phys. Rev. D , 104(8):084057, 2021. arXiv:2107.05046 , doi:10.1103/PhysRevD.104.084057 . I [21] K. S. Virbhadra and George F. R. Ellis. Schwarzschild black hole lensing. Phys. Rev. D , 62:084003, 2000. arXiv:astro-ph/9904193 , doi:10.1103/PhysRevD.62.084003 . I, IV, IV [22] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett. , 875:L1, 2019. arXiv:1906.11238 , doi:10.3847/2041-8213/ ab0ec7 . I [23] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. II. Array and Instrumentation. Astrophys. J. Lett. , 875(1):L2, 2019. arXiv:1906.11239 , doi:10.3847/2041-8213/ab0c96 . [24] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. III. Data Processing and Calibration. Astrophys. J. Lett. , 875(1):L3, 2019. arXiv:1906.11240 , doi:10.3847/2041-8213/ ab0c57 . [25] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole. Astrophys. J. Lett. , 875(1):L4, 2019. arXiv:1906.11241 , doi: 10.3847/2041-8213/ab0e85 . [26] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring. Astrophys. J. Lett. , 875(1):L5, 2019. arXiv:1906.11242 , doi:10.3847/2041-8213/ ab0f43 . [27] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole. Astrophys. J. Lett. , 875(1):L6, 2019. arXiv:1906.11243 , doi:10.3847/ 2041-8213/ab1141 . [28] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VII. Polarization of the Ring. Astrophys. J. Lett. , 910(1):L12, 2021. arXiv:2105.01169 , doi:10.3847/2041-8213/abe71d . [29] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon. Astrophys. J. Lett. , 910(1):L13, 2021. arXiv:2105.01173 , doi:10.3847/ 2041-8213/abe4de . [30] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett. , 930(2):L12, 2022. doi:10.3847/2041-8213/ac6674 . [31] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration. Astrophys. J. Lett. , 930(2):L13, 2022. doi:10.3847/2041-8213/ac6675 . [32] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole. Astrophys. J. Lett. , 930(2):L14, 2022. doi:10.3847/ 2041-8213/ac6429 . [33] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass. Astrophys. J. Lett. , 930(2):L15, 2022. doi:10.3847/2041-8213/ ac6736 . [34] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole. Astrophys. J. Lett. , 930(2):L16, 2022. doi:10.3847/2041-8213/ac6672 . [35] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric. Astrophys. J. Lett. , 930(2):L17, 2022. doi:10.3847/2041-8213/ac6756 . I [36] Heino Falcke, Fulvio Melia, and Eric Agol. Viewing the shadow of the black hole at the galactic center. Astrophys. J. Lett. , 528:L13, 2000. arXiv:astro-ph/9912263 , doi:10.1086/312423 . I [37] Clarissa-Marie Claudel, K. S. Virbhadra, and G. F. R. Ellis. The Geometry of photon surfaces. J. Math. Phys. , 42:818-838, 2001. arXiv:gr-qc/0005050 , doi:10.1063/1.1308507 . [38] Ernesto F. Eiroa, Gustavo E. Romero, and Diego F. Torres. Reissner-Nordstrom black hole lensing. Phys. Rev. D , 66:024010, 2002. arXiv:gr-qc/0203049 , doi:10.1103/PhysRevD.66.024010 . [39] K. S. Virbhadra. Relativistic images of Schwarzschild black hole lensing. Phys. Rev. D , 79:083004, 2009. arXiv:0810.2109 , doi:10.1103/PhysRevD.79.083004 . [40] Akifumi Yumoto, Daisuke Nitta, Takeshi Chiba, and Naoshi Sugiyama. Shadows of Multi-Black Holes: Analytic Exploration. Phys. Rev. D , 86:103001, 2012. arXiv:1208.0635 , doi:10.1103/PhysRevD. 86.103001 . [41] Shao-Wen Wei and Yu-Xiao Liu. Observing the shadow of Einstein-Maxwell-Dilaton-Axion black hole. JCAP , 11:063, 2013. arXiv:1311.4251 , doi:10.1088/1475-7516/2013/11/063 . [42] Alexander F. Zakharov. Constraints on a charge in the Reissner-Nordstrom metric for the black hole at the Galactic Center. Phys. Rev. D , 90(6):062007, 2014. arXiv:1407.7457 , doi:10.1103/PhysRevD. 90.062007 . [43] Farruh Atamurotov, Sushant G. Ghosh, and Bobomurat Ahmedov. Horizon structure of rotating Einstein-Born-Infeld black holes and shadow. Eur. Phys. J. C , 76(5):273, 2016. arXiv:1506.03690 , doi:10.1140/epjc/s10052-016-4122-9 . [44] Pedro V. P. Cunha, Carlos A. R. Herdeiro, Burkhard Kleihaus, Jutta Kunz, and Eugen Radu. Shadows of Einstein-dilaton-Gauss-Bonnet black holes. Phys. Lett. B , 768:373-379, 2017. arXiv:1701.00079 , doi:10.1016/j.physletb.2017.03.020 . [45] Sara Dastan, Reza Saffari, and Saheb Soroushfar. Shadow of a Kerr-Sen dilaton-axion Black Hole. 10 2016. arXiv:1610.09477 . [46] Muhammed Amir, Balendra Pratap Singh, and Sushant G. Ghosh. Shadows of rotating fivedimensional charged EMCS black holes. Eur. Phys. J. C , 78(5):399, 2018. arXiv:1707.09521 , doi:10.1140/epjc/s10052-018-5872-3 . [47] Mingzhi Wang, Songbai Chen, and Jiliang Jing. Shadow casted by a Konoplya-Zhidenko rotating nonKerr black hole. JCAP , 10:051, 2017. arXiv:1707.09451 , doi:10.1088/1475-7516/2017/10/051 . [48] Ali Ovgun, ˙ Izzet Sakallı, and Joel Saavedra. Shadow cast and Deflection angle of Kerr-Newman-Kasuya spacetime. JCAP , 10:041, 2018. arXiv:1807.00388 , doi:10.1088/1475-7516/2018/10/041 . [49] Volker Perlick, Oleg Yu. Tsupko, and Gennady S. Bisnovatyi-Kogan. Black hole shadow in an expanding universe with a cosmological constant. Phys. Rev. D , 97(10):104062, 2018. arXiv:1804.04898 , doi:10.1103/PhysRevD.97.104062 . [50] Rahul Kumar, Sushant G. Ghosh, and Anzhong Wang. Shadow cast and deflection of light by charged rotating regular black holes. Phys. Rev. D , 100(12):124024, 2019. arXiv:1912.05154 , doi:10.1103/ PhysRevD.100.124024 . [51] Tao Zhu, Qiang Wu, Mubasher Jamil, and Kimet Jusufi. Shadows and deflection angle of charged and slowly rotating black holes in Einstein-Æther theory. Phys. Rev. D , 100(4):044055, 2019. arXiv: 1906.05673 , doi:10.1103/PhysRevD.100.044055 . [52] Liang Ma and H. Lu. Bounds on photon spheres and shadows of charged black holes in EinsteinGauss-Bonnet-Maxwell gravity. Phys. Lett. B , 807:135535, 2020. arXiv:1912.05569 , doi:10.1016/ j.physletb.2020.135535 . [53] Akash K. Mishra, Sumanta Chakraborty, and Sudipta Sarkar. Understanding photon sphere and black hole shadow in dynamically evolving spacetimes. Phys. Rev. D , 99(10):104080, 2019. arXiv: 1903.06376 , doi:10.1103/PhysRevD.99.104080 . [54] Xiao-Xiong Zeng, Hai-Qing Zhang, and Hongbao Zhang. Shadows and photon spheres with spherical accretions in the four-dimensional Gauss-Bonnet black hole. Eur. Phys. J. C , 80(9):872, 2020. arXiv: 2004.12074 , doi:10.1140/epjc/s10052-020-08449-y . [55] Xiao-Xiong Zeng and Hai-Qing Zhang. Influence of quintessence dark energy on the shadow of black hole. Eur. Phys. J. C , 80(11):1058, 2020. arXiv:2007.06333 , doi:10.1140/epjc/ s10052-020-08656-7 . [56] K. Saurabh and Kimet Jusufi. Imprints of dark matter on black hole shadows using spherical accretions. Eur. Phys. J. C , 81(6):490, 2021. arXiv:2009.10599 , doi:10.1140/epjc/s10052-021-09280-9 . [57] Rittick Roy and Sayan Chakrabarti. Study on black hole shadows in asymptotically de Sitter spacetimes. Phys. Rev. D , 102(2):024059, 2020. arXiv:2003.14107 , doi:10.1103/PhysRevD.102.024059 . [58] Peng-Cheng Li, Minyong Guo, and Bin Chen. Shadow of a Spinning Black Hole in an Expanding Universe. Phys. Rev. D , 101(8):084041, 2020. arXiv:2001.04231 , doi:10.1103/PhysRevD.101. 084041 . [59] Rahul Kumar, Sushant G. Ghosh, and Anzhong Wang. Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes. Phys. Rev. D , 101(10):104001, 2020. arXiv:2001.00460 , doi:10.1103/PhysRevD.101.104001 . [60] Ming Zhang and Jie Jiang. Shadows of accelerating black holes. Phys. Rev. D , 103(2):025005, 2021. arXiv:2010.12194 , doi:10.1103/PhysRevD.103.025005 . [61] Merce Guerrero, Gonzalo J. Olmo, Diego Rubiera-Garcia, and Diego G'omez S'aez-Chill'on. Light ring images of double photon spheres in black hole and wormhole spacetimes. Phys. Rev. D , 105(8):084057, 2022. arXiv:2202.03809 , doi:10.1103/PhysRevD.105.084057 . [62] K. S. Virbhadra. Distortions of images of Schwarzschild lensing. 4 2022. arXiv:2204.01879 . I [63] Fabian Schmidt. Weak Lensing Probes of Modified Gravity. Phys. Rev. D , 78:043002, 2008. arXiv: 0805.4812 , doi:10.1103/PhysRevD.78.043002 . I [64] Jacek Guzik, Bhuvnesh Jain, and Masahiro Takada. Tests of Gravity from Imaging and Spectroscopic Surveys. Phys. Rev. D , 81:023503, 2010. arXiv:0906.2221 , doi:10.1103/PhysRevD.81.023503 . [65] Kai Liao, Zhengxiang Li, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu. The Distance Duality Relation From Strong Gravitational Lensing. Astrophys. J. , 822(2):74, 2016. arXiv: 1511.01318 , doi:10.3847/0004-637X/822/2/74 . [66] Prieslei Goulart. Phantom wormholes in Einstein-Maxwell-dilaton theory. Class. Quant. Grav. , 35(2):025012, 2018. arXiv:1708.00935 , doi:10.1088/1361-6382/aa9dfc . [67] J. R. Nascimento, A. Yu. Petrov, P. J. Porfirio, and A. R. Soares. Gravitational lensing in blackbounce spacetimes. Phys. Rev. D , 102(4):044021, 2020. arXiv:2005.13096 , doi:10.1103/PhysRevD. 102.044021 . [68] Xin Qin, Songbai Chen, and Jiliang Jing. Image of a regular phantom compact object and its luminosity under spherical accretions. Class. Quant. Grav. , 38(11):115008, 2021. arXiv:2011.04310 , doi:10.1088/1361-6382/abf712 . [69] Haroldo C. D. Lima Junior, Jian-Zhi Yang, Lu'ıs C. B. Crispino, Pedro V. P. Cunha, and Carlos A. R. Herdeiro. Einstein-Maxwell-dilaton neutral black holes in strong magnetic fields: Topological charge, shadows, and lensing. Phys. Rev. D , 105(6):064070, 2022. arXiv:2112.10802 , doi:10.1103/ PhysRevD.105.064070 . [70] Shafqat Ul Islam, Jitendra Kumar, and Sushant G. Ghosh. Strong gravitational lensing by rotating Simpson-Visser black holes. JCAP , 10:013, 2021. arXiv:2104.00696 , doi:10.1088/1475-7516/ 2021/10/013 . [71] Naoki Tsukamoto. Gravitational lensing by two photon spheres in a black-bounce spacetime in strong deflection limits. 5 2021. arXiv:2105.14336 . [72] Gonzalo J. Olmo, Diego Rubiera-Garcia, and Diego S'aez-Chill'on G'omez. New light rings from multiple critical curves as observational signatures of black hole mimickers. Phys. Lett. B , 829:137045, 2022. arXiv:2110.10002 , doi:10.1016/j.physletb.2022.137045 . I [73] Stuart L. Shapiro and Saul A. Teukolsky. Formation of naked singularities: The violation of cosmic censorship. Phys. Rev. Lett. , 66:994-997, 1991. doi:10.1103/PhysRevLett.66.994 . I [74] P. S. Joshi and I. H. Dwivedi. Naked singularities in spherically symmetric inhomogeneous TolmanBondi dust cloud collapse. Phys. Rev. D , 47:5357-5369, 1993. arXiv:gr-qc/9303037 , doi:10.1103/ PhysRevD.47.5357 . [75] Tomohiro Harada, Hideo Iguchi, and Ken-ichi Nakao. Naked singularity formation in the collapse of a spherical cloud of counter rotating particles. Phys. Rev. D , 58:041502, 1998. arXiv:gr-qc/9805071 , doi:10.1103/PhysRevD.58.041502 . [76] Pankaj S. Joshi, Naresh Dadhich, and Roy Maartens. Why do naked singularities form in gravitational collapse? Phys. Rev. D , 65:101501, 2002. arXiv:gr-qc/0109051 , doi:10.1103/PhysRevD.65. 101501 . [77] Rituparno Goswami and Pankaj S Joshi. Spherical gravitational collapse in N-dimensions. Phys. Rev. D , 76:084026, 2007. arXiv:gr-qc/0608136 , doi:10.1103/PhysRevD.76.084026 . [78] Narayan Banerjee and Soumya Chakrabarti. Self-similar scalar field collapse. Phys. Rev. D , 95(2):024015, 2017. arXiv:1701.04235 , doi:10.1103/PhysRevD.95.024015 . [79] Kaushik Bhattacharya, Dipanjan Dey, Arindam Mazumdar, and Tapobrata Sarkar. New class of naked singularities and their observational signatures. Phys. Rev. D , 101(4):043005, 2020. arXiv: 1709.03798 , doi:10.1103/PhysRevD.101.043005 . I [80] K. S. Virbhadra and G. F. R. Ellis. Gravitational lensing by naked singularities. Phys. Rev. D , 65:103004, 2002. doi:10.1103/PhysRevD.65.103004 . I [81] K. S. Virbhadra and C. R. Keeton. Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities. Phys. Rev. D , 77:124014, 2008. arXiv:0710.2333 , doi:10.1103/PhysRevD.77.124014 . [82] Galin N. Gyulchev and Stoytcho S. Yazadjiev. Gravitational Lensing by Rotating Naked Singularities. Phys. Rev. D , 78:083004, 2008. arXiv:0806.3289 , doi:10.1103/PhysRevD.78.083004 . [83] Satyabrata Sahu, Mandar Patil, D. Narasimha, and Pankaj S. Joshi. Can strong gravitational lensing distinguish naked singularities from black holes? Phys. Rev. D , 86:063010, 2012. arXiv:1206.3077 , doi:10.1103/PhysRevD.86.063010 . [84] Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar. On Strong Gravitational Lensing in Rotating Galactic Space-times. 4 2018. arXiv:1804.07030 . [85] Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, and Tapobrata Sarkar. Analytical approach to strong gravitational lensing from ultracompact objects. Phys. Rev. D , 99(10):104040, 2019. arXiv: 1903.08211 , doi:10.1103/PhysRevD.99.104040 . I, IV A, IV B [86] Suvankar Paul. Strong gravitational lensing by a strongly naked null singularity. Phys. Rev. D , 102(6):064045, 2020. arXiv:2007.05509 , doi:10.1103/PhysRevD.102.064045 . [87] Naoki Tsukamoto. Gravitational lensing by a photon sphere in a Reissner-Nordstrom naked singularity spacetime in strong deflection limits. Phys. Rev. D , 104(12):124016, 2021. arXiv:2107.07146 , doi: 10.1103/PhysRevD.104.124016 . I [88] M. Born and L. Infeld. Foundations of the new field theory. Proc. Roy. Soc. Lond. A , 144(852):425-451, 1934. doi:10.1098/rspa.1934.0059 . I [89] Tanay Kr. Dey. Born-Infeld black holes in the presence of a cosmological constant. Phys. Lett. B , 595(1-4):484-490, 2004. arXiv:hep-th/0406169 , doi:10.1016/j.physletb.2004.06.047 . I, II [90] Rong-Gen Cai, Da-Wei Pang, and Anzhong Wang. Born-Infeld black holes in (A)dS spaces. Phys. Rev. D , 70:124034, 2004. arXiv:hep-th/0410158 , doi:10.1103/PhysRevD.70.124034 . I, II [91] Sharmanthie Fernando and Don Krug. Charged black hole solutions in Einstein-Born-Infeld gravity with a cosmological constant. Gen. Rel. Grav. , 35:129-137, 2003. arXiv:hep-th/0306120 , doi: 10.1023/A:1021315214180 . I [92] Rabin Banerjee, Sumit Ghosh, and Dibakar Roychowdhury. New type of phase transition in Reissner Nordstr ˜ A ¶ m-AdS black hole and its thermodynamic geometry. Phys. Lett. B , 696:156-162, 2011. arXiv:1008.2644 , doi:10.1016/j.physletb.2010.12.010 . [93] De-Cheng Zou, Shao-Jun Zhang, and Bin Wang. Critical behavior of Born-Infeld AdS black holes in the extended phase space thermodynamics. Phys. Rev. D , 89(4):044002, Feb 2014. URL: https: //link.aps.org/doi/10.1103/PhysRevD.89.044002 , arXiv:1311.7299 , doi:10.1103/PhysRevD. 89.044002 . [94] Seyed Hossein Hendi, Behzad Eslam Panah, and Shahram Panahiyan. Einstein-Born-Infeld-Massive Gravity: adS-Black Hole Solutions and their Thermodynamical properties. JHEP , 11:157, 2015. arXiv:1508.01311 , doi:10.1007/JHEP11(2015)157 . [95] Xiao-Xiong Zeng, Xian-Ming Liu, and Li-Fang Li. Phase structure of the Born-Infeld-anti-de Sitter black holes probed by non-local observables. Eur. Phys. J. C , 76(11):616, 2016. arXiv:1601.01160 , doi:10.1140/epjc/s10052-016-4463-4 . [96] Shoulong Li, H. Lu, and Hao Wei. Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory in Diverse Dimensions. JHEP , 07:004, 2016. arXiv:1606.02733 , doi:10.1007/JHEP07(2016)004 . [97] Jun Tao, Peng Wang, and Haitang Yang. Testing holographic conjectures of complexity with BornInfeld black holes. Eur. Phys. J. C , 77(12):817, 2017. arXiv:1703.06297 , doi:10.1140/epjc/ s10052-017-5395-3 . [98] Amin Dehyadegari and Ahmad Sheykhi. Reentrant phase transition of Born-Infeld-AdS black holes. Phys. Rev. D , 98(2):024011, 2018. arXiv:1711.01151 , doi:10.1103/PhysRevD.98.024011 . [99] Peng Wang, Houwen Wu, and Haitang Yang. Thermodynamics and Phase Transitions of Nonlinear Electrodynamics Black Holes in an Extended Phase Space. JCAP , 04(04):052, 2019. arXiv:1808. 04506 , doi:10.1088/1475-7516/2019/04/052 . [100] Kangkai Liang, Peng Wang, Houwen Wu, and Mingtao Yang. Phase structures and transitions of BornInfeld black holes in a grand canonical ensemble. Eur. Phys. J. C , 80(3):187, 2020. arXiv:1907.00799 , doi:10.1140/epjc/s10052-020-7750-z . [101] Qingyu Gan, Guangzhou Guo, Peng Wang, and Houwen Wu. Strong cosmic censorship for a scalar field in a Born-Infeld-de Sitter black hole. Phys. Rev. D , 100(12):124009, 2019. arXiv:1907.04466 , doi:10.1103/PhysRevD.100.124009 . [102] Peng Wang, Houwen Wu, and Haitang Yang. Thermodynamics and Phase Transition of a Nonlinear Electrodynamics Black Hole in a Cavity. JHEP , 07:002, 2019. arXiv:1901.06216 , doi:10.1007/ JHEP07(2019)002 . [103] Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Einstein-Born-Infeld black holes. Phys. Rev. D , 103(10):104012, 2021. arXiv:2012.01066 , doi:10.1103/PhysRevD.103.104012 . I [104] Guangzhou Guo, Yuhang Lu, Peng Wang, Houwen Wu, and Haitang Yang. Black Holes with Multiple Photon Spheres. 12 2022. arXiv:2212.12901 . I, II, VI [105] M. Novello, V. A. De Lorenci, J. M. Salim, and Renato Klippert. Geometrical aspects of light propagation in nonlinear electrodynamics. Phys. Rev. D , 61:045001, 2000. arXiv:gr-qc/9911085 , doi:10.1103/PhysRevD.61.045001 . III [106] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon ring and observational appearance of a hairy black hole. Phys. Rev. D , 104(4):044049, 2021. arXiv:2105.11770 , doi: 10.1103/PhysRevD.104.044049 . III [107] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon spheres and spherical accretion image of a hairy black hole. Phys. Rev. D , 104(2):024003, 2021. arXiv:2104.08703 , doi:10.1103/ PhysRevD.104.024003 . VI [108] Guangzhou Guo, Xin Jiang, Peng Wang, and Houwen Wu. Gravitational lensing by black holes with multiple photon spheres. Phys. Rev. D , 105(12):124064, 2022. arXiv:2204.13948 , doi:10.1103/ PhysRevD.105.124064 . VI [109] Yiqian Chen, Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Appearance of an infalling star in black holes with multiple photon spheres. Sci. China Phys. Mech. Astron. , 65(12):120412, 2022. arXiv:2206.13705 , doi:10.1007/s11433-022-1986-x . III [110] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Quasinormal modes of black holes with multiple photon spheres. JHEP , 06:060, 2022. arXiv:2112.14133 , doi:10.1007/JHEP06(2022)060 . III [111] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Echoes from hairy black holes. JHEP , 06:073, 2022. arXiv:2204.00982 , doi:10.1007/JHEP06(2022)073 . III [112] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Superradiance Instabilities of Charged Black Holes in Einstein-Maxwell-scalar Theory. 1 2023. arXiv:2301.06483 . III [113] D. Pugliese, H. Quevedo, and R. Ruffini. Circular motion of neutral test particles in ReissnerNordstrom spacetime. Phys. Rev. D , 83:024021, 2011. arXiv:1012.5411 , doi:10.1103/PhysRevD. 83.024021 . III [114] V. Bozza. Gravitational lensing in the strong field limit. Phys. Rev. D , 66:103001, 2002. arXiv: gr-qc/0208075 , doi:10.1103/PhysRevD.66.103001 . IVA, IVA, IVA [115] Naoki Tsukamoto. Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime. Phys. Rev. D , 95(6):064035, 2017. arXiv:1612.08251 , doi: 10.1103/PhysRevD.95.064035 . IVA [116] Shao-Wen Wei, Ke Yang, and Yu-Xiao Liu. Black hole solution and strong gravitational lensing in Eddington-inspired Born-Infeld gravity. Eur. Phys. J. C , 75:253, 2015. [Erratum: Eur.Phys.J.C 75, 331 (2015)]. arXiv:1405.2178 , doi:10.1140/epjc/s10052-015-3556-9 . IVA [117] Pedro V. P. Cunha, Carlos A. R. Herdeiro, Eugen Radu, and Helgi F. Runarsson. Shadows of Kerr black holes with and without scalar hair. Int. J. Mod. Phys. D , 25(09):1641021, 2016. arXiv: 1605.08293 , doi:10.1142/S0218271816410212 . V [118] Hai-Shan Liu, Zhan-Feng Mai, Yue-Zhou Li, and H. Lu. Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality. Sci. China Phys. Mech. Astron. , 63:240411, 2020. arXiv:1907.10876 , doi:10.1007/s11433-019-1446-1 . VI [119] Michael D. Johnson et al. Universal interferometric signatures of a black hole's photon ring. Sci. Adv. , 6(12):eaaz1310, 2020. arXiv:1907.04329 , doi:10.1126/sciadv.aaz1310 . VI [120] Elizabeth Himwich, Michael D. Johnson, Alexandru Lupsasca, and Andrew Strominger. Universal polarimetric signatures of the black hole photon ring. Phys. Rev. D , 101(8):084020, 2020. arXiv: 2001.08750 , doi:10.1103/PhysRevD.101.084020 . [121] Samuel E. Gralla, Alexandru Lupsasca, and Daniel P. Marrone. The shape of the black hole photon ring: A precise test of strong-field general relativity. Phys. Rev. D , 102(12):124004, 2020. arXiv: 2008.03879 , doi:10.1103/PhysRevD.102.124004 . VI similar to the single-peak scenario. However, when the height of the inner peak surpasses that of the outer peak, a distant source can generate a total of eight n th -order relativistic images due to strong gravitational lensing near the inner and outer photon spheres. The light rays responsible for these relativistic images are categorized based on their impact parameter b , ϑ = ϑ out < ± n . Note that the angular position of the images, ϑ out >, in > ± n and ϑ in < ± n , can be computed using eqns. (29) and (39), respectively. Moreover, the deflection angle of light rays with b < b out has been previously shown to be [85] where Here, r m is the critical turning point when b is very close to b out . Since the divergent parts of the deflection angles (37) and (41) share the same form, one can utilize eqn. (39) to calculate ϑ out < ± n with the values of ¯ a and ¯ b from eqn. (42). Similarly, in the aforementioned astrophysical scenario of Born-Infeld naked singularities with a single photon sphere, the angular positions of the relativistic images can be estimated numerically. Specifically, we present the values of ∆ ϑ in ≷ ± n ≡ ϑ in ≷ ± n -ϑ in ±∞ and ∆ ϑ out ≷ ± n ≡ ϑ out ≷ ± n -ϑ out ±∞ for a/M 2 = 0 . 2, 1 and 1 . 5 in Table II. Here, ϑ in ±∞ and ϑ out ±∞ are the angular positions of the relativistic images formed at the inner and outer photon spheres, respectively. We also assume a resolution of 0 . 01 microarcseconds, which enables the resolution of the first-order image in a Schwarzschild black hole. For a singularity with a/M 2 = 0 . 2, the inner potential peak is significantly sharper and higher than the outer peak. Consequently, the relativistic images at ϑ = ϑ in ≷ ± n are closely situated near the inner critical curve at ϑ = ϑ in ±∞ , making them indistinguishable on the image plane. However, the images near the outer critical curve at ϑ = ϑ out ±∞ are well-separated from the outer critical curve, allowing for the distinction of the first-order images with b > b out and the first three orders ( n ≤ 3) images with b < b out . When a/M 2 = 1, the inner potential peak becomes flatter, resulting in greater separation among the images near the inner critical curve. It should be emphasized that, due to the inner potential peak not being significantly higher than the outer peak, the formulas for ϑ in > ± n and ϑ out < ± n in the strong deflection limit may exhibit substantial errors. For the singularity with a/M 2 = 1 . 5, the inner peak is lower than the outer peak, leading to the existence of relativistic images solely near the outer critical curve.", "pages": [ 14, 15, 16, 17 ] }, { "title": "V. CELESTIAL SPHERE IMAGES", "content": "In this section, we investigate gravitational lensing by a Born-Infeld naked singularity through the image of a luminous celestial sphere centered at the singularity and surrounding the observers. To obtain the image of the celestial sphere, we use the numerical backward ray-tracing method to calculate light rays from the observer to the celestial sphere. Light rays can be described either by numerically integrating eqn. (17) or eight first-order differential equations where λ represents the affine parameter, and ˜ Γ µ ρσ are the Christoffel symbols compatible with the effective metric. Numerically solving eqn. (43) for light rays enables us to avoid the need to account for turning points during the integration, resulting in improved numerical accuracy. Thus, we use eqn. (43) to calculate the light rays connecting the observer with the celestial sphere. For a static observer located at ( t o , r o , θ o , φ o ), we introduce a tetrad basis which span the tangent bundle at the observer. To obtain initial conditions for eqn. (43), a photon captured by the observer is considered, whose local 4-momentum ( p ( t ) , p ( r ) , p ( θ ) , p ( φ ) ) in the tetrad basis is related to the 4-momentum p µ o = dx µ /dλ | ( t o ,r o ,θ o ,φ o ) as The observation angles Θ and Φ, as defined in [117], are given by which p = √ p ( r )2 + p ( θ )2 + p ( φ )2 . We express p ( r ) , p ( θ ) and p ( φ ) in terms of p , Θ and Φ as Moreover, the condition ˜ g µν p µ o p ν o = 0 and eqn. (45) give Using eqn. (45), we can rewrite p µ o in terms of p , Θ and Φ, which, together with the coordinates of the observer, provide initial conditions for eqn. (43). Without loss of generality, we set p = 1 in what follows. The Cartesian coordinates ( x, y ) of the image plane of the observer is defined by where the sign convention for Φ leads to the minus sign in the x definition. Note that the direction pointing to the singularity corresponds to the zero observation angles (0 , 0). As discussed previously, when a light ray travels through the singularity, its affine parameter λ becomes divergent at r = 0, posing a challenge for our numerical implementation. To circumvent this issue, we introduce a small sphere of radius Mϵ to enclose the singularity. Outside the sphere, we can solve eqn. (43) numerically, ensuring the accuracy and stability of the light ray calculation. Within the sphere, we can use the expansions in eqn. (20) to describe the light ray and provide a connection formula for its entry and exit points from the sphere. Specifically, we consider the light ray entering and leaving the sphere at ( t in , r in , θ in , φ in ) and ( t out , r out , θ out , φ out ), respectively. With eqns. (21) and (22), one has In this section, we employ the leading terms of eqn. (50) to connect the ingoing and outgoing branches. To explore the numerical error caused by the finite size of ϵ , we investigate a light ray on the equatorial plane of a Born-Infeld naked singularity with a/M 2 = 1 and √ Q 2 + P 2 /M = 1 . 05. The light ray originates from ( r e , φ e ) = (25 M,φ ϵ ), and an observer located at ( r o , φ o ) = (10 M,π ) captures it with observation angles (Θ , Φ) = (0 , 3 / 20). To obtain the coordinate φ ϵ , we trace the light ray backward from the observer to r e = 25 M while connecting the ingoing branch with the outgoing one at r = Mϵ . We present the relative error ( φ ϵ -φ 0 ) /φ 0 as a function of ϵ in FIG. 4, where φ 0 ≡ φ ϵ =10 -3 . To maintain numerical precision and efficiency, we set ϵ = 10 -1 in the following numerical simulations, for which the relative error is well below 10 -3 . To illustrate gravitational lensing by Born-Infeld naked singularities, we position a luminous celestial sphere at r CS = 25 M , while an observer is situated at x µ o = (0 , 10 M,π/ 2 , π ). The celestial sphere is divided into four quadrants, each distinguished by a different color, and a white dot is placed in front of the observer. Additionally, we overlay a grid of black lines representing constant longitude and latitude, where adjacent lines are separated by π/ 18. To generate an observational image, we vary the observer's viewing angle and numerically integrate 2000 × 2000 photon trajectories until they intersect with the celestial sphere. The resulting image of the celestial sphere in Minkowski spacetime is presented in FIG. 5. FIG. 6 displays images of the celestial sphere in both RN and Born-Infeld naked singularities, both of which possess a single photon sphere. The dashed circular lines in the images correspond to the critical curve formed by light rays originating from the photon sphere. Beyond this critical curve, the images of the celestial sphere in naked singularities bear resemblance to those observed in black hole spacetime. Notably, unlike shadows observed in black hole images, the celestial sphere images persist within the critical curve due to the absence of an event horizon. Additionally, higher-order images of the celestial sphere can be observed both inside and outside this critical curve. The left panel of FIG. 6 illustrates the image of the celestial sphere in a RN naked singularity characterized by √ Q 2 + P 2 /M = 1 . 05. Inside the critical curve, three distinct white rings can be observed. These rings correspond to the Einstein ring generated by the white dot positioned on the celestial sphere. Specifically, the innermost white ring originates from light rays emitted by the white dot, undergoes reflection at the potential barrier, and eventually reaches the observer after experiencing an angular coordinate change of ∆ φ = π . Within this innermost white ring, FIG. 6. Images of the celestial sphere in naked singularities featuring a single photon sphere. The observer is situated at x µ o = (0 , 10 M,π/ 2 , π ) with a field of view of π/ 4. The dashed lines depict the critical curve formed by photons escaping from the photon sphere. Left: The RN naked singularity with √ Q 2 + P 2 /M = 1 . 05. The image within the critical curve is generated by light rays that rebound off the infinitely high potential barrier at the singularity. Three white rings, representing the Einstein ring of the white dot on the celestial sphere, can be observed within the critical curve. Right: The Born-Infeld naked singularity with √ Q 2 + P 2 /M = 1 . 2 and a/M 2 = 2. The image within the critical curve is formed by light rays passing through the singularity. A central white dot is visible, surrounded by two white rings. the reflections from the infinitely high potential barrier at the singularity produce a mirror image of the celestial sphere. Moreover, the middle and outermost white rings arise from light rays with angular coordinate changes of ∆ φ = 3 π and 5 π , respectively. In the right panel of FIG. 6, the image captured in a Born-Infeld naked singularity with √ Q 2 + P 2 /M = 1 . 2 and a/M 2 = 2 is displayed. As expected, light rays passing through the singularity and undergoing an angular coordinate change of ∆ φ = π result in a white dot positioned at the center of the image. Additionally, two white rings emerge within the critical curve, representing photons that traverse the singularity with angular coordinate changes of ∆ φ = 3 π and 5 π , respectively. white rings can be observed positioned between the inner and outer critical curves. These rings originate from photons that are reflected by the potential barrier located between the inner and outer potential peaks with ∆ φ = 3 π and 5 π . In the right panel, with √ Q 2 + P 2 /M = 1 . 05 and a/M 2 = 1 . 5, a white ring emerges between the two critical curves. This ring arises from photons that undergo reflection at the potential barrier with an angular coordinate change of ∆ φ = 5 π . Additionally, a white ring is observed inside the inner critical curve, which occurs due to photons passing through the singularity and experiencing an angular coordinate change of ∆ φ = 3 π .", "pages": [ 17, 18, 19, 20, 21, 22 ] }, { "title": "VI. CONCLUSIONS", "content": "This paper investigated the phenomenon of gravitational lensing by Born-Infeld naked singularities, which are solutions of a (3 + 1)-dimensional gravity model coupled to a Born-Infeld electromagnetic field. Owing to the nonlinearity inherent in Born-Infeld electrodynamics, photons follow null geodesics of an effective metric, deviating from the background metric, and remarkably, they are capable of traversing naked singularities. Additionally, we demonstrated that Born-Infeld naked singularities can exhibit the presence of two photon spheres with distinct sizes within specific parameter ranges. The existence of these double photon spheres, combined with the transparency of naked singularities, significantly impacts the gravitational lensing of light sources, leading to various effects such as the emergence of new relativistic images. Consequently, these findings provide a potent tool for detecting and studying Born-Infeld naked singularities through their distinctive gravitational lensing signatures. Naked singularities with double photon spheres have been infrequently reported; however, it has been discovered that asymptotically flat black holes can possess two photon spheres outside the event horizon [104, 107, 118]. A recent investigation focused on studying the relativistic images produced by point-like light sources and luminous celestial spheres in the presence of black holes with either a single or double photon spheres [108]. The key findings regarding strong gravitational lensing by black holes and naked singularities can be summarized as follows: Although current observational facilities lack the capability to distinguish higher-order relativistic images within the Born-Infeld naked singularity spacetime, the next-generation Very Long Baseline Interferometry has emerged as a promising tool for this purpose [119-121]. Furthermore, it has been demonstrated that relativistic images located inside the critical curves are more readily detectable compared to those outside the critical curves. Hence, it would be highly intriguing if our analysis could be extended to encompass more astrophysically realistic models, such as the rotating Born-Infeld naked singularity solution and the imaging of accretion disks.", "pages": [ 22, 23, 24 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to Qingyu Gan and Xin Jiang for useful discussions and valuable comments. This work is supported in part by NSFC (Grant No. 12105191, 12275183, 12275184 and 11875196). Houwen Wu is supported by the International Visiting Program for Excellent Young Scholars of Sichuan University. rotating regular black holes. Phys. Rev. D , 100(12):124024, 2019. arXiv:1912.05154 , doi:10.1103/ PhysRevD.100.124024 . 95(2):024015, 2017. arXiv:1701.04235 , doi:10.1103/PhysRevD.95.024015 . in the extended phase space thermodynamics. Phys. Rev. D , 89(4):044002, Feb 2014. URL: https: //link.aps.org/doi/10.1103/PhysRevD.89.044002 , arXiv:1311.7299 , doi:10.1103/PhysRevD. 89.044002 .", "pages": [ 24, 28, 30, 31 ] }, { "title": "10.1103/PhysRevD.104.044049 . III", "content": "polarimetric signatures of the black hole photon ring. Phys. Rev. D , 101(8):084020, 2020. arXiv: 2001.08750 , doi:10.1103/PhysRevD.101.084020 . [121] Samuel E. Gralla, Alexandru Lupsasca, and Daniel P. Marrone. The shape of the black hole photon ring: A precise test of strong-field general relativity. Phys. Rev. D , 102(12):124004, 2020. arXiv: 2008.03879 , doi:10.1103/PhysRevD.102.124004 . VI", "pages": [ 33 ] } ]

[ { "title": "Perturbations of bimetric gravity on most general spherically symmetric spacetimes", "content": "David Brizuela 1 , ∗ , Marco de Cesare 2 , 3 , † , and Araceli Soler Oficial 1 , ‡ 1 Department of Physics and EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain 2 Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy 3 INFN, Sezione di Napoli, Italy", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present a formalism to study linear perturbations of bimetric gravity on any spherically symmetric background, including dynamical spacetimes. The setup is based on the Gerlach-Sengupta formalism for general relativity. Each of the two background metrics is written as a warped product between a twodimensional Lorentzian metric and the round metric of the two-sphere. The different perturbations are then decomposed in terms of tensor spherical harmonics, which makes the two polarity (axial and polar) sectors decouple. In addition, a covariant notation on the Lorentzian manifold is used so that all expressions are valid for any coordinates. In this theory, there are seven physical propagating degrees of freedom, which, as compared to the two degrees of freedom of general relativity, makes the dynamics much more intricate. In particular, we discuss the amount of gauge and physical degrees of freedom for different polarities and multipoles. Finally, as an interesting application, we analyze static nonbidiagonal backgrounds and derive the corresponding perturbative equations.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Bimetric theory, as formulated in Refs. [1,2], is a modified gravity theory that extends general relativity (GR) by considering the existence of two coupled dynamical metrics. In particular, the corresponding interaction potential has a certain specific form in order to ensure the absence of the Boulware-Deser ghost. In this context, it has been shown that bimetric gravity is stable and well behaved in certain regions of parameter space [3]. The relevance of this theory lies in its potential to address cosmological questions, such as the accelerated expansion of the Universe and the nature of dark matter. In this sense, it is known that viable cosmological solutions that fit the expansion history of the accelerating Universe exist [4-7] and that the massive spin-two field can play the role of dark matter [8-11]. Moreover, constraints on the parameters of the theory have been derived through observational [12-15] and analytical [16] methods. The stability and viability of black-hole solutions within bimetric gravity have been widely addressed in the literature [17,18]. Static and spherically symmetric black-hole solutions split into two different branches [19]. In the first branch, a coordinate system exists in which the two metrics can be simultaneously diagonalized. These types of solutions are known as bidiagonal solutions. However, this is not possible in general, and therefore there exists a second branch of nonbidiagonal solutions. For bidiagonal solutions, the corresponding equations of motion cannot, in general, be solved analytically. Nevertheless, some exact black-hole solutions have been found in spherical symmetry [20, 21], all of them corresponding to the standard GR solutions (i.e., Schwarzschild, Schwarzschild-de Sitter, and Schwarzschild-anti-de Sitter). In fact, using analytical and numerical techniques, in Ref. [18] it was shown that, within the bidiagonal ansatz, all black-hole solutions with flat or de Sitter asymptotics correspond to GR solutions, with both metrics being conformal (see also Ref. [22]). This, together with the fact that it is known that bidiagonal solutions where both metrics are Schwarzschild are dynamically unstable [18, 23-25], suggests that static and spherically symmetric bidiagonal solutions cannot represent the end point of gravitational collapse [18]. In contrast, with a nonbidiagonal ansatz, both metrics obey the Einstein equations and thus correspond to standard GR solutions [19]. However, the correspondence with black holes in GR only holds at the background level, and it is broken by perturbations. In particular, previous work [26] proved the stability of a particular subclass of nonbidiagonal static black-hole solutions against generic linear perturbations, although not for general nonbidiagonal black holes. In this work, we present the equations for linear perturbations around a completely general (including dynamical) spherically symmetric background within bimetric theory. To this end, we use the Gerlach-Sengupta formalism [27-29], based on a 2+2 decomposition of the spacetime separating the spherical symmetry orbits from a general two-dimensional Lorentzian manifold. Making use of the tensor spherical harmonics, this allows us to use a compact and covariant description of the perturbative equations both on the Lorentzian manifold and on the two-sphere, which is valid for any coordinate choice. As noted, the formalism describes the evolution of the perturbations on any spherical static black-hole or star [30] backgrounds, but could also be used in the dynamical case to study, for instance, the stability during a spherically symmetric gravitational collapse [31,32]. Here, as an interesting application, we specialize the obtained equations to a general nonbidiagonal background with a static physical metric. In this case, the analytical form of the background can be solved up to a function that satisfies a nonlinear partial differential equation [22, 33]. In order to perform the computations of the present paper, we have made extensive use of the different packages of the xAct project [34] for Wolfram Mathematica, and particularly of xPert [35]. The remainder of this paper is organized as follows. In Sec. 2 we present the formulation of linear perturbations of bimetric gravity. In Sec. 3 we take spherically symmetric background spacetimes and introduce the 2+2 decomposition characteristic of the Gerlach-Sengupta formalism. Then, in Sec. 4, we decompose the metric perturbations in tensor spherical harmonics. In Sec. 5 we discuss the gauge freedom of the theory, and obtain the equations for the linear perturbations for any two spherically symmetric background metrics, both for the axial and the polar sectors. We specialize these expressions to nonbidiagonal backgrounds in Sec. 6. Finally, in Sec. 7, we review and discuss the main results of the paper. Notations and conventions: We assume the metric signature ( -+++) and units with the speed of light c = 1. The symmetrization of indices is denoted by round brackets and includes a factor of 1 / 2, that is, T ( ab ) := 1 2 ( T ab + T ba ).", "pages": [ 1, 2 ] }, { "title": "2 Linear perturbations of bimetric gravity", "content": "The bimetric gravity theory proposed by Hassan and Rosen [36] is based on the existence of two dynamical and nonlinearly interacting metrics, ˜ g µν and ˜ f µν , on the four-dimensional spacetime manifold. The action is given by the linear combination of the Einstein-Hilbert term for each metric, complemented with a coupling term where R (˜ g ) and R ( ˜ f ) are the Ricci scalars of the metrics ˜ g µν and ˜ f µν , respectively. The coupling constants M g , M f , and m have dimensions of mass, while the β n are dimensionless. Finally, the e n are symmetric polynomials of scalar combinations of the matrix [37] and are explicitly defined as [38,39] with Tr[ ˜ S ] = ˜ S µ µ . For a d × d matrix ˜ S , e n ( ˜ S ) = 0 for any n > d and e d ( ˜ S ) = det ( ˜ S ). Therefore, one could also write e 4 ( ˜ S ) = det ( ˜ S ). By the relation it is straightforward to see that the bimetric action (2.1) is invariant under the simultaneous replacements which means that both metrics are treated on the same footing in the pure gravity theory. However, such a symmetry is broken by matter fields, which typically are only coupled to a single metric [40,41] (see also the review [42]). Therefore, here we will also assume that matter sources couple only to the metric g µν , and are described by the corresponding stress-energy tensor T µν . This leads to the equations of motion where G (˜ g ) µν and G ( ˜ f ) µν are the Einstein tensors of the corresponding metrics, while α := M f /M g measures the ratio between the gravitational couplings. The interaction between the two metrics is encoded in the potential where the matrices Y ( i ) read as In the following we will consider the effective stress-energy tensors so that the equations of motion (2.6) formally take the same form as the Einstein equations, Now, in order to perform a perturbative analysis of the theory, we write where the metrics g µν and f µν are exact solutions of the equations (2.11) and will be referred to as the background . In turn, h ( g ) µν and h ( f ) µν encode the perturbations and will be assumed to be small. That is, in order to obtain their equations of motion, one simply substitutes the ansatz (2.12) into (2.11), and regards any term quadratic in the perturbations as negligible. Let us define the operator ∆ as providing the linear term in h ( g ) µν and h ( f ) µν of any object; for instance, t (˜ g ) µν = t ( g ) µν +∆[ t ( g ) µν ]. In this way, the linear equations of motion for h ( g ) µν and h ( f ) µν can be written as The left-hand side are the perturbations of the Einstein tensor of each metric, whose form is well known, where the semicolon ' ; ' denotes the covariant derivative associated to g µν , and R ( g ) αβ is its Ricci tensor. The perturbation of the Einstein tensor of the metric ˜ f µν can be computed analogously. Therefore, the nontrivial part of the present computation will be to obtain the linear version of the effective stress-energy tensors t (˜ g ) µν and t ( ˜ f ) µν . In particular, this requires us to compute the perturbation of the matrix ˜ S µ ν . By definition, we have Replacing the expansions (2.12) and ˜ S µ α = S µ α +∆[ S µ α ] in this expression, the term linear in perturbations yields which can be rewritten as Hence, in order to obtain ∆[ S ρ σ ] explicitly in terms of h ( g ) µν and h ( f ) µν , one would need to compute the inverse of the expression in brackets above. Although this does not seem feasible for generic backgrounds, on a spherically symmetric background the problem can be simplified by decomposing ∆[ S ρ σ ] in a basis of tensor spherical harmonics, as we will show in Sec. 4.2.1. Before we move on to analyze perturbations around specific backgrounds, let us comment on the gauge freedom and the number of propagating degrees of freedom. In vacuum GR the only dynamical field is the metric, which, being a rank-two symmetric tensor field, in principle encodes ten independent local degrees of freedom. However, there are eight first-class constraints: four corresponding to the generators of diffeomorphisms (the so-called Hamiltonian and diffeomorphism constraints), and four more corresponding to the vanishing of the conjugate momenta of lapse and shift. Each of these constraints removes one degree of freedom (i.e., two phase-space dimensions per spacetime point), which leaves a total of two propagating degrees of freedom, corresponding to two independent polarizations for the graviton. Considering two decoupled copies of GR, hence with two independent groups of diffeomorphisms (each acting independently on a single metric sector), both the number of degrees of freedom and constraints would double, and one would have four propagating degrees of freedom. However, when the two metrics are coupled, a set of four first-class constraints of the system is removed, due to the now common diffeomorphism invariance. For generic choices of the potential describing the interaction between the two metrics, this would lead to eight propagating degrees of freedom in total. In the Hassan-Rosen theory though, given by the action (2.1), the coupling term is chosen in such a way that there appears a couple of second-class constraints [2, 38, 43] that remove one degree of freedom, the so-called Boulware-Deser ghost, leaving seven propagating degrees of freedom [44-46].", "pages": [ 2, 3, 4 ] }, { "title": "3 Spherically symmetric background", "content": "Any four-dimensional spherically symmetric manifold is given as a direct product M 2 ×S 2 , where M 2 is a two-dimensional Lorentzian manifold and S 2 is the two-sphere. The background metric tensors can then be written in block-diagonal form, where Greek indices take values from 0 to 3, capital Latin indices from 0 to 1, and lowercase Latin indices run from 2 to 3. The tensor is the unit metric on the two-sphere, while g AB and f AB are Lorentzian metrics in M 2 . With this decomposition, the matrix S µ ν defined by Eq. (2.2) is also diagonal by blocks with S a b = r f r g δ a b . For future convenience, we define the determinant of the block in the M 2 sector as D := det ( S A B ) and the ratio between the two area radii as ω := r f /r g . The nonvanishing components of the bimetric equations (2.11) for any general spherically symmetric spacetimes, which define our gravitational background, read as where we have introduced the label i ∈ { g, f } to write collectively the two metric sectors, with g ( g ) AB = g AB and g ( f ) AB = f AB . In addition, we have defined the vector fields and Q i := γ ab r 2 i t ( i ) ab on M 2 , while R ( i ) stands for the Ricci scalar of the corresponding two-dimensional metrics g AB and f AB . We have also introduced ( g ) ∇ and ( f ) ∇ as the covariant derivatives of the metrics g AB and f AB , respectively. This notation will be used throughout the paper. Moreover, in expressions with a label i , and wherever repeated capital Latin indices appear, the ensuing contraction should be understood as being performed with the corresponding metric g ( i ) AB . Finally, the components of the effective stress-energy tensors on M 2 are explicitly given by while the traces of the angular components read as where we have defined the contribution from the matter sector as Q m := γ ab T ab 8 πM 2 g r 2 g .", "pages": [ 5 ] }, { "title": "4.1 Tensor spherical harmonics", "content": "The usual scalar spherical harmonics Y m l = Y m l ( x a ) are defined as the eigenfunctions of the Laplacian operator acting on scalars, where ' : ' is the covariant derivative associated with γ ab , while l and m are integers such that l ≥ | m | . These special functions form a basis on the sphere, and thus any scalar function F = F ( x a ) can be written as a linear combination with certain complex constants F m l . Making use of the metric γ ab , its covariant derivative, and the antisymmetric tensor 1 /epsilon1 ab on S 2 , it is possible to generalize this basis to tensors of any rank (see Ref. [47] for more details). For instance, a basis for vectors on the sphere is given by the two vectors Z m l a := ∂ a Y m l and X m l a := /epsilon1 a b Z m l b , which are irrotational and divergence-free, respectively. Thus, any vector F a ( x b ) can be decomposed as, where ˜ F m l and ˆ F m l are constants. In the theory under consideration there are also rank-two tensors, for which we will use the basis { Z m l ab , X m l ab , γ ab Y m l , /epsilon1 ab Y m l } , with Z m l ab := Y m l : ab + l ( l +1) 2 γ ab Y m l and X m l ab := 1 2 ( X m l a : b + X m l b : a ) being symmetric and trace-free. The tensor harmonics have different polarity properties and they are divided into polar (or even) and axial (or odd) polarities. In the case of scalar functions, only polar components appear, and, in order to have a more uniform notation, we will denote Z m l := Y m l . In this way, all the terms multiplying a Z are polar, while those multiplying an X are axial. It is a well-known result in GR that different polarities decouple at the linear level, so long as the background is spherically symmetric. This holds true also in bimetric gravity, as explicitly shown below. Finally, we note that different harmonics are defined for a different range of values of l . More precisely, while scalar harmonics are defined for l ≥ 0, vector harmonics Z m l a and X m l a are exactly vanishing for l = 0, and thus only contribute for l ≥ 1, while tensor harmonics Z m l ab and X m l ab are nonvanishing only for l ≥ 2.", "pages": [ 6 ] }, { "title": "4.2 Decomposition of the perturbations into tensor spherical harmonics", "content": "The components of the metric perturbations h ( i ) µν have different tensorial rank in S 2 , which can be easily identified in terms of their indices. Namely, h ( i ) AB is a scalar, h ( i ) Ab is a vector, and h ( i ) ab is a symmetric rank-two tensor. Therefore, one needs to use a suitable basis, given by tensor spherical harmonics of the appropriate rank, as explained above. In this way, we introduce the following decompositions: for i ∈ { f, g } . For each set of labels ( ( i ) , l, m ) with l ≥ 2 there are ten new independent functions: these are encoded in a symmetric two-tensor H ( i ) m l AB , two vectors { H ( i ) m l A , h ( i ) m l A } , and three scalars { K ( i ) m l , G ( i ) m l , h ( i ) m l } , all of which only depend on coordinates of M 2 . From these, seven are polar { H ( i ) m l AB , H ( i ) m l A , K ( i ) m l , G ( i ) m l } and three are axial { h ( i ) m l A , h ( i ) m l } . Note that for l = 0 we only have the four polar components H ( i )0 0 AB and K ( i )0 0 , whereas, for l = 1, in addition to H ( i ) m 1 AB and K ( i ) m 1 , two polar components H ( i ) m 1 A and two axial components h ( i ) m 1 A are also present. Similarly, linear perturbations of the effective stress-energy tensors read as Note that, in the g -sector, according to the definition of the effective stress-energy tensors (2.9), besides the perturbations of the bigravity interaction term V (˜ g ) µν , the harmonic components defined above also include the contribution of the perturbations of the matter stress-energy tensor T µν . Hence, for future convenience, we also introduce the following decompositions Finally, we will also need the decomposition into spherical harmonics of the components of the perturbation ∆[ S µ ν ] of the matrix S , Note that, in general, neither S nor its perturbations are symmetric. This is why the harmonic coefficients S mA l and ˜ S m l B , and also s mA l and ˜ s m l A , are in principle different, and ˇ S m l is in general nonvanishing. In the following, we remove the harmonic labels l and m to make the notation lighter.", "pages": [ 6, 7 ] }, { "title": "4.2.1 The expression of ∆[ S µ ν ] in terms of metric perturbations", "content": "As explained in Sec. 2, in order to obtain the perturbed components of the matrix S in terms of the perturbations of g and f , we need to solve Eq. (2.17). Projecting this equation on the two-sphere, the scalar components of ∆[ S µ ν ] can be solved explicitly, and read as However, this is no longer the case for the vector and tensor harmonic components. In general, we have where ( M -1 ) A B is the inverse of the matrix M A B = ωδ A B + S A B . Note that such inverse is well defined because S A B cannot have real negative eigenvalues [37]. Analogously, we obtain whereas, for the two-tensor S A B , we can just write", "pages": [ 7, 8 ] }, { "title": "5.1 Gauge and physical degrees of freedom", "content": "At linear level, the perturbative gauge freedom can be parametrized in a vector field ξ µ , which defines the gauge transformation of the perturbation ∆[ T ] of any background tensor field T as In this sense, for any vector field ξ µ , ∆[ T ] and ∆[ T ] physically represent the same perturbation of T . If we perform the corresponding harmonic decomposition of the vector field, we observe that, for l ≥ 1, there are three polar { Ξ m l A , Ξ m l } and one axial { ξ m l } gauge degrees of freedom, while, for l = 0, there are only two polar components encoded in Ξ 0 0 A . Applying the above transformation to the metric perturbations h ( i ) µν , it is straightforward to obtain the gauge transformation of the different harmonic coefficients, where the barred objects are the harmonic coefficients of h ( i ) µν . Note, in particular, that the transformation (5.5c) is defined for all l ≥ 0, but, for l = 0, one should understand Ξ = 0. For l ≥ 2 a standard choice in GR, where only one metric is present, say g µν , is the Regge-Wheeler gauge, which corresponds to H ( g ) A = 0, G ( g ) = 0, and h ( g ) = 0. One can also construct gauge-invariant variables associated to this particular gauge [48]: these are defined by Eqs. (5.5) with ξ = -h ( g ) / 2, Ξ = -r 2 g G ( g ) / 2, and Ξ A = -H ( g ) A + G ( g ) | A / 2, which are the components of the generator of the infinitesimal transformation from a generic gauge to the Regge-Wheeler one. However, in the vacuum bimetric theory there are twice as many perturbative variables as in GR, but the same amount of gauge degrees of freedom. Therefore, it is not possible to simultaneously choose the Regge-Wheeler gauge for the perturbations of both metrics. In fact, since it is not clear a priori what gauge might be the most convenient one for the different applications of the formalism, we will refrain from imposing a specific gauge choice at the outset and we will present the equations of motion for any generic gauge. Concerning the number of physical propagating degrees of freedom, one needs to take into account that, for l ≥ 2, there are 12 first-class constraints in the theory, which can be classified in 3 four-vectors and thus contain 9 polar and 3 axial components. The two rank-two tensors h ( f ) µν and h ( g ) µν have a total of 20 (14 polar and 6 axial) components. Each first-class constraint removes 1 degree of freedom. In addition, the pair of second-class constraints characteristic of bimetric theory kills the scalar (Boulware-Deser) ghost, which is polar. In this way, for l ≥ 2, the theory contains 7 (4 polar and 3 axial) physical propagating degrees of freedom. Lower values of l need separate consideration. Since tensor spherical harmonics are not defined for l = 1, the above numbers differ, and, while the amount of first-class constraints remains as for l ≥ 2 (9 polar and 3 axial), there are only 16 (12 polar and 4 axial) metric perturbations. Therefore, for l = 1, there are 2 polar and 1 axial propagating degrees of freedom. Finally, for l = 0, neither tensor nor vector harmonics are defined and there are no axial degrees of freedom. In this case, the 6 polar first-class constraints and a couple of second-class constraints remove 7 degrees of freedom from the 8 possible, which leaves just 1 propagating physical degree of freedom.", "pages": [ 8, 9 ] }, { "title": "5.2 Perturbative equations of motion", "content": "As commented above, except for matter couplings, the action of the theory is invariant under the transformation (2.5). Making use of such a symmetry, it is straightforward to obtain the quantities associated to one metric from the quantities associated to the other. In this perturbative setup, it is clear how to implement (2.5) both for background objects and the harmonic coefficients of metric perturbations h ( i ) µν . Concerning the interaction terms, since (2.5) maps the matrix S to its inverse S -1 , the perturbations of S will be mapped to those of S -1 . From perturbing the relation ˜ S µ ν ( ˜ S -1 ) ν α = δ µ α , one obtains and from this expression one can read the harmonic coefficients of ∆[( S -1 ) µ ν ] from those of ∆[ S µ ν ]. In this section we will explicitly provide the equations of motion for the perturbations of the metric g µν , whereas the equations for the f -sector can be readily obtained using the symmetry (2.5) discussed above and removing matter variables. More precisely, apart from obvious changes in the labels g → f , in order to obtain the equations for the perturbations of f µν , one should perform the changes, of background objects, while the harmonic coefficients of ∆[ S µ ν ] must be changed as follows, In addition, since we are assuming matter coupled only to the g -sector, the perturbations of the matter stressenergy tensor ∆[ T µν ] must be taken to be identically zero to reproduce the equations for the f -sector, Ψ AB → 0 , Ψ A → 0 , ψ A → 0 , Ψ → 0 , ˜ Ψ → 0 , ψ → 0 . (5.9) The rest of the section is divided in two subsections where we analyze the axial (Sec. 5.2.1) and polar (Sec. 5.2.2) sectors separately. We recall that the differential part of the perturbative equations (2.13) corresponds to the usual first-order perturbed Einstein tensor in spherical symmetry, whereas bimetric effects are encoded in the linearized effective stress-energy tensor.", "pages": [ 9, 10 ] }, { "title": "5.2.1 Axial sector", "content": "For l = 0, all axial tensor spherical harmonics are identically zero, and the axial equations of motion are trivial. For l ≥ 1, the axial part of the ( Ab ) component of Eq. (2.13), that is the equation for ∆[ G ( i ) Ab ], gives where the potential V ( i ) l reads as The source term for the g -sector is while the source t ( f ) A can be obtained directly applying the rules (5.7)-(5.9) to the expression (5.12). For l = 1, the axial part of the equation for ∆[ G ( i ) ab ] is not defined, whereas for l ≥ 2, using the background equation (3.5), it reads as with and t ( f ) can be derived using (5.7)-(5.9). Therefore, the evolution of the axial sector is completely determined by Eqs. (5.10) and (5.13). As commented in the previous section, there is one gauge degree of freedom, which one can fix. With the equations at hand, we can analyze more explicitly the number of propagating degrees of freedom in this sector. For l = 1, there are four equations, all of them contained in the relation (5.10). Making explicit the second-order derivative terms and expanding in a generic chart x A = ( x 0 , x 1 ), they can be combined to give, schematically for i = { f, g } , where /squaresolid stands for background terms, while the dots encode first-order derivatives and terms with no derivatives. Since there are no second-order time derivatives, Eqs. (5.15a) are constraint equations, while Eqs. (5.15b) can be understood as evolution equations for the two functions h ( f ) 1 and h ( g ) 1 . However, the remaining axial gauge degree of freedom kills one of those, for instance by choosing h ( f ) 1 = 0, which leaves one single propagating axial degree of freedom for l = 1. Now, for l ≥ 2, in addition to the four equations (5.15), one also has (5.13), with principal part These can be understood as two evolution equations for h ( g ) and h ( f ) . There is the same amount of gauge freedom as for l = 1, and thus one ends up with three propagating axial degrees of freedom for l ≥ 2. Nonetheless, it is highly nontrivial to obtain the corresponding master variables that would obey unconstrained hyperbolic equations and would thus encode complete physical information on the problem. Following the procedure presented by Gerlach-Sengupta [27], one can define the following scalar functions 2 so that, taking then the curl of Eq. (5.10), yields for l ≥ 1, where Since in GR there is only one copy of equation (5.18), say for i = g , introducing the new matter invariant φ A = t ( g ) A -Q g 2 h ( g ) A [27], one can use the remaining gauge freedom to set h ( g ) = 0 (for l ≥ 2). In this way, in GR this equation is uncoupled to the rest of the metric perturbations and thus Π ( g ) follows an unconstrained evolution equation, which, for vacuum, reduces to the Regge-Wheeler equation [49]. However, in bimetric gravity there is not enough gauge freedom to set both h ( i ) to zero and, in addition, the sources t ( i ) A do not only correspond to matter perturbations, but they are complicated functions (cf. Eq. (5.12)) of the metric perturbations. Therefore, the variables Π ( i ) defined by (5.17) do not obey unconstrained master equations uncoupled to other metric perturbations.", "pages": [ 10, 11 ] }, { "title": "5.2.2 Polar sector", "content": "In this subsection, we provide the set of equations for the polar perturbations of the g -sector, while the equations corresponding to the f -sector can be obtained by applying the rules (5.7)-(5.9). On the one hand, the equation for ∆[ G ( i ) AB ] gives, for l ≥ 0, with On the other hand, from the equation for ∆[ G ( i ) Ab ], and for l ≥ 1, one obtains with Finally, ∆[ G ( i ) ab ] gives, for l ≥ 2, with and, for l ≥ 0, where Again, the polar components of ∆[ t ( f ) µν ] can be derived using (5.7)- (5.9). The number of propagating degrees of freedom in this sector can be analyzed following the same rationale as used in the axial sector. However, the polar case is much more involved, due to the greater number of equations and variables. Concerning master equations, we would like to note that the construction of a polar master variable for a generic background is an open question even in GR, and there are results only for certain specific backgrounds, like the Zerilli variable for vacuum [50].", "pages": [ 11, 12, 13 ] }, { "title": "6 Static backgrounds", "content": "Next, we proceed to apply the formalism developed in previous sections to specific backgrounds of interest. In this section we will assume that the background metric g µν is static, that is, it contains a hypersurfaceorthogonal Killing field ∂ t . Since exact bidiagonal solutions have been shown to lead to instabilities [18,23-25], such backgrounds will not be treated. Here we will focus instead exclusively on nonbidiagonal backgrounds, thus assuming that there does not exist a chart such that the metrics f µν and g µν are both diagonal. As it is well known [22, 33], imposing a staticity condition on g µν implies that f µν is also static, and has a Killing vector field ∂ T = 1 ˙ T ∂ t that is collinear with ∂ t . (Here and in the following an overdot is used to denote a derivative with respect to t .) We exhibit the general equations of motion for perturbations around such a static nonbidiagonal background in vacuo, obtained as a particular case of the equations derived in Sec. 5. Finally, we discuss the special case where the Killing vector fields of both metrics coincide, that is for ˙ T = constant .", "pages": [ 13 ] }, { "title": "6.1 Nonbidiagonal background metrics with a static g µν", "content": "Following Ref. [33], let us thus begin with the most general nonbidiagonal ansatz with a static form for g µν : /negationslash /negationslash /negationslash where r f is positive, C = 0, and the chart is valid for U = 0 and V = 0. Since g µν is diagonal and independent of t , its Einstein tensor G ( g ) µ ν is also diagonal. Moreover, it follows from the equations of motion (2.6a) (with T µν = 0) that V ( g ) µ ν must also be diagonal on solutions. This implies the following algebraic constraint /negationslash Since we are considering nonbidiagonal solutions with C = 0, this equation translates into the condition r f = ωr , with ω a positive root of Moreover, the Bianchi constraint ∇ ( g ) µ V ( g ) µν = 0 implies Thus, leaving aside the particular choice of parameters ( β 2 + ωβ 3 ) = 0, the combination of terms in square brackets must vanish. 3 This leads us to the following relation in terms of the metric functions Note, in particular, that the reality of the metric restricts the right-hand side of this expression to be strictly non-negative. Next, imposing (6.4) and (6.6), it can be shown that the equations of motion (2.6a) for the background at hand boil down to the Einstein equations, with the effective cosmological constant Λ g := β 0 + 2 ωβ 1 + ω 2 β 2 defined in terms of the parameters of the theory. Therefore, the standard Birkhoff theorem with cosmological constant applies, and the solution for the metric coefficients is which completely determines g µν as the Schwarzschild-(anti)de Sitter metric, depending on the sign of Λ g . Now, under the above assumptions, the equations of motion for f µν (2.6b) are decoupled from g µν and they also reduce to the Einstein equations, with the corresponding cosmological constant given by Λ f := 1 ω 2 ( β 2 +2 ωβ 3 + ω 2 β 4 ). In addition, the metric functions must also obey the nonbidiagonal condition (6.6). In order to solve these equations, it is convenient to change to new coordinates ( T, r f ), with T = T ( t, r ), where the metric f µν becomes diagonal, The solution of Eq. (6.9) in these new coordinates is once again the diagonal form of the Schwarzschild-(anti)de Sitter metric with Σ f = 1 -2 µ f r f -m 2 Λ f 3 α 2 r 2 f . Transforming back to the original ( t, r ) coordinates, one finds the relations which, upon substitution into Eq. (6.6), yield the following partial differential equation for the unknown function T = T ( t, r ), Here we have defined ˙ T := ∂T/∂t and T ' := ∂T/∂r . Note that, in general, the function T will depend on both ( t, r ). In fact, for C to be nonvanishing, so as to ensure a nonbidiagonal form of the metrics, neither ˙ T nor T ' can vanish. In particular, this excludes the case where the two metrics describe black holes with the same mass and cosmological constant, since that would imply Σ g = Σ f and thus, following (6.13), T ' = 0. Since (6.13) is a nonlinear partial differential equation, there is no systematic procedure to obtain its general solution T = T ( t, r ). In addition, the reality conditions imply that the right-hand side of (6.13) must be non-negative, which, in general, will impose certain restrictions on ˙ T (or, if one had a general solution at hand, on the corresponding integration constants). Interestingly, in regions where Σ f Σ g < 0, the right-hand side of (6.13) is positive definite, and thus ˙ T is unrestricted by this condition. Note also that, in terms of the function T , for this nonbidiagonal ansatz, the matrix S can be written in the following compact form, From these expressions, it is straightforward to conclude that the matrix S will be real as long as T is real. There is, however, one specific interesting case where Eq. (6.13) can be solved. Namely, if one assumes that the Killing vector field of both metrics coincide, and thus ˙ T is constant, the equation can then be reduced to the quadrature, with c an integration constant. Owing to the reality conditions discussed above, this integration constant is not completely free in general, and it is constrained so that the argument of the square root is positive, a condition that will depend on the specific parameters (mass and cosmological constant) of the black holes and on the range of r . Remarkably, the choice c 2 = ω 2 is the only one that reduces the argument of the square root to a perfect square, and therefore it is valid for any parameter of the black holes and any range of r . Furthermore, we note that the background geometry considered in Ref. [26] can be obtained as a particular case of our more general (6.15) with c = ω and Λ g = Λ f = 0. At background level, the interaction between the two metric sectors only manifests itself through the cosmological constants Λ g and Λ f , so that the two metrics are effectively decoupled. Therefore, one could treat both metrics as independent and take a different coordinate frame for each, for instance, such that both are diagonal (i.e., ( t, r ) for g µν and ( T, r f ) for f µν ). In this sense, c does not have a physical impact on the background geometry, and, in particular, no curvature invariant depends on c . Hence, at the background level, this constant only appears when one relates the two metrics. For instance, it affects the relative tilt of the lightcones of g µν and f µν (for an analysis of the causal structure in the general case see Refs. [32,37]). However, at a perturbative level the two sectors are indeed coupled, and the constant c appears in the equations of motion in a nontrivial way.", "pages": [ 13, 14, 15 ] }, { "title": "6.2 Linear perturbations on a static nonbidiagonal background", "content": "Here we compute the source terms in the equations of motion for linear perturbations around nonbidiagonal static backgrounds. The Killing vector field of g µν is ∂ t . Then, under these conditions, f µν is also static, though its Killing vector field ∂ T generically does not coincide with ∂ t , but is instead defined in terms of the function T = T ( t, r ) that solves Eq. (6.13). For a general static nonbidiagonal spherically symmetric ansatz, the expressions for the axial harmonic components of the perturbed effective stress-energy tensor for the metric g µν , Eqs. (5.12) and (5.14), imposing T µν = ∆[ T µν ] = 0, take the form where we have introduced the two-by-two matrix At this point, it is clear that when the Killing vector fields of both metrics coincide (and therefore ˙ T = c ), the constant c will appear explicitly in the equations of motion through the source terms. Similarly, for the axial harmonic components of ∆[ t ( f ) µν ], we have As for the polar components, Eqs. (5.21), (5.23),(5.25), and (5.27), boil down to with the following two matrices: The corresponding source terms for the f -sector read as In the above expressions one can explicitly check that, as commented previously, for the particular case β 1 + ωβ 2 = 0, the metric sectors are decoupled also at the linear level. Finally, we would like to remark that the case | ˙ T | = ω analyzed in Ref. [26] is, at first sight, a very particular choice that considerably simplifies the source terms. Even more, as shown in the mentioned reference, for this choice both metrics can be conveniently written in the advanced Eddington-Finkelstein form, simplifying even more the expressions above.", "pages": [ 15, 16 ] }, { "title": "7 Conclusion", "content": "We have presented the equations to describe the evolution of linear perturbations of bimetric gravity on a completely general spherically symmetric background spacetime. In order to obtain a covariant setup, valid for any coordinate choice, we have followed the formalism by Gerlach-Sengupta. More precisely, we have performed a 2+2 decomposition of the manifold, so that the background metric is written as a warped product between a two-dimensional metric on a Lorentzian manifold and the metric of the two-sphere. Then we have decomposed all perturbative variables in the natural basis given by tensor spherical harmonics. This removes the dependence on the angles from the different equations and defines two polarity sectors (axial and polar), which evolve independently at the linear level. In the bimetric theory, there are two sets of equations for linear perturbations, one set for each metric, that couple through effective stress-energy tensors determined by the bimetric interaction potentials. That is, in addition to the contribution of ordinary matter fields, each metric sees the other effectively behaving as a source in the field equations. Hence, the difference with respect to GR, where the matter stress-energy tensor is independently prescribed and matter perturbations are defined independently of the geometry, lies in the fact that here one needs to obtain the explicit expressions for the perturbed effective stress-energy tensors in terms of the perturbations of the two metrics. Such expressions are presented in Sec. 5, and represent one of the main results of this paper. Owing to the fact that there are twice as many variables as in GR, the dynamical content of the theory is much more intricate and, instead of two, there are seven propagating degrees of freedom. In particular, we have discussed the number of propagating degrees of freedom for each polarity sector and for each multipole l = 0, l = 1, and l ≥ 2. However, the construction of explicit master equations to describe these physical degrees of freedom in the general case is far from trivial. In GR, for a general spherical background, only the Gerlach-Sengupta master equation is known in the axial sector, but there is no such result for the polar one. For the bimetric theory, we have followed the construction by Gerlach-Sengupta for the axial sector and shown that the obstruction to obtain an unconstrained independent equation for the Gerlach-Sengupta master variable lies, on the one hand, in the coupling between the perturbations of the two metrics, and, on the other hand, in the fact that, unlike in GR, there is not enough gauge freedom to remove certain variables. This formalism is valid for any spherically symmetric background, which, in general, might be dynamical. Even so, as an interesting application, in the last section we have considered the case of a nonbidiagonal static background. More precisely, we have assumed that one of the background metrics ( g µν ) contains a hypersurfaceorthogonal Killing field, and that there is no chart where both metrics are simultaneously diagonal. These assumptions imply that both background geometries are solutions of the Einstein equations, and thus they correspond to the Schwarzschild-(anti)de Sitter geometry with collinear Killing vector fields, while deviations from GR become manifest at the perturbative level. In addition to the two masses and the two cosmological constants, the only freedom at the background level corresponds to the norm of the Killing field of f µν , which is encoded in the function T = T ( t, r ) that obeys Eq. (6.13). It is not possible to obtain the general analytic solution for this equation, and thus we have left T ( t, r ) unspecified in the evolution equations for the perturbations, so as to ensure that our results are valid for any static nonbidiagonal solution and are presented in a form suitable for future studies.", "pages": [ 16, 17 ] }, { "title": "Acknowledgements", "content": "This work has been supported by the Basque Government Grant IT1628-22 and by the Grant PID2021123226NB-I00 (funded by MCIN/AEI/10.13039/501100011033 and by 'ERDF A way of making Europe'). ASO acknowledges financial support from the fellowship PIF21/237 of the UPV/EHU. MdC acknowledges support from INFN (iniziative specifiche QUAGRAP and GeoSymQFT).", "pages": [ 17 ] } ]

[ { "title": "On the (non-)degeneracy of massive neutrinos and elastic interactions in the dark sector", "content": "Jose Beltr'an Jim'enez, 1, ∗ David Figueruelo, 1, † and Florencia Anabella Teppa Pannia 2, ‡ 1 Departamento de F'ısica Fundamental and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain. 2 Departamento de Matem'atica Aplicada a la Ingenier'ıa Industrial, Universidad Polit'ecnica de Madrid, E-28006 Madrid, Spain. (Dated: March 6, 2024) Cosmological models featuring an elastic interaction in the dark sector have been shown to provide a promising scenario for alleviating the σ 8 tension. A natural question for these scenarios is whether there could be a degeneracy between the interaction and massive neutrinos since they suppress structures in a similar manner. In this work we investigate the presence of such a degeneracy and show that the two effects do not exhibit strong correlations.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Our current cosmological model, namely ΛCDM, has received a remarkable support from the different data that have been collected in the last decades and which include observations of the Cosmic Microwave Background [1], Supernovae [2, 3], Baryon Acoustic Oscillations [4-6], Large Scale Structures or Weak Lensing [7, 8], etc. As the instruments improve and the amount of data increase so does the precision on the measured cosmological parameters and some tensions between different datasets have commenced to appear [9, 10]. One of these tensions is the apparent discrepancy in the clustering of matter as predicted by CMB measurements [11] and those based on low redshift observations [12, 13]. This tension is usually described in terms of the parameter σ 8 (or the related S 8 ) that parameterises the amplitude of matter fluctuations on spheres of 8 Mpc /h . As usual, the tension could be driven by unknown systematics, but it could also be signalling the need for physics beyond ΛCDM. Since the tension seems to indicate that the clustering in the late-time universe appears to be smaller than what the CMB suggests, it is natural to appeal to some mechanism that erases structures or prevents the clustering at low redshift. This idea is realised in scenarios where the dark sector features some interaction between dark matter and dark energy (see Ref. [14] for a review on interactions between dark energy and dark matter). Among all the plethora of interacting models, those with a mechanism that naturally operates at late times, when precisely dark energy becomes relevant and then such a mechanism will naturally emerge, would naturally accommodate the lower clustering suggested by late probes. Moreover, if the interaction effectively provides dark matter with a pressure, that mechanism will prevent its clustering due to such gained pressure. Furthermore, in order to leave the background cosmology un- ected, so we do not worsen the Hubble tension [15, 16] as typically happens, one can consider the interaction to be elastic. This results in that there is only momentum exchange between the dark components at linear order. The described scenario has been investigated in several versions and in all its variants (see Refs. [17-25] for several examples where a momentum exchange takes place), being all in an agreement that the elastic interaction is efficient in alleviating the σ 8 tension. Furthermore, it has been repeatedly reported that, when including measurements of S 8 , not only the σ 8 tension can be alleviated, but the non-interacting case is excluded at several sigma s and a detection of the interaction could be inferred. Also, if the interaction is indeed there, it has been shown that the dipole of the matter power spectrum might provide a smoking gun [26]. On the other hand, it is known that the presence of massive neutrinos also suppresses the growth of structures on small scales and at relatively low redshift when they become non-relativistic [27, 28]. Thus, a natural and pertinent question to ask is whether the effects of the interaction could be degenerated with massive neutrinos since both appear to have similar effects on the matter power spectrum, i.e., they both tend to suppress the growth of structures at late times and on small scales. This degeneracy would introduce a degradation on the measured value of the interaction parameter and, hence, it would reduce the significance of the previous findings in the literature seemingly pointing towards a detection of the interaction. The goal of this work is to analyse the presence of such a degeneracy and unveil whether allowing for a varying neutrinos mass could actually degrade the measurement of the interaction parameter. The work is structured as follows. In Sec. II we briefly review the model that we will consider and we will compare its effects on the matter power spectrum with those of massive neutrinos. We then will perform a MCMC analysis in Sec. III to confront different datasets to the predictions of the interacting model while allowing for a varying neutrino mass. Finally, we will conclude in Sec. IV with a discussion of our main results.", "pages": [ 1 ] }, { "title": "II. THE DARK ELASTIC INTERACTING MODEL", "content": "We will consider the model with an elastic interaction introduced in Ref. [19] that has been subsequently analysed in Refs. [22, 29-31]. The general idea is to modify the conservation equations of the dark sector by introducing an interaction that only affects the perturbations. This is achieved by exploiting the existence of a common rest frame on large scales for all the components in our universe. In this scenario, the interaction is assumed to be proportional to the relative 4-velocities of the dark fluids 1 so the conservation equations read with u µ dm and u µ de the 4-velocities of cold dark matter and dark energy respectively and α a constant parameter that controls the strength of the interaction. As desired, this interaction will modify the standard evolution only when the relative velocity between the dark components is nonnegligible. The parameter α has dimension 5 in natural units and the natural scale associated to it is ρ c H 0 so, from now on, we will work with a dimensionless parameter α that will be understood to be normalised with 3 H 3 0 8 πG . This will be the only new parameter of this scenario and it will control both the strength of the interaction as well as the redshift at which it becomes important. In the described scenario, the density contrast δ and velocity perturbations θ in the dark sector will be coupled not only in the usual indirect way through the gravitational potential, but they will also feature a direct coupling via the interaction. In the Newtonian gauge, the perturbation equations in the dark sector read [19, 29]: where w is the dark energy equation of state parameter and c 2 de its adiabatic sound speed squared. In the above equations, we have also introduced the convenient quantities Γ and R defined as ̸ These are the physically relevant quantities because Γ measures the effective interaction rate in the dark sector, while R gives the relative fraction of dark energy to dark matter. Notice that we need w = -1 to have an effect. In the strict w = -1 case, the dark energy component does not have perturbations and, hence, our scenario fails. 2 As advertised, the interaction simply adds a new term to the Euler equations of the coupled fluids. This term formally resembles that of the Thomson scattering between baryons and photons before decoupling. For this reason, we refer to our scenario as covariantised dark Thomson-like scattering. It is clear from the equations that the interaction requires the presence of peculiar velocities between the dark components, something that only occurs on sub-Hubble scales, as well as an interaction rate larger than the Hubble expansion rate, something that parameterically occurs when Γ ≳ H , i.e., at late times. From this discussion, it is clear that the interaction will affect the small scales at low redshift. For those scales, the interaction provides the dark matter component with an effective pressure originated from the pressure of dark energy. This pressure will work against the dark matter clustering, thus leading to a suppression of the matter power spectrum on small scales and at late times. This suppression of the matter power spectrum looks similar to the effect of massive neutrinos when they become non-relativistic, thus it is natural to wonder if both effects could be degenerate. In Figure 1 and Figure 2 we show the effects of the interaction and massive neutrinos in the matter power spectrum for different cosmologies. We can see that both the interaction and the massive neutrinos produce a similar suppression of the matter power spectrum on small scales. However, we can already see that, while the interaction does not modify the background, the massive neutrinos cosmologies also have an impact in the background evolution and this is reflected in a modification of the power spectrum on very large scales. Of course, this can be corrected by varying other background cosmological parameters. On the other hand, the scale-dependence of the suppression of the matter power spectrum also differs in both scenarios. These different effects can help breaking the potential degeneracies between both scenarios. In particular, a full shape analysis of the power spectrum should allow to break possible degeneracies. Another distinctive feature of the elastic interaction is that it modifies the dipole of the matter power spectrum and this effect will clearly allow to distinguish the interaction from massive neutrinos, which do not produce such an effect [26]. It should be clear then that the potential degeneracy between both scenarios can eventually be broken. In this work, however, we are interested in analysing if the results already obtained in the literature for the elastic interacting scenarios are prone to a degeneracy with massive neutrinos because some observables are not sensitive to the discussed effects or they are not sufficiently precise to see the differences between both scenarios. This is important because the existing studies point towards a possible detection of the interaction pa- rameter that is significantly favoured with respect to the non-interacting scenario. It is then important to analyse if allowing for a varying neutrino mass could alter these findings. Before proceeding to the observational constraints on the interacting scenarios with massive neutrinos, let us also comment on the effects on the CMB power spectrum. In Figure 3 we show the joint effects of the interaction and the neutrino mass in the cosmic microwave background. The power spectrum for temperature is most significantly modified at large scales through the late Integrated Sachs-Wolfe (ISW) effect (as expected because the interaction is relevant at very late times) and the reduction of the weak lensing effect due to the neutrino mass. The amplitude of the peaks is also modified due to the impact of the total neutrino mass in the background and perturbations evolution. An increase in neutrino mass also produces a decrease in the late ISW effect, reducing the CMB temperature spectrum at low l s. In general, we can see that the massive neutrinos have a bigger impact on the CMB than the interaction and this is in part due to the fact that the interaction does not affect the background evolution.", "pages": [ 2, 3, 4 ] }, { "title": "III. MCMC RESULTS", "content": "In order to address the possible degeneracy between the elastic interaction and massive neutrinos, we will perform Markov Chain Monte Carlo (MCMC) analyses that extend those performed in [19, 22, 29, 30] by allowing for a varying neutrino mass. For that, we will use a modified version of the Boltzmann solver for cosmological perturbations CLASS [34, 35] that includes the inter- action [29], 3 so that we can use the MCMC code MontePython [38, 39] for sampling the parameter space. In our analyses we are going to consider the following two different cosmologies: Consequently, the cosmological parameters to be sampled are baryon density as 100Ω b h 2 , the dark matter density as Ω dm h 2 , the angle of the comoving sound horizon at recombination as 100 θ s , the amplitude of primordial perturbations as ln(10 10 A s ), the scalar spectral index n s , the optical depth τ reio , the neutrino mass m ν and the equation of state of dark energy w . In addition to that, we will also have the coupling parameter of the covariantised dark Thomson-like model α . As derived parameters we will have the redshift of reionisation z reio , the total matter abundance Ω m , the primordial Helium fraction Y He , the Hubble parameter H 0 and the root-mean-square of density fluctuations inside spheres of 8 h -1 Mpc radius σ 8 . We will set flat priors on the parameters with bounds only for α and w as α ∈ [ -0 . 01 , 100] for the coupling parameter and w > -1 for the equation of state of dark energy due to stability reasons. We will also use the conservative bound on the neutrino mass of m ν ∈ [0 , 2] eV. For the datasets to be used, we will consider the following combinations: using the results from the DES survey third year release consisting of S DES -Y3 8 , obs = 0 . 776 ± 0 . 017 [48]. The use of this data has some caveats (see e.g. the discussion in this respect in Ref. [22]), but it is a common approach and we will also adopt it here. This is however a very important point because it is precisely this data and used in this manner what permits to constrain the interaction parameter. We could have also introduced SunyaevZeldovich data [13] or KIDS data [49] in the same way, but it would not change substantially our results and it will not be necessary for our purpose here, which is to show the impact of varying the neutrino mass. In all the scenarios, we will make use of the GelmanRubin criteria [50] satisfying that | R -1 | < 0 . 01 in order to ensure the convergence of the chains. The results that we obtain are given in Table I and in Figure 5 where we show the constraints for some relevant parameters in the ν -ΛCDMand ν -α CDMscenarios. A general conclusion inferred from the results is that, as expected by the very nature of the interaction, most of the parameters remain unchanged in the presence of the momentum transfer. The other outcome from the MCMC analyses is that the interaction can only be detected once we add low redshift data, in our case the third year DES data, in agreement with previous findings [22, 29, 30]. As a matter of fact, when no low-redshift data is used, the results show no lower constraint for the coupling param- and we only obtain the bounds: However, when we add the low-redshift data in the very simple form of the Gaussian likelihood explained above, we are able to establish both a lower and an upper constraint on the coupling parameter of the covariantised dark Thomson-like scattering, resulting in the following value: Let us notice that the coupling parameter is detected to be different from zero with a 2 σ confidence level only once the low-redshift data are used. Similar occurrences have been found using other datasets, such as the CFHTLenS [51] or the Planck Sunyaev-Zeldovich clusters counts [13]. In order to understand the previous result, one has to first consider that the covariantised dark Thomson-like scattering is a very late-Universe interaction and only occurs on small scales, while the previous surveys are not, except for DES-Y3. Therefore, they are unable to set strong constraints on the strength of the interaction as they are not sensitive to it, i.e., both high redshift perturbations and the background cosmology are oblivious to the interaction. The second reason for this trend relates to the nature of the interaction and the σ 8 / S 8 tension. Those previous surveys, used in the very simple form of a Gaussian likelihood, capture the low-redshift indication for less structures in our current Universe than the amount of structures suggested by early Universe datasets. Therefore, a low-redshift mechanism capable of reducing the clustering, as the momentum transfer, will naturally accommodate those values for S 8 , as we have in turn obtained. These results permit us to answer our question, namely: there is no strong degeneracy between the interaction and the mass of the neutrinos so that allowing for a varying neutrino mass does not degrade the potential detection of the interaction. This is clearly seen in the α -m ν plane in Figure 5 where we see that the constraint on α is insensitive to the value of m ν . Remarkably, we see that, although more massive neutrinos could lower the value of σ 8 (or S 8 ), it is the effect of the interaction what mainly drives its value. In fact, we can see the clear correlation between α and σ 8 , while the values of σ 8 and m ν are not correlated. In order to illustrate this more clearly, in Figure 9 we show the σ 8 -α plane coloured with the value of m ν . In that figure it is apparent that the mass of the neutrinos plays no role in the value of σ 8 and that it is exclusively the interaction what drives σ 8 towards smaller values. This is also shown in Figure 10 where the plane α -m ν coloured with the value of σ 8 is shown. Apart from this, there are certain parameters that deserve further explanations: In addition to the parameter constraints that we have discussed, it remains to analyse the goodness of the fit. In Table I we provide the best-fit values of χ 2 together with the AIC [52] information criteria. Comparing the results to the cases with only massless neutrinos or with one massive neutrino fixed to m ν = 0 . 06 eV displayed on Table II, the preferences are weakened, as expected because we have fewer free parameters and, then, the AIC criteria penalises less. However, the important result to highlight is that ν -α CDM is preferred over ν -ΛCDM when we add the low-redshift (DES-Y3) data. Thus, this means that even when we allow for the neutrino masses to vary, the interacting scenario is favoured over the noninteracting scenario.", "pages": [ 4, 6, 7 ] }, { "title": "IV. CONCLUSIONS", "content": "In this paper we have investigated the possible presence of a degeneracy between the neutrino mass and the coupling parameter of a momentum transfer interaction in the realm of the covariantised dark Thomson-like scattering. This model has been investigated in previous works and it has been shown to be a promising scenario for alleviating the σ 8 tension and, furthermore, the addition of low-redshift data seems to signal the presence of the interaction. Since these scenarios give a suppression of the matter power spectrum on small scales and this effect is to some extent shared by scenarios with massive neutrinos, it is important to unveil whether there are degeneracies between both effects. We have first studied the effects of the interaction and the neutrino mass in standard observables like the matter power spectrum and the cosmic microwave background. We have shown how the small scale suppression has a different scale dependence in both cases, being therefore a first probe of the non-existence of the α -m ν degeneracy. Furthermore, as we have argued, the massive neutrinos also affect the background cosmology, while the elastic interaction, by construction, leaves the background unaffected and this could be another way of breaking the degeneracies. However, there are observables that are not very sensitive to the different scale dependence of the suppression and the modifications on the background evolution could be compensated with variations of other background quantities and, thus, there could still be some residual degeneracies. In order to set clearly the existence or not of the degeneracy we have performed several MCMC analyses using the latest available datasets. In particular, cosmic microwave background and baryonic acoustic oscillations data, while we also use the results from DES third year for the S 8 parameter as a Gaussian prior in our analyses. As already found in previous studies, the interaction can only be detected when the latest dataset, DES-Y3, is considered. Without that dataset, or other low-redshift probes, we can only put an upper bound on the coupling parameter of the covariantised dark Thomson-like scattering. Once low-redshift information is added, we find a detection of the interaction at more than 2 σ confidence level. Such detection is intrinsically related to a lower value of the σ 8 (or S 8 ) parameter, since the interaction induces a suppression of structures which is precisely captured by that parameter. Low-redshift datasets are continuously suggesting there are less structures in our current Universe than the expected ones from early Universe probes like the cosmic microwave background. Consequently, a late-time interaction, like the one studied here, can naturally accommodate both early and late Universe probes to a compatible value for the σ 8 or S 8 parameters and, thus, solving the corresponding σ 8 or S 8 tension. The other cosmological parameters remain oblivious to the presence of the interaction. Regarding the possible degeneracy between the neutrino mass and the coupling parameter, we have been able to establish in a clear manner that such a degeneracy does not exist. Instead, we discovered that the well-known correlation in the σ 8 -H 0 plane with the neutrino mass disappears once the interaction is considered. Given this interaction is able to alleviate or even solve the σ 8 or S 8 tension, one can envision a mechanism to solve the H 0 tension which combined with the momentum transfer models could simultaneously account for both tensions. This particular situation is something that does not happen in the standard ΛCDM model or in a broad plethora of alternative cosmological descriptions. As a final comment, let us emphasise that our elastic interacting scenario serves as a proxy for more general cosmologies featuring a pure momentum exchange, so our findings are expected to be also valid for those scenarios. Acknowledgements: We thank Dario Bettoni for useful discussions and collaboration in related works. JBJ, DF and FATP acknowledge support from the Atracci'on del Talento Cient'ıfico en Salamanca programme, from Project PID2021-122938NB-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by 'ERDF A way of making Europe', and Ayudas del Programa XIII by USAL. DF acknowledges support from the programme Ayudas para Financiar la Contrataci'on Predoctoral de Personal Investigador (ORDEN EDU/601/2020) funded by Junta de Castilla y Le'on and European Social Fund. m", "pages": [ 7, 8, 9, 12 ] } ]

[ { "title": "Improved binary black hole searches through better discrimination against noise transients", "content": "Sunil Choudhary, 1, 2, 3, ∗ Sukanta Bose, 1, 4, † Sanjeev Dhurandhar, 1, ‡ and Prasanna Joshi 5, 6, § 3 Department of Physics, University of Western Australia, Crawley WA 6009, Australia 6 Leibniz Universitat Hannover, 30167 Hannover, Germany Short-duration noise transients in LIGO and Virgo detectors significantly affect the search sensitivity of compact binary coalescence (CBC) signals, especially in the high mass region. In a previous work by the authors [1], a χ 2 statistic was proposed to distinguish them, when modeled as sineGaussians, from non-spinning CBCs. The present work is an extension where we demonstrate the better noise-discrimination of an improved χ 2 statistic - called the optimized sine-Gaussian χ 2 - in real LIGO data. The extension includes accounting for the initial phase of the noise transients and use of a well-informed choice of sine-Gaussian basis vectors selected to discern how CBC signals and some of the most worrisome noise-transients project differently on them [2]. To demonstrate this improvement, we use data with blip glitches from the third observational run (O3) of LIGO-Hanford and LIGO-Livingston detectors. Blips are a type of short-duration non-Gaussian noise disturbance known to adversely affect high-mass CBC searches. For CBCs, spin-aligned binary black hole signals were simulated using the IMRPhenomPv2 waveform and injected into real LIGO data from the same run. We show that in comparison to the sine-Gaussian χ 2 , the optimized sine-Gaussian χ 2 improves the overall true positive rate by around 6% in a lower-mass bin ( m 1 , m 2 ∈ [20 , 40] M ⊙ ) and by more than 3% in a higher-mass bin ( m 1 , m 2 ∈ [60 , 80] M ⊙ ). On the other hand, we see a larger improvement - of more than 20% - in both mass bins in comparison to the traditional χ 2 .", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Gravitational-wave (GW) astronomy has achieved several feats in recent years - following up on the first detection of the binary black hole merger GW150914 [3]. After that first breakthrough detection, two LIGO detectors (in Livingston and Hanford) [4] along with the Virgo detector (in Cascina) [5] have observed more than 90 compact binary coalescence (CBC) signals from various kinds of binaries involving black holes (BHs) and neutron stars (NSs) in their first three observation runs [6]. The fourth observation (O4) run began in 2023, and is expected to include KAGRA [7]. The GW community is expecting the CBC detection rate to increase significantly in O4. It is, therefore, important to find ways to effectively handle data quality and detector characterization to improve the search sensitivity so as not to miss interesting signals. Currently, high-mass CBC searches (for component mass > 60 M ⊙ ) are adversely affected by noise transients [8, 9] and some works have developed techniques to improve the search sensitivity in that part of the CBC parameter space [1, 2, 8, 10-14]. These works include statistical, instrumental and, recently, a few machine-learning efforts. All studies about CBC search sensitivity in the high-mass region typically mention the impact of blip glitches [15] as a major source of deterioration. These glitches are a type of short-duration non-Gaussian noise artifact found in both LIGO detectors as well as in the Virgo detector. The duration of these glitches is around 10ms. In the frequency domain they are over 100Hz wide. Studies on the blips in O2 and O3 [15, 16] mention that they occur 2-3 times per hour in both LIGO detectors. The reason behind blips affecting the CBC search sensitivity is that their time-frequency morphol- ogy has a lot of similarity with GW signals from CBCs with high total mass. These are essentially signals from binary black holes (BBHs). According to recent blip studies, their source is still not fully known [15]. These types of glitches do not show much correlation with any of the auxiliary channels (i.e., noise source monitoring channels). Therefore, it is tricky to confirm them as non-astrophysical in origin and remove them from short-duration signal searches. One way to veto blips from GW data is to develop a statistical test that can differentiate them from CBC signals based on their different time-frequency characteristics, e.g., in spectrograms. There are χ 2 statistics, such as the traditional (or power) χ 2 [17] and sine-Gaussian χ 2 [8], that are implemented in GW search pipelines to tackle glitches, with, especially, the latter showing some success in discriminating against blips. There are yet other χ 2 s that check for signal consistency by employing expected SNR variation in time or across a CBC templatebank [18-21]. Still there remains room for improving current blip discrimination methods. In this work, we exploit the unified χ 2 formalism [1] to develop a new χ 2 statistic that incorporates information about how blip glitches and BBH signals project differently on a basis of sine-Gaussian functions. Following that work, we call it the optimized sine-Gaussian χ 2 statistic. We also tune it in real data to specifically reduce the adverse impact of blip glitches on the sensitivity of spinning BBH searches. Previous work on this χ 2 statistic in Ref. [1] had targeted simulated sine-Gaussian transients and non-spinning BBH signals. This paper is organized as follows. In Sec. II, we discuss theoretical aspects of the optimized SG χ 2 , including a brief introduction to the general framework of χ 2 statistics. Sec. III describes the procedure for constructing the optimized SG χ 2 . In particular, we show how to identify the basis vectors for this statistic and how to employ singular-value decomposition to choose from them the most effective ones - a limited few. Sec. IV presents the results and performance of optimized SG χ 2 in discriminating simulated spin-aligned BBH signals in real LIGO data from its third observation run (O3), which includes thousands of real blip glitches. Finally, in Sec. V we discuss the future applicability and prospects of this work.", "pages": [ 1, 2 ] }, { "title": "A. General Framework", "content": "The general framework for χ 2 discriminators has been described in Ref. [22]. It shows how the various χ 2 discriminators can be unified into a single discriminator, which can be appropriately termed as the unified χ 2 . In this framework, a data train x ( t ) defined over a time interval [0 , T ] is viewed as a vector x . Such data trains form a vector space D . Vectors in D will be denoted in boldface, namely, x , y ∈ D . Since the detector strain is typically sampled at a high rate, of O (10 3 ) Hz, and the signals studied here can be as long as O (10 3 ) sec, the data vectors can have large number of components, i.e., N ∼ 10 6 or larger. Hence, D is essentially the N -dimensional real set R N . When additional structure is added to D , namely, that of a scalar product, then it becomes a Hilbert space. Next consider the detector noise n ( t ), which is a stochastic process defined over the time segment [0 , T ]. It has an ensemble mean of zero, and is stationary in the wide sense. A specific noise realisation is a vector n ∈ D , where n is in fact a random vector. Its one-sided power spectral density (PSD) is denoted by S n ( f ). If ˜ x ( f ) and ˜ y ( f ) are the Fourier representations of the vectors x and y , respectively, then the scalar product of two vectors x and y in D is given by: where the integration limits usually demarcate the signal band of interest, [ f lower , f upper ]. We have used an integral for the scalar product because the number of components of a data vector is very large, as argued above, and the continuum limit may be taken from a sum to an integral. The χ 2 discriminator is a mapping from D to positive real numbers and is defined so that its value for the signal is zero and for Gaussian noise has a χ 2 distribution with a reasonable number of degrees of freedom, p . Typically, the number of degrees of freedom is a few tens to a hundred. If a template h is triggered, then the χ 2 for h is defined by choosing a finite-dimensional subspace S of dimension p that is orthogonal to h , i.e., for any y ∈ S , we must have ( y , h ) = 0. Then the χ 2 for the template h is defined as just the square of the L 2 norm of the data vector x projected onto S . Specifically, we perform the following operations. Take a data vector x ∈ D and decompose it as: where S ⊥ is the orthogonal complement of S in D . x S and x S ⊥ are projections of x into the subspaces S and S ⊥ , respectively. We may write D as a direct sum of S and S ⊥ , that is, D = S ⊕ S ⊥ . Then the required statistic χ 2 is, The χ 2 statistic so defined has the following properties. Given any orthonormal basis of S , say e α , with α = 1 , 2 , ..., p and ( e α , e β ) = δ αβ , we obtain the following: Observe that the random variables ( n , e α ) are independent and Gaussian, with mean zero and variance unity. This is because ⟨ ( e α , n )( n , e β ) ⟩ = ( e α , e β ) = δ αβ , where the angular brackets denote ensemble average (see [23] for proof). Thus, χ 2 ( n ) possesses a χ 2 distribution with p degrees of freedom. For convenience, one is free to choose any orthonormal basis of S . In an orthonormal basis the statistic is manifestly χ 2 since it can be written as a sum of squares of independent Gaussian random variables, with mean zero and variance unity. In the context of CBC searches, however, we have a family of waveforms that depend on several parameters, such as masses, spins and other kinematical parameters. We denote these parameters by λ a , a = 1 , 2 , ..., m . The templates corresponding to these waveforms are normalized, i.e., ∥ h ( λ a ) ∥ = 1. Then the templates trace out a manifold P - the signal manifold - which is a submanifold of D . We now associate a p -dimensional subspace S orthogonal to the template h ( λ a ) at each point of P - we have a p -dimensional vector-space 'attached' to each point of P . When done in a smooth manner, this construction produces a vector bundle with a p -dimensional vector space attached to each point of manifold P . We have, therefore, found a very general mathematical structure for the χ 2 discriminator. Any given χ 2 discriminator for a signal waveform h ( λ a ) is the square of the L 2 norm of a given data vector x projected onto the subspace S at h ( λ a ). It can be easily shown that the traditional χ 2 falls under the class of unified χ 2 . This is done by exhibiting the subspaces S or by exhibiting the basis vector field for S over P ; the conditions mentioned above must be satisfied by S . In [24] such a basis field has been exhibited explicitly.", "pages": [ 2, 3 ] }, { "title": "B. Optimizing the χ 2 discriminator", "content": "The χ 2 discriminator must produce as large a value as possible for a glitch in the data. In our framework we achieve this, on average, given the collection of glitches. The optimization is therefore carried out for a family of glitches, say, G . Here we will model the glitches as sine-Gaussians and select a family of such glitches based on the ranges of the parameters describing the sineGaussians. The subspace S then must be chosen in such a way as to have maximum projection on an average. Also one must keep in mind that S must be orthogonal to the trigger template. These two criteria essentially guide us to obtain the subspaces S . The third criterion is that its dimension should be kept small in order to keep the computational cost at a reasonable level. More specifically for a given trigger template h , we perform the following steps: component parallel to h from each of the sample vectors spanning V G . Thus if v ∈ V G , then we define v ⊥ = v -( v , h ) h . These vectors v ⊥ by construction are orthogonal to h . The space spanned by these clipped vectors v ⊥ is called V ⊥ . Steps 1 and 2 above were also described in the construction of χ 2 tests in Ref. [25] where the vectors v were taken to be gravitational waveforms . In our formulation, however, the sampled vectors in principle could be any vectors in D , the only condition being that they be orthogonal to h . Further, in order to construct an effective χ 2 , the vital step is to choose these vectors in the direction of the glitches . This is distinct and different from the waveforms suggested in Ref. [25]. For more details on our formulation, we refer to Ref. [24], where we also proved that statistically independent χ 2 discriminators, such as the traditional one and the optimized SG χ 2 , can be straightforwardly added to form new discriminators with a larger number of degrees of freedom that continue to have the χ 2 distribution in Gaussian noise. Such discriminators will tend to be effective against a broader class of glitches. In the next section, Sec. III A, we describe how one can define a metric on G to obtain the sample vectors that span V G .", "pages": [ 3, 4 ] }, { "title": "A. Selection of vectors that span V G", "content": "In order to form an optimal χ 2 , it is important that we select appropriate vectors in V G to project the GW data on. In order to improve the sensitivity of CBC searches, we specifically target the blip glitches in this work, which are a major source of reduction in CBC search sensitivity. A recent work [1] demonstrates how transient bursts represented by sine-Gaussian waveform [26] can be vetoed with the help of an optimal χ 2 from the GW data. Since the blip glitches are also known to have a time-domain morphology similar to the sineGaussian waveforms, we use these waveforms to form the vectors in V G for constructing the optimized SG χ 2 here. To allow for an arbitrary phase in the noise transient, we use a complex-valued sine-Gaussian waveform. This is an improvement over our earlier work [1], especially relevant in dealing with the general scenario. In the time domain the sine-Gaussian waveform can be defined as, In the frequency domain, it is where t 0 is central time, f 0 is central frequency, Q is the quality factor, and the amplitudes are A = ( 8 πf 2 0 Q 2 ) 1 4 and ˜ A = ( Q 2 2 πf 2 0 ) 4 . It is important to note that this model for the glitches is different from the one considered in [1] because this model considers sine-Gaussians with arbitrary phase, which is more general. Therefore, the results that follow here are markedly different, and the resulting discriminator is effective more generally. To calculate the metric in the ( t 0 , f 0 , Q ) space we begin by considering two neighboring sine-Gaussian waveforms in that space, namely, ψ 1 ( f ; t 0 , f 0 , Q ) and ψ 2 ( f ; t 0 + dt 0 , f 0 + df 0 , Q + dQ ). A metric may then be introduced on this space as a map from the differences in the parameters of these waveforms to the fractional change in their match: We do not consider the noise power spectral density (PSD) to calculate the metric since it has a negligible effect on the arrangement of the vectors in V G [26]. The above metric (in Eq. (8)) can be reduced to its diagonal form using the transformations, and In the new coordinates, ( t 0 , ω 0 , Q ), the metric takes the form In comparison to the metric in Ref. [1], this metric has no ω 0 term multiplying dt 2 0 . This results from our accounting for the aforementioned arbitrary phase of the sine-Gaussian waveform in this work. As mentioned in Ref. [27], a CBC template is triggered with a time lag t d after the occurrence of a glitch, i.e., after t 0 . The time t d is given by [27], where the second term inside the parentheses determines the magnitude of the 'correction' beyond the chirp time τ 0 , which is given by and ζ is negative of the logarithmic derivative of the noise PSD ( S h ( f )) evaluated at f 0 . The metric in Eq. (11) can be reduced to a more simplified form with the help of transformations such that By examining Eq. (12), one may reckon that for Q ∼ 2 the t d would be significantly affected. This is in fact so but it turns out that it makes little difference to the metric. This can be seen as follows. In Eq. (12) the correction term is inversely proportional to Q 2 ; therefore, for high Q , say, with Q ≳ 5, it is negligible. Also, at high central frequencies, i.e., f 0 ≳ 500 Hz, the factor ( ζ +2 / 3) is small for the aLIGO design PSD. Thus, in either case one has t d ≈ τ 0 . The remaining case is one where f 0 ∼ 100 Hz and Q ∼ 2. Now the term arising from t d is the second term multiplying dz 2 in the metric expression Eq. (15). It turns out that this term is small compared to the first one in the same metric expression - about 14% at this frequency. Hence, even if the t d is changed significantly, it makes a small difference to the metric. To summarize, in most of the parameter space we consider, the metric given in Eq. (15) applies and our sampling based on this metric valid to a good approximation. From a broader perspective, the main idea is to sample the parameter space of glitches adequately, so that they are not misidentified. Thus, any inadequacy resulting from inaccuracies of the metric can be easily remedied by more densely sampling the parameter space. This can be achieved by just increasing the match between neighbouring points. We have explored this possibility and find that it does not lead to dissimilar results. To choose the appropriate vectors in V G for the optimalχ 2 , we take help from the sine-Gaussian projection maps introduced in Ref. [2]. The SGP maps are a projection of GW data onto the sine-Gaussian parameter space. We experimented with several real blips and simulated CBC signals to model their projections on the sineGaussian parameter space. As seen in Fig. 1, blips and CBC signals show projection in distinct regions of the sine-Gaussian parameter space. Blips project strongly in the frequency region above 100 Hz, whereas for the CBC signals with component masses above 10 M ⊙ the projection lies mostly below 100 Hz. Along the Q coordinate, the CBC signals show more elongated features than the blips. This difference in the projections of blips and CBCs on sine-Gaussians paves the way for selecting appropriate vectors in V G to formulate an optimalχ 2 following the unified χ 2 formalism [22]. We choose parameter ranges such that the blips have a high projection on these vectors that lead to higher values of the χ 2 statistic for blips than CBC signals. In our case, we find f 0 ∈ [100 , 500] Hz and Q ∈ [2 , 8]. To construct the V G vectors in the chosen region of the sine-Gaussian parameter space we first use Eq. (15) to sample points in the z -y space. The coefficients of dz 2 depend on Q , f 0 and the chirpmass M through the parameter z . To get a flat metric in z -y space, we fix Q = 8 and f 0 = 500 Hz. The choice of Q and f 0 is made after observing that it provides sufficiently dense sampled points such that the mismatch between two neighbouring vectors is not more than 0.20. After sampling points in the z -y space, we transform them back to the Q -f 0 space. The top panel of Fig. 2 shows points sam- pled in the z -y space, while the bottom panel shows the same points after transforming to the Q -f 0 space. SineGaussian waveforms are chosen corresponding to each of these sampled points. The number of total points can vary depending on the chirp mass M of the triggered template. This can lead to a large number of vectors. To reduce that number, note that most of them are not linearly independent. We, therefore, use singular value decomposition (SVD) (discussed below in Sec. III B) to obtain the (much smaller number of) basis vectors from that sample. The resultant SVD vectors are then used to project the GW data upon them using Eq. (1). As it turns out, we require just about three vectors - on which the data show maximum projection - to compute the optimalχ 2 , which is then defined as, where x is the data vector and g α are the three aforementioned basis vectors.", "pages": [ 4, 5, 6 ] }, { "title": "B. Reducing the number of degrees of freedom of the χ 2", "content": "The selected vectors (sine-Gaussians) as described in Section III A usually turn out to be quite large in number. After time-shifting the sine-Gaussians and clipping their components parallel to the template, they span the subspace V ⊥ (see Section II B). We could in principle use V ⊥ on which to project the data vector and compute the χ 2 statistic. But the dimension of V ⊥ is usually large and in practice it would involve too much computational effort and slow down the search pipeline - the χ 2 would involve too many degrees of freedom, namely, the dimension of V ⊥ . Therefore, we look for the best p -dimensional approximation to V ⊥ , where p is reasonably small. The SVD algorithm allows us to achieve just this - this is the essence of the Eckart-Young-Mirsky theorem [28]. However, the SVD cannot be applied directly to the selected vectors in V G . The input matrix, say, M needs to be first prepared. We briefly describe the procedure here since the full details can be found in [1]. The fol- lowing are the salient steps needed: This is however not the end of the story. We have to also modify the output - which are the right singular vectors. We need to unwhiten these vectors by multiplying by the factor √ S h ( f ). The unwhitened singular vectors are orthonormal in the weighted scalar product. The SVD algorithm also yields singular values σ i , i = 1 , 2 , ..., r . The Frobenius norm of the input matrix is just ∥ M ∥ F = ∑ r i =1 σ 2 i . The number of degrees of freedom p are chosen so that: where δ may be chosen to be, say, 0 . 1 or 10 %. The subspace S is generated by the first p right singular vectors and so has dimension p . This also means that we have a projection of about 90%, on the subspace S . S is the best p dimensional approximation to V ⊥ - it is essentially a p -dimensional least-square fit to V ⊥ (see [1] for more discussion). We have also succeeded in reducing the number of degrees of freedom of the χ 2 to p . Typically, for the ranges of parameters f 0 , Q, M , etc., considered here, the number of selected vectors in V G is about a few hundred while p < 10. Thus, p is much smaller than the number of selected vectors in V G .", "pages": [ 6, 8 ] }, { "title": "A. Performance on Blips", "content": "In this study, we tuned the optimized SG χ 2 specifically for the blip glitches. In order to test the effectiveness of that χ 2 in differentiating blips from aligned spin BBH signals, we chose real blips from LIGO's O3 run, as identified by the Gravity Spy tool [29-33]. Blips were selected from both Hanford and Livingston detectors when they had a confidence level of 0.6 on a scale of 0 to 1, as rated by Gravity Spy. A total of 4000 strain data segments, each of length 16 sec and containing a blip with a matched-filtering signal-to-noise ratio (SNR) between 4 and 12, as registered by the loudest BBH template, were chosen. The BBH data sample is prepared by simulating BBH signals using the family of IMRPhenomPv2 waveforms [34]. These simulated signals also span the same SNR range, namely, 4 to 12, uniformly. We first divide the signals into two bins based on their component masses: One bin consists of signals with component masses m 1 , m 2 ∈ [20 , 40] M ⊙ and the other consists of those with m 1 , m 2 ∈ [60 , 80] M ⊙ . In both cases, the aligned spins s 1 z and s 2 z are distributed uniformly in the range [0 . 0 , 0 . 9]. The purpose of this division is to compare the performance of the optimal SG χ 2 in the two mass bins and use it to understand which part of the BBH parameter space benefits more from its use in reducing the adverse impact of the blips. For the studies in this section, we needed 3 (complex) basis vectors in Eq. (16) for the construction of the optimized SG χ 2 , which amounts to 6 degrees of freedom for that statistic. To compute the matched-filtering SNRs of both blips and spin-aligned BBH signals, we use different CBC template banks for the two mass bins. For the lower mass bin, m 1 , m 2 ∈ [20 , 40], we use a template bank with component masses ( m 1 , m 2 ) bank ∈ [10 , 50] M ⊙ and for the higher mass bin, m 1 , m 2 ∈ [60 , 80] M ⊙ , we use a template bank with component masses ( m 1 , m 2 ) bank ∈ [50 , 90] M ⊙ . The template banks are intentionally chosen to cover a broader range of masses than the injections. For both mass bins, the templates had the same range of spins as the injections, namely, s 1 z , s 2 z ∈ [0 . 0 , 0 . 9]. Templates banks were generated with a minimum mismatch of 0.97 using the stochastic bank code of PyCBC [35]. We used the same IMRPhenomPv2 waveform for templates as was used for simulating the CBC injections. The BBH signals so prepared are injected into 16 sec long data segments from O3 that are not known to have any astrophysical signals in them. These data segments are multiplied with the Tukey window to make a smooth transition to zero strain at the edges, which mitigates spectral leakage during the analysis. We use the receiver-operating characteristic (ROC) curves to assess the performance of the optimal χ 2 and compare it with that of the traditional χ 2 and sineGaussian χ 2 statistics. First, the optimal, traditional and sine-Gaussian χ 2 s are computed for the chosen sam- of BBHs and blips, in addition to their respective SNRs. We then use the ranking statistic defined in Ref. [36, 37] to rank the triggers in case of traditional χ 2 . On the other hand, for sine-Gaussian and optimal χ 2 we use the new ranking statistic defined in Ref. [8]. The ranking statistics used here for the traditional χ 2 and the sine-Gaussian χ 2 are the usual ones, respectively. The decision to use the new ranking statistic for the optimized SG χ 2 was made after finding how both ranking statistics performed for the optimal χ 2 . As one can see from the ROC curves in Fig. 3, prima facie , for these choices of the ranking statistics the optimized SG χ 2 appears to perform better than the traditional and sine-Gaussian χ 2 at all false alarm probabilities in both mass bins. Specifically, the optimized SG χ 2 achieves an improvement in sensitivity of around 4% over the traditional χ 2 at a false alarm probability of 10 -2 . For higher masses m 1 , m 2 ∈ [60 , 80] M ⊙ too we see that the optimized SG χ 2 shows notable improvement in at and around the same false alarm probability. Here, it is important to note that one is not discounting the possibility that further tuning of the Power and sine-Gaussian χ 2 s on the same injections and glitches would improve their performance. In fact, we did not exhaustively tune the optimized SG χ 2 either. To check if the aforementioned performance improvement is limited to only a specific SNR bin, we repeated the above study for somewhat higher SNRs, namely SNR ∈ [8 , 12] in Fig. 4 (as compared to SNR ∈ [8 , 12] in Fig. 3). To further test if the improvement of the new statistic was limited in source distance, we analysed its performance in a couple of broad distance bins in Fig. 5. With the aforementioned caveat about tuning, we observe that the optimized SG χ 2 is found to improve the sensitivity in each of these cases, although in the more distant bin the improvement tends to vanish at low false-alarm probabilities, as expected: Weak signals are noise dominated and are difficult to discern from noise transient by any statistic. We also compare the volume-time sensitivities [38-40] as functions of the inverse false-alarm rate for three different χ 2 statistics in Figs. 6 and 7. For this study, we performed signal injections in O3 data, for various component mass ranges, up to a maximum distance of 6 Gpc, uniformly distributed within the comoving volume. The volume-time sensitivity is then calculated by dividing the number of recovered injections by the total number of injected signals, as detailed in [38]. As can be observed in these results, the optimized SG χ 2 performs at least as good as or somewhat better than the other statistics. This is likely because it accounts for both the differences and the similarities between blips and BBH waveforms quantitatively by utilizing the metric in the sine-Gaussian space [1]. The performance im- rovement is superior for higher mass BBHs because those signals occupy a lower part of the frequency band compared to the blips. The improvement over the sineGaussian χ 2 can be understood in terms of the choice of Q and f 0 range while sampling the sine-Gaussian waveforms to construct the V G vectors and then subtracting the triggered templates from them. This work makes a few advances compared to the previous implementation of the optimal χ 2 [1]. For instance, here we utilized the complex form of the sine-Gaussian waveforms to account for the phase of the signal. Moreover, the sine-Gaussian projection maps [2] helped us in identifying appropriate regions in the sine-Gaussian parameter space to choose the V G vectors from for the construction of the optimized SG χ 2 statistic. These advances also helped in bringing about appreciable improvement in CBC search sensitivity over traditional and sine-Gaussian χ 2 . It must be emphasized that just like for the traditional χ 2 , the computation of the optimized SG χ 2 uses the parameter values of the triggered template. Therefore, the choice of template-bank boundaries while doing the matched-filtering operation can play an important role in its effectiveness. As we can see in the Fig. 8, blips can trigger a wide variety of templates, and a narrow template bank choice can adversely affect the performance of the optimized SG χ 2 .", "pages": [ 8, 10, 11, 12 ] }, { "title": "B. Performance on other noise transients", "content": "So far we focused on applying the optimized SG χ 2 for distinguishing BBH signals from the blip glitch and demonstrated its utility via ROC curves. In realistic observation scenarios, however, noise transients that trigger BBH templates are of a wider variety, and not limited to blips. To test the performance of the optimized SG χ 2 on such a class, we compared the same three χ 2 s statistics as before on a set of mixed glitches, which include 500 each of koifish, tomte, low-frequency blip, scattered light glitches, in addition to 1000 blips - which are known to occur more frequently. The performance comparison is shown as ROC plots in Fig. 9. We observe that even in the presence of other kinds of noise transients in significant numbers, the optimized SG χ 2 does slightly better than other two χ 2 s overall. Fig. 10 shows the improvement in the true positive rate by optimized SG χ 2 over SG χ 2 and traditional χ 2 . It is important to mention here that this improvement of optimized SG χ 2 over other two χ 2 s is found without any tuning specific to these additional glitches. To construct the basis vectors in this study, the sine-Gaussian waveforms were chosen from the region, Q ∈ [2 , 8] and f 0 ∈ [100 , 500], as was done in the previous sections for blips.", "pages": [ 12 ] }, { "title": "V. DISCUSSION", "content": "Owing to their time-frequency morphological similarity with high-mass BBH signals, the short-duration noise transients like blips affect the search sensitivity of those signals adversely. In this study, we showed how the optimized SG χ 2 can be constructed in real LIGO data so as to reduce that impact. The previous version of this χ 2 statistic, introduced in Ref. [1], is extended here to better discriminate blip glitches in searches for spinaligned BBH signals in real data. A few advances have been made here compared to the past work in Ref. [1] that possibly helped in achieving further improvement. The first one was accounting for the phase of the sig- nal in the construction of the basis vectors - by using complex sine-Gaussian waveforms. The second contributor was the use of projection maps [2], which guide us in locating the region in sine-Gaussian parameter space from which the initial sine-Gaussians should be chosen for constructing the basis vectors used in computing the optimized SG χ 2 . The computational cost per trigger of the optimized SG χ 2 is higher than that of the traditional χ 2 and the sine-Gaussian χ 2 . A major fraction of this cost arises from the construction of orthonormal basis vectors described in Sec. III B. This makes the implementation of the optimized SG χ 2 in an online search less efficient computationally. A straightforward way to reduce this inefficiency is to construct the orthonormal basis vectors beforehand for a set of template masses. In case of a high-mass template-bank, this advance preparation can be done for all the templates, as their number is relatively small. In case of low-mass template-banks, a more sparsely-populated fraction of templates can be selected for pre-computation of the orthonormal basis vectors. The optimized SG χ 2 is then computed using the basis vectors of the nearest template. As noted before, there have been some attempts to veto blip glitches from the data with the help of a χ 2 like statistic [8] and machine learning networks [2, 41] - claiming varying degrees of improvement in the BBH search sensitivity. Some of these works exploit certain insights provided by the unified χ 2 formalism [22] but so far none of them fully leverages the power afforded by it. The work reported in this paper attempted to bridge that gap - for non-spinning as well as spin-aligned BBH signals. One could possibly extend this work to explore possible mitigation of the blips' effect on more general spinning CBC searches.", "pages": [ 12, 14 ] }, { "title": "VI. ACKNOWLEDGMENT", "content": "We would like to thank Sudhagar Suyamprakasam and Tanmaya Mishra for discussions related to LIGO data. We also thank Bhooshan Gadre for reviewing the manuscript and sharing some useful comments. SVD acknowledges the support of the Senior Scientist Platinum Jubilee Fellowship from the National Academy of Sciences, India (NASI). The data analysis and simulations for this work were carried out at the IUCAA computing facility Sarathi. This research has made use of data, software, and/or web tools obtained from the Gravitational Wave Open Science Center [42, 43], a service of LIGO Laboratory, the LIGO Scientific Collaboration, and the Virgo Collaboration. Some of that material and LIGO are funded by the National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN), and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. We also use some of the modules from the PyCBC an open source software package [36]. This work was funded in part by NSF under Grant PHY-2309352. 024001 (2015), arXiv:1408.3978 [gr-qc].", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Research in Astron. Astrophys. Vol.0 (20xx) No.0, 000-000 http://www.raa-journal.org http://www.iop.org/journals/raa (L A T E X: ms2023-0280.tex; printed on October 16, 2023; 0:47) R esearch in A stronomy and A strophysics", "pages": [ 1 ] }, { "title": "Early phases of star formation: testing chemical tools", "content": "N. C. Martinez 1 and S. Paron 1 CONICET-Universidad de Buenos Aires. Instituto de Astronom'ıa y F'ısica del Espacio, Ciudad Universitaria, (C1428EGA) Ciudad Aut'onoma de Buenos Aires, Argentina Received 20xx month day; accepted 20xx month day Abstract The star forming processes strongly influence the ISM chemistry. Nowadays, there are available many high-quality databases at millimeter wavelengths. Using them, it is possible to carry out studies that review and deepen previous results. If these studies involve large samples of sources, it is preferred to use direct tools to study the molecular gas. With the aim of testing these tools such as the use of the HCN/HNC ratio as a thermometer, and the use of H 13 CO + , HC 3 N, N 2 H + , and C 2 H as 'chemical clocks', we present a molecular line study towards 55 sources representing massive young stellar objects (MYSOs) at different evolutive stages: infrared dark clouds (IRDCs), high-mass protostellar objects (HMPOs), hot molecular cores (HMCs) and ultracompact HII regions (UCHII). We found that the use of HCN/HNC ratio as an universal thermometer in the ISM should be taken with care because the HCN optical depth is a big issue that can affect the method. Hence, this tool should be used only after a careful analysis of the HCN spectrum, checking that no line, neither the main nor the hyperfine ones, present absorption features. We point out that the analysis of the emission of H 13 CO + , HC 3 N, N 2 H + , and C 2 H could be useful to trace and distinguish regions among IRDCs, HMPOs and HMCs. The molecular line widths of these four species increase from the IRDC to the HMC stage, which can be a consequence of the gas dynamics related to the star-forming processes taking place in the molecular clumps. Our results do not only contribute with more statistics regarding to probe such chemical tools, useful to obtain information in large samples of sources, but also complement previous works through the analysis on other types of sources. Key words: Stars: formation - ISM: molecules - ISM: clouds", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The sites in which the stars form are characterized by a rich and complex chemistry. The smallest gaseous fragments within a molecular cloud, known as hot molecular cores (HMCs), which are related to the formation of massive stars, are the chemically richest regions in the interstellar medium (ISM) (e.g. Herbst & van Dishoeck 2009; Bonfand et al. 2019). Molecules and chemistry are ubiquitous along all the stages that a forming star goes through, and moreover, the star-forming processes strongly influence the chemistry of such environments (Jørgensen et al. 2020). For instance, as material collapses and becomes ionized by the young massive stars and shocked by jets and outflows, temperature and density can change drastically, leading to the formation and destruction of molecular species. Thus, observing molecular lines and studying their emission and chemistry is important to shed light on the different stages of star formation and to characterize physical and chemical conditions. To explore the star-forming processes in different environments, one should work with accurate values of the physical properties, such as the case of the kinetic temperature (T K ). For example, calculating the excitation temperature (T ex ) of carbon monoxide 12 C 16 O, the T K of a molecular cloud can be roughly estimated if it is assumed that there is a complete thermalization of the lines (T ex = T K ). Ammonia (NH 3 ) has been found in different interstellar environments: from dark quiescent clouds, circumstellar envelopes, and early stages of high and low luminosity star formation to planetary atmosphere and external galaxies (Martin & Ho 1979; Betz et al. 1979; Ho & Barrett 1980; Rathborne et al. 2008; Takano et al. 2013; B\"ogner et al. 2022). Estimating the NH 3 rotational temperature usually results in a reliable indicator of the kinetic temperature. Hydrogen cyanide (HNC) and isocyanide (HNC) are two of the most simple molecules in the ISM, first detected almost fifty years ago (Snyder & Buhl 1971, 1972). These isomers have a linked chemistry, and differences in the spatial distributions in which they lie within a cloud can reflect the gas chemical conditions and the evolution of the star-forming regions (Schilke et al. 1992). Recently, Hacar et al. (2020) proposed the HCN-to-HNC integrated intensity ratio as a direct and efficient thermometer of the ISM with an optimal working range 15 K ≲ T K ≤ 40 K. The authors performed an analysis of such isomers throughout the Integral Shape Filament in Orion deriving an empirical correlation between the HCN/HNCratio and the kinetic temperature T K . Based on the analysis of such correlation towards many dense molecular clumps from the MALT90 survey, they proposed that the HCN/HNC thermometer can be extrapolated for the analysis of the ISM in general, particularly in star-forming sites, aiming to explore it towards different regions and sources along the ISM. Molecular species that emerge and destroy during the birth of stars can be used to track the starforming processes within molecular clumps and cores (Stephens et al. 2015; Urquhart et al. 2019). Comparisons between column densities and molecular abundance ratios that can be used to estimate the age and mark the evolutionary stages of star-forming regions are known as 'chemical clocks'. As Sanhueza et al. (2012) pointed out, only molecules that show differential abundances with time can be used to evaluate the evolutionary status of a star-forming region. In general, chemical clocks have been studied in depth in low-mass star-forming regions, but it has been less developed in the context of highmass star-forming regions. As shown by Yu & Wang (2015), in the context of an analysis of chemical clocks it is important not only studying abundance ratios of such molecules, but also the integrated line intensities, the line widths, among other parameters. For instance, the molecular line widths ( ∆ v FWHM ) are related to the gas kinematics of the molecular clump interior, regarding turbulence, outflows, and shocks among other processes (Sanhueza et al. 2012) which can give information about the evolutive stage of a MYSO. Some interesting molecular species for probing physical and chemical properties of star-forming regions are the diazenylium (N 2 H + ) and ethynyl radical (C 2 H). Both molecules seem to be good tracers of dense gas in the early stages of the star-forming evolution (Beuther et al. 2008; Sanhueza et al. 2012, 2013), N 2 H + traces cold gas due to its resistance to depletion at low temperatures (Li et al. 2019), and the latter, additionally can indicate the presence of a photodissociation region (PDR), where UV photons from young and hot massive stars irradiate acetylene to produce C 2 H (Fuente et al. 1993; Nagy et al. 2015; Garc'ıa-Burillo et al. 2017). HCO + and H 13 CO + (formylium species) are usually employed to investigate infall motions and outflow activity (Rawlings et al. 2004; Veena et al. 2018), and HC 3 N is helpful to explore gas associated with hot molecular cores (Bergin et al. 1996; Taniguchi et al. 2016; Duronea et al. 2019). The mentioned molecules are among the brightest lines, they were called as molecular fingerprints in a study of a large sample of molecular clumps (Urquhart et al. 2019). As the authors pointed out, such molecules are able to trace a large range of physical conditions including cold and dense gas (HNC, H 13 CO + , HCN, HN 13 C, H 13 CN), outflows (HCO + ), early chemistry (C 2 H), gas associated with protostars and YSOs (HC 3 N, and cyclic molecules). Thus, the analysis of such molecules gives us a significant amount of scope to search for differences in the chemistry as a function of the evolutionary stage of the star formation taking place within molecular clumps. For instance, Yu & Wang (2015) studied 31 extended green objects (EGOs) clumps with data from the MALT90 aiming to better understanding the chemical processes that take place in the evolution of massive star formation. They classified the sources and made a molecular line study over 20 massive young stellar objects (MYSOs) and 11 HII regions. Through the comparison of integrated intensities, line widths, and column densities, derived from the emission of N 2 H + and C 2 H with those of the formylium (H 13 CO + ) and cyanoacetylene (HC 3 N), they suggested that N 2 H + and C 2 H could act as efficient chemical clocks. They found that the N 2 H + and C 2 Hcolumn densities decrease from MYSOs to HII regions, and the [N 2 H + ]/[H 13 CO + ] and [C 2 H]/[H 13 CO + ] abundance ratios also decrease with the evolutionary stage of the EGO clumps. In addition they found that the velocity widths of N 2 H + , C 2 H, H 13 CO + , and HC 3 N are comparable to each other in MYSOs, while in HII regions the velocity widths of N 2 H + and C 2 H tend to be narrower than those of H 13 CO + and HC 3 N. Nowadays, there are available many high-quality databases generated from observations obtained with the most important (sub)millimeter telescopes such as the IRAM 30 m Telescope and the Atacama Large Millimeter Array (ALMA), among others. For instance, using this kind of data, it is possible to carry out new chemical studies that, in turn, review and deepen previous results. If these studies involve large samples of sources, it is preferred to use direct tools to study and probe the molecular gas as those presented in Hacar et al. (2020) and Yu & Wang (2015). Infrared dark clouds (IRDCs) are massive, dense, and cold clumps that may harbor budding stars, while high mass protostellar objects (HMPOs) are already protostars accreting material from their surroundings; at this stage, both temperature and density increase, but it is thought that at the beginnings, they are chemically poor from an evolutionary point of view. Hot molecular cores (HMCs) are considered hotter sources where the chemistry is prolific as a consequence of embedded and evolved HMPOs (Giannetti et al. 2017) until they eventually reach the last stage here considered: UCHII regions. The stars responsible for the UCHII regions have generally finished their accretion process and have begun to ionize the gas around them. All the processes involved in this evolutive path impact on the molecular gas that eventually can be investigated through the emission of molecular lines. It is important to highlight that the mentioned phases in the massive star formation may not have well-defined limits, and sometimes one determined source may have some physical conditions overlapping with those of another kind of source (e.g., Beuther et al. 2007; Gerner et al. 2014). Moreover, different kind of sources may be embedded within the same molecular clump. Given this complex scenario, comparative studies with many sources are needed to analyze the involved physics and chemistry. With the aim of testing the presented tools concerning the use of the HCN/HNC ratio as a thermometer (following the methodology presented in Hacar et al. 2020), and the analysis of H 13 CO + , HC 3 N, N 2 H + , and C 2 H (following the methodology presented in Yu & Wang 2015), we present this study towards 55 sources representing MYSOs at different evolutive stages as described above. After the presentation of the data and the analyzed sources (Sect. 2), the paper is structured as follows: a presentation of the results regarding each tested tool (Sect. 3), their respective discussion (Sect. 4), and a summary of the main results (Sect. 5).", "pages": [ 1, 2, 3 ] }, { "title": "2 DATA AND ANALYZED SOURCES", "content": "We made use of the catalog 'IRAM 30 m reduced spectra of 59 sources' J/A+A/563/A97 in the ViZieR database 1 (Gerner et al. 2014), from which the molecular data were obtained and the sources were selected to perform our study. The particular molecular lines analyzed in our work are presented in Table 1. These data, obtained by Gerner et al. (2014), with the 30 m IRAM telescope, are single pointing spectra observed toward each source. The data used here correspond to the 86-94 GHz band, which have an angular and spectral resolution of 29 '' and 0.6 km s -1 , and typical 1-sigma rms values of about 0.03 K. The sources, classified as IRDCs, HMPOs, HMCs, and UCHII, are presented in Table ?? , where the coordinates and the distances are included. The classification is based on the guidelines introduced by Beuther et al. (2007) and Zinnecker & Yorke (2007) according to the physical conditions of the evolutionary sequence in the formation of high-mass stars. IRDCs are sources that consist of cold and dense", "pages": [ 3 ] }, { "title": "N. C. Martinez & S. Paron", "content": "gas and dust that emit mainly at (sub)millimeter wavelengths. According to Gerner et al. (2014), the IRDCs of this sample consist of starless IRDCs as well as IRDCs already starting to harbor point sources at µ m-wavelengths. HMPOs are sources hosting an actively accreting massive protostar(s), which shows an internal emission source at mid-infrared. HMCs are much warmer than HMPOs and can be distinguished from a chemical point of view. At this stage, the central source(s) heats the surroundings evaporating molecular-rich ices. Finally, the UV radiation from the embedded massive protostar(s) ionizes the surrounding gas and gives rise to an UCHII region. Even though there might be overlaps among HMPO, HMC, and even UCHII stage, such classification was followed by Gerner et al. (2014) based on physical quantities, especially the temperature, which rises from IRDCs to HMPOs to UCHII regions, and based on the chemistry that can differentiate HMPOs (chemical poorer sources) from HMCs (molecular richer chemistry sources). The sample includes 19 IRDCs, 20 HMPOs, 7 HMCs, and 9 UCHII regions. In addition to the possible overlaps among the type of sources, it is possible that within the beam of the observational data lies more than one type of source. This is the case of HMC034.26, a region hosting several HII regions and a hot core. Thus, it is necessary to be careful with any analysis in relation to the evolutionary stages. The quality of these IRAM data and the variety of sources in the sample included in such a catalog allow us to perform a new analysis of several molecular species in order to probe the molecular gas conditions and the chemistry related to IRDCs, HMPOs, HMCs, and UCHII regions with the aim of probing chemical tools. Additionally, we used infrared (IR) data to complement the information obtained from the molecular lines with the aim of probing, in these wavelengths, the activity of the star-forming regions. Data at the Ks band extracted from the UKIRT Infrared Deep Sky Survey (UKIDSS) (Lawrence et al. 2007), and IRAC-Spitzer 4.5 µ mdata obtained from the GLIMPSE survey (Churchwell et al. 2009) were used.", "pages": [ 4 ] }, { "title": "3.1 Infrared and submillimeter continuum emission", "content": "Given that jets and outflows affect the star-forming regions' chemistry, we looked for evidence of such processes by using IR data to have additional information on each source, besides its classification, to interpret and complement the molecular analysis. Data at the Ks band extracted from the UKIDSS were used to search for signs of jets (e.g. Paron et al. 2022) mainly in HMPOs, and also in HMCs and UCHII regions. IRAC-Spitzer 4.5 µ m data, obtained from the GLIMPSE survey, were employed to analyze whether the sources present extended 4.5 µ m, likely indicating outflow activity (e.g. Davis et al. 2007). Additionally, each source was checked if it is cataloged as an 'EGO - likely MYSO outflow candidate' according to the catalog of Cyganowski et al. (2008). In Table ?? (Cols. 6 and 7), we indicate the kind of emission at the Ks and 4.5 µ m bands observed within the beam size of the IRAM data centered at the position of each source. If one or several point sources appear within the IRAM beam, it is indicated as 'points'; extended emission, suggesting the presence of jets and outflows, is indicated as 'ext.', and if this extended emission present a jet-like mor-", "pages": [ 4 ] }, { "title": "Molecules and star formation", "content": "phology, it is stated as 'jets'. Diffuse emission without any clear morphology is indicated as 'diff.'. While 'no' means that there is no emission, an 'x' indicates there is no data at the corresponding wavelength. In the case of the 4.5 µ memission, if the source is cataloged as an 'EGO - likely MYSO outflow candidate', this is indicated with 'EGO' in Col. 7. As an example, Fig. 1 displays two sources in the Ks band: on the left HMC 45.47 shows a jet-like feature and some extended emission at the 4.5 µ m band, on the right HMC 45.47 appears with extended emission at both bands, and it is cataloged as an EGO. Additionally, in Table ?? (Col. 8), we include the peak flux of the submillimeter emission obtained from the ATLASGAL compact source catalog (Contreras et al. 2013; Urquhart et al. 2014) for sources that lie within the IRAM beam centered at the coordinates indicated in Cols. 2 and 3.", "pages": [ 5, 6 ] }, { "title": "3.2 HCN/HNC ratio and T K", "content": "Following the same procedure as done by Hacar et al. (2020) when they test the HCN/HNC kinetic temperature in a large sample of dense molecular clumps extracted from the MALT90 survey (see their Sect. 4.4), we derived the T K for each source of the sample presented here. The correlation to derive T K proposed by the authors is as follows: if the integrated intensity ratio of the isomers HCN and HNC (hereafter I(HCN)/I(HNC)) is ≤ 4, T K is estimated as T K = 10 × I(HCN)/I(HNC); and if I(HCN)/I(HNC) > 4, it is used T K = 3 × (I(HCN)/I(HNC) -4)+40 . As done by Hacar et al. (2020), we integrated the HCN (J=1-0), including all hyperfine components and HNC (J=1-0) over a given velocity interval (in Appendix. A we include some spectra indicating the integrated area in both isomers). The obtained integrated emission values, and the respective ratios, are presented in Table 3 (Cols. 2, 3, and 4, respectively), and the obtained kinetic temperatures following the mentioned relations are presented in Table 5. Typical errors in I(HCN) and I(HNC) are about 0.2 and 0.1 K km s -1 , respectively, which yields errors between 1 and 5%, and in some cases at most 10%, in the T K . From now on, the obtained errors in the measured and derived parameters appear as bars in the figures, and for a better display, we do not include them in the tables. In the further analysis and comparisons, following Hacar et al. (2020), we do not consider values of kinetic temperatures obtained from the HCN/HNC ratio (hereafter T K (HCN/HNC)) lower than 15 K. These values are due to I(HCN)/I(HNC) ratios close to the unity, which according to the authors, the uncertainties in the method grow up for such cases, likely due to the combination of excitation and opacity effects. In Sect. 4.1.1, based on the inspection of the HCN and HNC spectra of sources that T K (HCN/HNC) < 15 K, we discuss this issue. In order to compare the T K (HCN/HNC), we sought the dust temperature of each source. Dust temperature ( T dust ) values were obtained from the maps 2 generated by the point process mapping (PPMAP) algorithm (Marsh et al. 2015) done to the Hi-GAL maps in the wavelength range 70-500 µ m (Marsh et al. 2017). T dust values are included in Table 5, and they represent average values on the dust temperature map over the IRAM beam size centered at the source position. Additionally, we also compared with the ammonia kinetic temperature ( T K (NH 3 )) derived by Urquhart et al. (2011) in several sources of the sample analyzed here that are contained in the Red MSX Source survey (Hoare et al. 2005; Mottram et al. 2006; Urquhart et al. 2008). Figure 2 exhibits the comparisons between T K (NH 3 ) and T dust (upper panel), T dust and T K (HCN/HNC) (middle panel), and T K (NH 3 ) and T K (HCN/HNC) (bottom panel). Table 6 presents the average temperature values with their errors. As mentioned above, sources with values T K (HCN/HNC) < 15 K, were not considered neither in the figures nor in the calculated average.", "pages": [ 6, 7 ] }, { "title": "3.3 Molecules as 'chemical clocks'?", "content": "Following Yu & Wang (2015), we performed a similar analysis using the HNC, C 2 H, HC 3 N, H 13 CO + and N 2 H + emissions measured toward the sources presented in Table ?? . This is: 1) we compared the molecular emissions with the flux at the sub-millimeter continuum, 2) we analyzed the line widths of the C 2 H, HC 3 N, H 13 CO + and N 2 H + , and 3) we analyzed the column densities (extracted from Gerner et al. 2014) of C 2 H and N 2 H + by comparing with that of the H 13 CO + . Figure 3 displays the integrated intensities of HNC, C 2 H, HC 3 N and N 2 H + (presented in Table 3) versus the 870 µ mpeak flux obtained from the ATLASGAL compact source catalog (presented in Col. 8 in Table ?? ). We fitted the line emission of C 2 H, N 2 H + , H 13 CO + , and HC 3 N with Gaussian functions to obtain the FWHM line widths ( ∆ v) in each source (in Appendix A we include some spectra showing the Gaussian fittings). In the case of the C 2 H and N 2 H + , which have hyperfine components, the fitting was done with multiple Gaussian functions, and the ∆ v values used in this analysis correspond to the main components. These values are presented in Table 4. The absence of values in some sources is due to a lack of emission or that the hyperfine components are completely blended with the main component. Table 7 presents the average line widths for each kind of source, and with the aim of analyzing their behavior, following the analysis presented in Yu & Wang (2015), we present plots that display relations among the measured ∆ v in Figure 4. From Gerner et al. (2014) we obtained the column densities of C 2 H, N 2 H + , and H 13 CO + for each source. We used the column densities obtained as 'iteration 1' according to the model used by the authors, which are the values derived with the mean temperatures from the best-fit models of 'iteration 0'. Values indicated as upper limits are not included in our analysis. Figure 5 displays the column densities of C 2 Hand N 2 H + versus the H 13 CO + column density. Error bars are not included, given the authors did not inform them in their work. Finally, Fig. 6 displays the relative abundance [C 2 H]/[H 13 CO + ] versus [N 2 H + ]/[H 13 CO + ].", "pages": [ 7, 8 ] }, { "title": "4 DISCUSSION", "content": "Unlike low-mass stars, the formation of high-mass stars is not yet fully understood, and different scenarios are proposed (see e.g. Motte et al. 2018; Tan et al. 2014). Probing the chemical conditions of highmass star-forming regions at different evolutive stages is an important issue to advance in the knowledge on the formation of this kind of stars, and to explore such chemical conditions, it is necessary to use efficient tools. In this context, we decided to test the HCN/HNC ratio as a thermometer of the gas, and the use of the emission of H 13 CO + , HC 3 N, N 2 H + , and C 2 Has 'chemical clocks'. The obtained results from each study are discussed in what follows.", "pages": [ 8 ] }, { "title": "4.1 Testing the use of HCN/HNC ratio as a thermometer", "content": "Employing simple molecules, ubiquitous in the star-forming regions, to derive their physical parameters is a helpful strategy for methodically analyzing many regions and avoiding complex calculations. In this context, we use the HCN/HNC intensity ratio formulation proposed recently by Hacar et al. (2020) to estimate the kinetic temperature (T K (HCN/HNC)). As done by the authors, we probed them through comparisons with dust temperature (T dust ) and T K obtained from the ammonia emission ( T K (NH 3 ) ), when those parameters were available towards the sources of the analyzed sample. It is worth noting that the useful kinetic temperature range derived from the ammonia is between 15 and 40 K (Ho & Townes 1983; Urquhart et al. 2011), in coincidence with the valid range indicated for T K (HCN/HNC). Despite the limited amount of sources satisfying that the obtained T K from the HCN/HNC ratio is > 15 K with measured values of T dust and T K (NH 3 ) (see Table 5) some conclusion can be extracted. Also, from the sources with T K (HCN/HNC) < 15 K or lower than expected, we can obtain important conclusions about this tool. The comparison between T dust and T K (NH 3 ) (see upper panel of Figure 2) indicates that, within a range of 0-5 K of difference between both temperatures, the molecular gas and the dust could be considered that are thermally coupled, allowing us to suggest that in general this property could be extrapolated to the whole sample. This supports that the comparison between T K (HCN/HNC), which is obtained from the gas emission, and the T dust can be made. Valid kinetic temperature values obtained following the HCN/HNC ratio (i.e. > 15 K) seem to be more accurate for IRDCs and HMPOs than for HMCs and UCHII regions. By analyzing the average values (see Table 6), T K obtained from the dust and ammonia emissions increases from IRDCs to UCHII regions as it is expected, but T K (HCN/HNC) increases only from IRDCs to HMPOs, and then it decreases Value obtained from a single source. ∗ in HMCs and UCHII regions. This suggests that in such sources, using the HCN-HNC tool to derive T K can be not appropriated, and it may underestimate the actual temperature value. Hacar et al. (2020) calculated the T K (HCN/HNC) towards a large sample of dense clumps extracted from the MALT90 survey (Foster et al. 2011; Jackson et al. 2013), and compare them with the T dust measurements derived by Guzm'an et al. (2015). At this point, they stated that the comparison between T K and T dust should be done with caution given the fact that the beam resolution of the MALT90 targets and the measurement sensitivity within this beam does not allow to resolve the temperature of individual clumps. Anyway, they pointed out that a correspondence between both temperatures prevails despite the mentioned observational caveats. However, they do not discuss the optical depth issue in the MALT90 sample as they did for their Orion results. This is an important matter to be considered, mainly when we work with a sample of different kinds of sources. As Hacar et al. (2020) mention, the increase of the HCN J=1-0 line opacity, while the HNC J=1-0 line likely remains optically thin, would reduce the HCN/HNC ratio, decreasing the values of T K (HCN/HNC). This phenomenon could be occurring in the sources with T K (HCN/HNC) < 15 K (see Sect. 4.1.1) and in some other sources that, although the T K obtained from the HCN-HNC tool is greater than 15 K, their values are well below the temperatures obtained from the dust and/or ammonia (about 10 K of difference). This is the case of HMCs and UCHII regions of our sample, in which, in general, it is observed that the HCN hyperfine line F=1-1 appears absorbed, indicating high optical depths (for an example, see Fig. A.5).", "pages": [ 9, 10, 11, 12 ] }, { "title": "4.1.1 Sources with T K (HCN/HNC) < 15 K", "content": "In our sample, there are several sources that the HCN/HNC ratio yields kinetic temperatures lower than 15 K, which were not included in the analysis nor the comparisons with T dust and T K (NH 3 ) performed above. These sources have I(HCN)/I(HNC) ratios close to the unity or even lower than one, which according to Hacar et al. (2020), in those cases, the uncertainties seem to increase, likely due to the combination of excitation and opacity effects. The percentages of sources with this issue is: 58%, 30%, 28%, and 11% for IRDCs, HMPOs, HMCs, and UCHII regions, respectively, suggesting that may be a condition that is more pronounced at earlier stages. Taking into consideration the HCN hyperfine line anomalies (Walmsley et al. 1982; Loughnane et al. 2012), which prevent obtaining reliable values of HCN opacities, we carefully analyzed each HCN and HNC spectrum to look for signatures of high optical depths. We found that the HCN spectra of sources with T K (HCN/HNC) < 15 K have pronounced saturation and/or self-absorption features in some cases at both the main component and hyperfine lines (see Fig. 7 for an example), and in other cases in the hyperfine line F=1-1. In the case of the HNC, we did not find such spectral features. We point out that the use of the HCN/HNC ratio as an universal thermometer in the ISM should be taken with care. We suggest that such a thermometer could be used only in some IRDCs and HMPOs (when the derived kinetic temperature is not lower than 15 K; see Sect. 4.1.1) and in more evolved regions, for instance, HMCs and UCHII regions, this tool underestimates the temperature. Thus, we conclude that the HCN-HNC tool as a kinetic temperature estimator should be used only after a careful analysis of the HCN spectrum, checking that no line, neither the main nor the hyperfine ones, presents absorption features; otherwise, it is not an useful tool for calculating kinetic temperatures.", "pages": [ 12, 13, 14 ] }, { "title": "4.2 An exploration of molecules as 'chemical clocks'", "content": "Molecular species such as HNC, HCN, H 13 CO + , C 2 H, and HC 3 Nare relatively easy to be observed and they give us a significant amount of scope to search for differences in the chemistry as a function of the evolutionary stage in massive star forming regions (Sanhueza et al. 2012; Yu & Wang 2015; Urquhart et al. 2019). After a similar analysis as presented by Yu & Wang (2015) performed to our sample of massive star forming regions, in what follows we discuss our results. Firstly, from the comparison between the molecular integrated intensities and the 870 µ m peak flux (see Fig. 3), we found that, in general, the integrated intensities of N 2 H + , C 2 H, HNC, and HC 3 N increase with the submillimeter peak flux. In the case of IRDCs, HMPOs, and HMCs, this increment correlates with the evolutive stage of the sources, i.e. the position of each kind of source in the plots seems to be sectorized. UCHII regions show the same trend, but their points are overlapped with those of the other sources in all cases except for the N 2 H + case. Yu & Wang (2015) found that the N 2 H + integrated intensities of HII regions tend to be lower that those for MYSOs. This behavior seems to occur with the UCHII regions of our sample in comparison with the others sources, suggesting that the destruction path due to electronic recombination, where N 2 H + is destroyed by free electrons from the surroundings (Vigren et al. 2012), could be ongoing in the UCHII regions. This result complements what was presented by Yu & Wang (2015) with sources at earlier stages, and suggests that the relation between the molecular integrated intensities and the 870 µ m peak flux could be useful to distinguish regions among IRDCs, HMPOs, and HMCs. The comparisons between the column densities obtained from Gerner et al. (2014) show also an interesting trend regarding the evolutive stage of the sources. Yu & Wang (2015) stated, based on astrochemical models (e.g. Bergin et al. 1997; Nomura & Millar 2004), that the H 13 CO + column density reflects the amount of H 2 density in a clump because its abundance seems to not vary much with the time. In line with their work, we compared N(C 2 H) and N(N 2 H + ) with N(H 13 CO + ), (see Fig. 5). The N(H 13 CO + ) values of our sample range in a wider interval of values (from some 10 12 cm -2 for IRDCs to values close to 10 14 cm -2 for HMCs) than the N(H 13 CO + ) presented by Yu & Wang (2015). An increase in the N(C 2 H) and N(N 2 H + ) with the N(H 13 CO + ) is observed. In the case of N(C 2 H) versus N(H 13 CO + ) relation, such an increment also presents a conspicuous correlation with the evolutive stage from IRDCs to HMCs in which the positions of each kind of source is sectorized in the plot. In the UCHII regions, both column densities have similar values compared with those of HMCs and the largest ones of HMPOs. A similar behavior, with larger dispersion, is observed in the comparison between N(N 2 H + ) versus N(H 13 CO + ). The N 2 H + and H 13 CO + column densities increment from IRDCs to HMCs could be explained by their progressive formation through the H + 3 mechanisms ( H + 3 + 13 CO → H 13 CO + +H 2 , and H + 3 +N 2 → N 2 H + +H 2 ; Jørgensen et al. 2004). Then, the brake on the constant increment in the column density values in the UCHII regions compared with the previous sources can be due to the beginning of the destruction of such molecules by electronic recombination (Yu & Xu 2016). It is worth noting that the discussed behavior in the column densities (which have molecular excitation assumptions) regarding the correlation with the evolutive stage from IRDCs to HMCs is quite similar to the comparison between the integrated intensities and the peak submillimeter flux (which are direct measurements). Thus, in the line of a chemical clock analysis, even though it is necessary more statistics, we propose that these relations, mainly the N(N 2 H + ) versus N(H 13 CO + ) and the I(C 2 H) versus F 870 µ m , in which the different kinds of sources (IRDCs, HMPOs, and HMCs) are clearly separated, could be used to differentiate them, which can be useful in works handling with a large amount of sources of unknown, or not completed known, nature. The comparison between the relative abundance ratios [C 2 H]/[H 13 CO + ] versus [N 2 H + ]/[H 13 CO + ] (Fig. 6) presents a high dispersion, mainly along the y-axis, i.e. in the N(C 2 H)/N(H 13 CO + ) ratio, and does not show a remarkable difference as it was found between MYSOs and HII regions by Yu & Wang (2015). However, along the x-axis, i.e. in the N(N 2 H + )/N(H 13 CO + ) ratio, some trend can be appreciated, mainly such a ratio seems to be larger in IRDCs in comparison with the other kind of sources, which in the line of a chemical clock analysis, we suggest that this abundance ratio can be used to differentiate the earliest stage of the star forming regions. The molecular line widths (FWHM) are usually associated with the kinematics of the molecular gas and can be affected by multiple events, such as shocks, outflows, rotation of the clump, and turbulence. In this work, we assume that line widths originate mainly from the increasing turbulence that arises as a consequence of evolving star formation stages (Sanhueza et al. 2012; Yu & Wang 2015). With this in mind, by analyzing the average line width of N 2 H + , HC 3 N, H 13 CO + and C 2 H (see Table 7), it can be appreciated that in the four species, the line width rises until reaching the HMC stage. Such an increase could be a consequence of the gas dynamics related to the star-forming processes that take place in the molecular clumps (Fontani et al. 2002; Pirogov et al. 2003). Particularly, our ∆ v(N 2 H + ) average values are in quite agreement with those presented by Sanhueza et al. (2012): the ∆ v(N 2 H + ) average value that we obtained for IRDCs is similar to that obtained by the authors in their so-called 'quiescent' and 'intermediate' clumps, while the value that we obtained for HMPOs is in agreement with the values of their 'active' clumps. Finally, our ∆ v(N 2 H + ) average values obtained for HMCs and UCHII regions are slightly larger than the value obtained for their 'red' clumps. Following Sanhueza et al. (2012), we can confirm that line widths of N 2 H + slightly increase with the evolution of the clumps. As found by Sanhueza et al. (2012) in their sample of clumps embedded in IRDCs, we observed that in our sources, the H 13 CO + and HC 3 N have slightly narrower line widths than N 2 H + (see panels d and f in Fig. 4). This hints that they trace similar optically thin gas emanating from the internal layers of the regions, and according to our findings, it seems to be independent of the kind of source. Yu & Wang (2015) found that the best ∆ v correlation is between C 2 H and N 2 H + for their sample of EGOs. In our case, we did not find such a well correlation between these molecular species (see panel e in Fig. 4). The C 2 H line widths are slightly narrower than those of the N 2 H + . Sanhueza et al. (2012) also remarked that the C 2 H appears to present the best correlation with N 2 H + among all the studied ∆ v relations. However, by inspecting their ∆ v(C 2 H) versus ∆ v(N 2 H + ) plot, we observe a similar behavior as our plot, where most of the points tend to be below the unity line. In the spectra of these four molecular species towards the whole sample, we did not find line wings that may probe outflows, except for the HC 3 N spectra of HMPO18247, HMC029.96 (see Fig. A.6), UCH10.30, and in the H 13 CO + spectrum of HMC34.26, in which small spectral wings appear. UCH10.30 is an EGO; the other sources present 4.5 µ m extended emission and diffuse emission at the Ks band. Besides these sources, it can be noticed that the near-IR evidence of jets or/and outflows have not any direct correlation with spectral features in the emission of N 2 H + , HC 3 N, H 13 CO + , and C 2 H.", "pages": [ 14, 15, 16, 17 ] }, { "title": "5 SUMMARY AND CONCLUDING REMARK", "content": "We presented a spectroscopic molecular line analysis of 55 high-mass star-forming regions with the aim of probing some chemical tools that were recently proposed to characterize such regions. The analyzed sources are classified as IRDCs, HMPOs, HMCs, and UCHII regions, according to an evolutionary progression in the high-mass star formation. The main results can be summarized as follows: 1. The emissions of the HCN and HNC isomers were used to estimate the kinetic temperature through a new 'thermometer' whose formulation was proposed by Hacar et al. (2020), and we investigated its use in the presented sample of sources. By comparing the T K derived from the HCN/HNC ratio with temperatures obtained from the dust and ammonia emission, we found that the use of such a ratio as a universal thermometer in the ISM should be taken with care. The HCN optical depth is a big issue to be taken into account. Line saturation may explain the derived temperature values below 15 K, which must be discarded, and mainly correspond to IRDCs and HMPOs. In addition, we found that, although the T K obtained from the HCN-HNC tool is greater than 15 K, their values could be far from the temperatures obtained from the dust or ammonia, yielding lower temperatures. This is the case of HMCs and UCHII regions of our sample, in which, in general, it is observed that the HCN hyperfine line F=1-1 appears absorbed. In conclusion, we point out that the HCN-HNC tool as a kinetic temperature estimator should be used only after a careful analysis of the HCN spectrum, checking that no line, neither the main nor the hyperfine ones, presents absorption features. In conclusion, this work explores chemical tools, some of them based on direct measurements, to be applied in different ISM environments. In that sense, we probed such tools in a new sample of sources with respect to previous works, and these results not only contribute to more statistics in the literature but also complement such works with other types of sources. Using direct tools like, for instance, the ratios of different molecular parameters that can be directly measured from the observations, can be very useful mainly when a large sample of sources is handled. If it is proven that such tools are reliable, they can be used to obtain important statistical information in a simple way. Our work points to it and encourages to perform similar works in larger samples of sources of different types.", "pages": [ 17 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank the anonymous referee for her/his useful comments pointing to improve our work. N.C.M. is a doctoral fellow of CONICET, Argentina. S.P. is a member of the Carrera del Investigador Cient'ıfico of CONICET, Argentina. This work was partially supported by the Argentina grant PIP 2021 11220200100012 from CONICET.", "pages": [ 18 ] }, { "title": "References", "content": "Bergin, E. A., Goldsmith, P. F., Snell, R. L., & Langer, W. D. 1997, ApJ, 482, 285, doi: 10.1086/ 304108 16 Bergin, E. A., Snell, R. L., & Goldsmith, P. F. 1996, ApJ, 460, 343, doi: 10.1086/176974 2 Betz, A. L., McLaren, R. A., & Spears, D. L. 1979, ApJ, 229, L97, doi: 10.1086/182937 2 Garc'ıa-Burillo, S., Viti, S., Combes, F., et al. 2017, A&A, 608, A56, doi: 201731862 2 Gerner, T., Beuther, H., Semenov, D., et al. 2014, A&A, 563, A97, doi: 10.1051/0004-6361/ 201322541 3, 4, 7, 14, 15, 16 Hoare, M. G., Lumsden, S. L., Oudmaijer, R. D., et al. 2005, in Massive Star Birth: A Crossroads of Astrophysics, ed. R. Cesaroni, M. Felli, E. Churchwell, & M. Walmsley, Vol. 227, 370-375, doi: 10. 1017/S174392130500476X 7 Jackson, J. M., Rathborne, J. M., Foster, J. B., et al. 2013, PASA, 30, e057, doi: 10.1017/pasa. 2013.37 11 Jørgensen, J. K., Belloche, A., & Garrod, R. T. 2020, ARA&A, 58, 727, doi: 10.1146/ annurev-astro-032620-021927 1 Jørgensen, J. K., Schoier, F. L., & van Dishoeck, E. F. 2004, A&A, 416, 603, doi: 10.1051/ 0004-6361:20034440 16 Lawrence, A., Warren, S. J., Almaini, O., et al. 2007, MNRAS, 379, 1599, doi: 10.1111/j. 1365-2966.2007.12040.x 4 Marsh, K. A., Whitworth, A. P., & Lomax, O. 2015, MNRAS, 454, 4282, doi: 10.1093/mnras/ stv2248 7 Marsh, K. A., Whitworth, A. P., Lomax, O., et al. 2017, MNRAS, 471, 2730, doi: 10.1093/mnras/ stx1723 7, 8 Martin, R. N., & Ho, P. T. P. 1979, A&A, 74, L7 2 Nagy, Z., Ossenkopf, V., Van der Tak, F. F. S., et al. 2015, A&A, 578, A124, doi: 10.1051/ 0004-6361/201424220 2 Nomura, H., & Millar, T. J. 2004, A&A, 414, 409, doi: 10.1051/0004-6361:20031646 16 Paron, S., Mast, D., Fari˜na, C., et al. 2022, A&A, 666, A105, doi: 10.1051/0004-6361/ 202243908 4 Pirogov, L., Zinchenko, I., Caselli, P., Johansson, L. E. B., & Myers, P. C. 2003, A&A, 405, 639, doi: 10.1051/0004-6361:20030659 16 Rathborne, J. M., Lada, C. J., Muench, A. A., Alves, J. F., & Lombardi, M. 2008, ApJS, 174, 396, doi: 10.1086/522889 2 Rawlings, J. M. C., Redman, M. P., Keto, E., & Williams, D. A. 2004, MNRAS, 351, 1054, doi: 10. 1111/j.1365-2966.2004.07855.x 2 Snyder, L. E., & Buhl, D. 1971, ApJ, 163, L47, doi: 10.1086/180664 2 Stephens, I. W., Jackson, J. M., Sanhueza, P., et al. 2015, ApJ, 802, 6, doi: 10.1088/0004-637X/ 802/1/6 2 Takano, S., Takano, T., Nakai, N., Kawaguchi, K., & Schilke, P. 2013, A&A, 552, A34, doi: 10.1051/ 0004-6361/201118593 2 Theory, ed. H. Beuther, H. Linz, & T. Henning, 381, doi: 7 10.1111/j. Urquhart, J. S., Moore, T. J. T., Csengeri, T., et al. 2014, MNRAS, 443, 1555, doi: 10.1093/mnras/ stu1207 6 Urquhart, J. S., Figura, C., Wyrowski, F., et al. 2019, MNRAS, 484, 4444, doi: 10.1093/mnras/ stz154 2, 14 Veena, V. S., Vig, S., Mookerjea, B., et al. 2018, ApJ, 852, 93, doi: 10.3847/1538-4357/aa9aef 2 Vigren, E., Zhaunerchyk, V., Hamberg, M., et al. 2012, ApJ, 757, 34, doi: 10.1088/0004-637X/ 757/1/34 15 Walmsley, C. M., Churchwell, E., Nash, A., & Fitzpatrick, E. 1982, ApJ, 258, L75, doi: 10.1086/ 183834 13 Yu, N., & Wang, J.-J. 2015, MNRAS, 451, 2507, doi: 10.1093/mnras/stv1058 2, 3, 7, 14, 15, 16, 17 Yu, N., & Xu, J. 2016, ApJ, 833, 248, doi: 10.3847/1538-4357/833/2/248 16 Zinnecker, H., & Yorke, H. W. 2007, ARA&A, 45, 481, doi: 10.1146/annurev.astro.44. 051905.092549 3", "pages": [ 18, 19, 20 ] }, { "title": "Appendix A: EXAMPLE SPECTRA", "content": "In this appendix we include some spectra of the analyzed molecular lines as an example of each kind of source, showing the integration area in the case of the HCN and HNC emissions and the Gaussian fittings for the H 13 CO + , C 2 H, HC 3 N, and N 2 H + . Spectra of IRDC 18151 are shown in Fig.A.1 and A.2, of HMPO 20126 in Figs. A.3 and A.4, of HMC 0.29.26 in Figs. A.5 and A.6, and of UCHII 45.45 in Figs. A.7 and A.8. Frequency [GHz] Frequency [GHz] Frequency [GHz] Frequency [GHz] Frequency [GHz] Frequency [GHz]", "pages": [ 21, 22, 23, 24, 26, 27, 28 ] } ]

[ { "title": "ABSTRACT", "content": ". Article . SPECIAL TOPIC: Month Year Vol. 66 No. 1: 000000 https: // doi.org / ??", "pages": [ 1 ] }, { "title": "A pseudoclassical theory for the wavepacket dynamics of the kicked rotor model", "content": "Zhixing Zou 1 , and Jiao Wang 1, 2* 1 Department of Physics and Key Laboratory of Low Dimensional Condensed Matter Physics (Department of Education of Fujian Province), Xiamen University, Xiamen 361005, Fujian, China; 2 Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, Gansu, China Received Month Day, Year; accepted Month Day, Year In this study, we propose a generalized pseudoclassical theory for the kicked rotor model in an attempt to discern the footprints of the classical dynamics in the deep quantum regime. Compared with the previous pseudoclassical theory that applies only in the neighborhoods of the lowest two quantum resonances, the proposed theory is applicable in the neighborhoods of all quantum resonances in principle by considering the quantum e ff ect of the free rotation at a quantum resonance. In particular, it is confirmed by simulations that the quantum wavepacket dynamics can be successfully forecasted based on the generalized pseudoclassical dynamics, o ff ering an intriguing example where it is feasible to bridge the dynamics in the deep quantum regime to the classical dynamics. The application of the generalized pseudoclassical theory to the PT -symmetric kicked rotor is also discussed. Quantum-classical correspondence, Kicked rotor model, Pseudoclassical theory, Wavepacket dynamics PACS number(s): 05.45.Mt, 03.65.Sq, 03.65.-w Citation: Zou Z X, Wang J, A pseudoclassical theory for the wavepacket dynamics of the kicked rotor model, Sci. China-Phys. Mech. Astron. 66 , 000000 (Year), https: // doi.org / ??", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Quantum-classical correspondence in a general system is the core issue of quantum chaos research or quantum chaology [1]. Based on the quantum-classical correspondence principle, the characteristics of a classically chaotic system are anticipated to manifest themselves in the corresponding quantum system in the semiclassical limit /planckover2pi1 → 0 ( /planckover2pi1 is the e ff ective Planck constant). Indeed, after 4 decades of intensive research, the general quantum manifestations of classical chaos, such as spectral statistics and the morphologies of wavefunctions, have been well revealed [2]. However, for quantum systems in the deep quantum regime opposite to the semiclassical limit, thus far, the relation between them and their classical counterparts have not been systematically addressed yet. The question arises: Are there any close connections between the classical chaotic dynamics and the quantum properties in the deep quantum regime? Without the guide of any physical law or principle, the answer is not immediate. An interesting and illuminating example is the pseudoclassical dynamics [3, 4] found in the kicked rotor model [5], a paradigm of quantum chaos with a free rotor subjected to periodic stroboscopic external kicks. Its properties significantly depend on the dimensionless parameter α = /planckover2pi1 T / I , where T is the kicking period and I is the rotational inertia of the rotor. For a general value of α , the quantum kicked rotor follows the classical di ff usive dynamics first and then the quantum dynamical localization eventually takes over [5,6]; how-", "pages": [ 1 ] }, { "title": "SCIENCE CHINA Physics, Mechanics & Astronomy", "content": "ever, by contrast, for α being a rational multiple of 4 π , that is, α = 4 π r / s with coprime integers r and s , a quantum state usually ballistically spreads, which is named as 'quantum resonance' [6, 7]. For low-order resonance with a small s value, the system is in the deep quantum regime; however, surprisingly, Fishman et al. [3,4] observed that when α is slightly detuned from the resonance condition by a nonzero δ , the quantum motion can be interpreted according to a certain fictitious classical system that is di ff erent from but closely related to the original classical counterpart. The fictitious classical system is named the pseudoclassical system, and the limit δ → 0 is the pseudoclassical limit. This result demonstrates a novel and unconventional aspect of quantum-classical correspondence. Unfortunately, the so-developed pseudoclassical theory only directly applies near the lowest two resonances, that is, s = 1 and 2. When the neighborhood of high-order resonance is considered, because a global pseudoclassical phasespace approximation regarding a unique classical Hamiltonian is impossible, one has to resort to local pseudoclassical approximations of di ff erent Hamiltonians related to the quasienergy bands of the considered quantum resonance [8]. Nevertheless, the interaction between the quasienergy bands may ruin the local pseudoclassical approximations to make their predictions invalid [9]. Recently [10], we realized that owing to the periodicity of the phase space, in a system with a spherical or cylindrical phase space, the free rotation of a wavepacket at quantum resonance may lead to the simultaneous presence of multiple wavepackets, in clear contrast to the usual scenario where only one wavepacket is found throughout. Such a wavepacket-multiplying e ff ect is a pure quantum interference e ff ect and has not been considered earlier. Considering this, the pseudoclassical theory may find wide applications when being generalized by considering this effect. This conjecture has been confirmed with the kicked top model [10], another paradigm of quantum chaos with a spherical phase space, which is closely connected with the kicked rotor [11, 12]. The objective of this work is to establish a generalized pseudoclassical theory for the kicked rotor model with a cylindrical phase space with regard to its significant role in quantum chaos research. This attempt is successful again. Based on our theory, for the lowest two resonances of s = 1 and 2, the free rotation does not lead to the wavepacket multiplying, and this is why the previous pseudoclassical theory is valid in these two cases. However, for higher-order resonance, particularly that of an odd s , the wavepacket multiplying can be shown to occur for certain, and the generalized theory is thus imperative. Our study might be a preliminary but positive attempt to address the quantum-classical correspondence issue in the deep quantum regime. This article is organized as follows: Sec. II briefly describes the kicked rotor model and discusses in detail the pseudoclassical limit of the wavepacket dynamics. Sec. III compares the quantum evolution of a wavepacket with the prediction of the pseudoclassical theory via numerical simulations, convincingly demonstrating that the latter is e ff ective. Sec. IV presents the extension to the PT -symmetric kicked rotor and confirmation by numerical simulations. Finally, Sec. V concludes this work.", "pages": [ 2 ] }, { "title": "2 The pseudoclassical limit of the kicked rotor", "content": "The Hamiltonian of the kicked rotor is Here, I , P , and θ are the rotational inertia, the angular momentum, and the conjugate angular coordinate of the rotor, respectively, whereas T and K are the kicking period and the dimensionless kicking strength, respectively. The integer parameter ω is introduced for our aim here, which is unity in the conventional kicked rotor model. The angular coordinate θ is imposed with a period of 2 π , such that the phase space is a cylinder. For the quantum kicked rotor, it is convenient to adopt the basis of the angular momentum eigenstates, {| n 〉 ; -∞ < n < ∞} , where n is an integer and | n 〉 meets P | n 〉 = n /planckover2pi1 | n 〉 . Because the Hamiltonian is periodic with time T , the evolution of the rotor for time T can be fulfilled by applying the Floquet operator to its current state just before a kick, where α ≡ /planckover2pi1 T / I and ˆ ν ≡ P / /planckover2pi1 . As stated in the Introduction, the quantum dynamics depends qualitatively on the fact if α is a rational multiple of 4 π . The case α = 4 π r / s with coprime integers r and s corresponds to quantum resonance, and the quantum state usually ballistically spreads. An exception is for s = 2 and ω being odd so that U 2 = 1, suggesting that the quantum state does not change after time 2 T , which is termed as 'quantum antiresonance' [7, 13]. When the system is slightly detuned away from a quantum resonance, a pseudoclassical theory has been developed to address the quantum dynamics via a classical map, the so-called pseudoclassical limit [3, 4]. However, this theory only works at the lowest two resonances. In the following of this section, we attempt to extend this theory to the neighborhoods of higher-order resonances with α = 4 π r / s + δ , where δ (incommensurate to π ) is a weak perturbation to the resonance condition. To perform a close comparison between the quantum and the classical dynamics, which is critical to our objective, we invoke the (squeezed) coherent state in the former, with regard to the advantages that the Husimi distribution of a coherent state has the minimum uncertainty in the phase space and its center point in the phase space represents exactly the classical counterpart of the coherent state. For convenience explained later, we use ( p , θ ) to denote a classical state, or a point in the phase space, with p ≡ P ( T / I )( δ/α ). The expression of the coherent state centered at ( p , θ ) is where c is the normalization factor. When δ is small, the uncertainty of p and θ is the same, δ p ≈ δθ ≈ √ δ/ 2. In the phase space, the Husimi distribution of the coherent state is a Gaussian function centered at ( p , θ ). In the limit of δ → 0, it collapses to the point ( p , θ ). Our task is to determine the one-step evolution for the classical state ( p , θ ) by analogy according to the corresponding quantum evolution for the coherent state | p , θ 〉 . To this end, note that for α = 4 π r / s + δ , the Floquet operator can be rewritten as where k = K δ/α . It includes two parts, i.e., U = UfU δ , with Uf = exp( -i 2 π r s ˆ ν 2 ) and U δ = exp ( -i δ 2 ˆ ν 2 ) exp( -i k δω cos( ωθ )). The former, Uf , represents a pure free rotation, while the latter, U δ , represents kicked rotor dynamics (see Eq. (2)) with α → δ and K → k . We thus break our task down into two steps. First, note that in the limit δ → 0, the quantum operation U δ has a well-defined semiclassical limit. This can be observed more clearly by imagining δ as a virtual Planck constant so that we can write down the classical Hamiltonian corresponding to U δ as Based on H δ , the classical Poincar'e map corresponding to U δ , denoted as M δ , can be derived straightforwardly. To be concrete, in terms of p and θ , the map M δ that evolves the state ( p , θ ) to ( ˜ p , ˜ θ ) reads Note that the phase space portrait created by M δ is periodic in both p and θ of period 2 π/ω . Due to this close relation between the quantum operation U δ and the classical map M δ in the limit of δ → 0, we assume that as well when δ is small, which is the only significant approximation we adopt for our theory. Equation (7) is the counterpart of the classical map M δ : ( p , θ ) → ( ˜ p , ˜ θ ) (see Eq. (6)), and thus, we finish the first step of our task. Afterward, we need to derive Uf | ˜ p , ˜ θ 〉 and work out its classical counterpart. The detailed calculation of Uf | ˜ p , ˜ θ 〉 is given in Appendix A, resulting in where Gl is the Gaussian sum [14] The physical meaning of Eq. (8) is evident: The intermediate coherent state | ˜ p , ˜ θ 〉 is mapped by Uf into s coherent states whose centers are located along the line of p = ˜ p in the phase space. They are separated in θ by 2 π r s (and its multiples) from each other, and each coherent state has an associated complex amplitude given by a Gaussian sum. This is a peculiar characteristic of the quantum rotor at resonance. Note that not all of these s coherent states exist necessarily. A coherent state disappears if the associated amplitude Gl disappears. Suppose that there are N nonzero amplitudes, Eq. (8) can be rewritten as Here, for the j th component coherent state, its amplitude Aj is a nonzero Gaussian sum Gl , and its position bias ∆ j is associated with the subscript of Gl by ∆ j = 2 π lr / s mod 2 π . Similarly, as in the limit δ → 0, a coherent state reduces to a point in phase space, Eq. (10) can be interpreted pseudoclassically. That is, the intermediate classical state ( ˜ p , ˜ θ ) is mapped by the pseudoclassical counterpart of Uf , denoted as M f , into a set of N states, and each of these is associated with a complex amplitude, This concludes the second step of our task. Therefore, formally, the pseudoclassical map corresponding to the quantum evolution U | p , θ 〉 that we are seeking for, denoted as M , can be expressed as M = M f M δ , that is, The intermediate state ( ˜ p , ˜ θ ) is related to ( p , θ ) by the map M δ (see Eq. (6)). This is the core result that we have obtained. As shown in the next section, it does allow the prediction of the quantum dynamics in such a pseudoclassical way. Here, we emphasize that the amplitudes { Aj } are crucial to this end. Specifically, | Aj | 2 has to be taken as the weight of the j th state ( ˜ p , ˜ θ + ∆ j ) that it is associated with to assess the expected value of a given observable. In addition, the phases encoded in these amplitudes have to be simultaneously considered to correctly trace the quantum evolution. Some remarks are in order. First, for the lowest two resonances s = 1 and s = 2, N = 1 and the amplitude of the only resultant state is unity. In addition, the map M reduces to that given by the original pseudoclassical theory [3,4], which has a seemingly pure classical form. For higher-order resonance, even though the map M is substantially more complex, there are no quantum operations and parameters that are explicitly involved. Despite this, the map M should be understood as a mix of the quantum and the classical dynamics according to our reasoning for deriving M . It might be appropriate to regard the seemingly simple form for the lowest two resonances as a coincidence. Second, for an odd s , all s Gaussian sums are nonzero [14] so that N = s ; for an even s , it can be shown that half of them must be zero so that N = s / 2 (see Appendix B). In either case, the main challenge for the implementation of the pseudoclassical map M lies in the rapid proliferation of the involved states when N ≥ 2. In general, their number exponentially increases, ∼ N t , as the number of iterations t (or the evolving time τ = tT ). This suggests why quantum dynamics is more complicated from such a novel perspective. Thus, for a general case, it can be hoped that in practice, our pseudoclassical theory only works in a short time to predict quantum evolution, which can be attributed to the intrinsic complexity of quantum dynamics. Third, however, for the following two cases equipped with an s - or s / 2-fold translational symmetry in the θ direction of the phase space, that is, the proliferation problem can be prevented because from the second iteration on ( t ≥ 2), two or more states can be mapped into one so that the total number of resultant states remains bounded by s in case C1 and by s / 2 in case C2. Appendix B provides a detailed discussion. In fact, note that for a general case, Uf and U δ do not commute. However, for case C1, they commute (see Appendix C), demonstrating that the operator for the t steps of iteration, ( UfU δ ) t , can be split into ( UfU δ ) t = U t f U t δ . As the state proliferation is exclusively caused by U t f = exp( -i 2 π t r s ν 2 ), we can conclude that the number of final resultant coherent states cannot surpass s for any time t . Moreover, the corresponding pseudoclassical map can be expressed as M t = M t f M t δ , based on which the computing of the pseudoclassical evolution can be substantially simplified. In case C2, Uf and U δ do not generally commute. However, U ' f and U ' δ commute, where U ' f = Uf exp( i π r ν ) and U ' δ = exp( -i π r ν ) U δ (see Appendix C). Consequently, ( UfU δ ) t = U ' f t U ' δ t , which has similar implications to case C1.", "pages": [ 2, 3, 4 ] }, { "title": "3 Verification of the pseudoclassical theory", "content": "The e ff ectiveness of the pseudoclassical theory is checked by the comparison of its predictions on the wavepacket dynamics and that directly obtained with the quantum Floquet operator. Three representative examples are examined, of which one is for a general case with r = 1, s = 4, and ω = 1, where the number of coherent states doubles after every step of iteration. The other two instances, one with r = 1, s = 3, and ω = 3 and another with r = 1, s = 4, and ω = 2, are for cases C1 and C2, respectively, where the proliferation of the coherent states is suppressed. As shown in the following of this section, the pseudoclassical theory e ff ectively works in all three cases. To conduct a close comparison between the quantum wavepacket evolution and its pseudoclassical counterpart, it is appropriate to visualize the quantum evolution in the phase space with the Husimi distribution [15]. For a given quantum state | ψ 〉 , at the given point ( p , θ ) in the phase space, the Husimi distribution H ( p , θ ) is defined as the expectation value of the density matrix ρ = | ψ 〉〈 ψ | with respect to the corresponding coherent state | p , θ 〉 . That is, First, for the first representative example of a general case where r = 1, s = 4, and ω = 1, based on Eq. (9), we have two nonzero Gl and as such N = 2. The pseudoclassical map is  with the complex amplitudes A 1 = 1 + i 2 and A 2 = 1 -i 2 . In addition, based on Eq. (6), ˜ p = p + k sin θ and ˜ θ = θ + ˜ p in this case. On the quantum aspect, for a given initial coherent state | ψ 0 〉 = | p 0 , θ 0 〉 , the state after t iterations, | ψ t 〉 = U t | ψ 0 〉 , can be numerically determined by repeatedly applying the evolution operator U . The contour plot of the corresponding Husimi distribution of | ψ t 〉 for t = 0 to 3 is presented in Fig. 1, where the wavepacket proliferation owing to quantum resonance can be clearly observed. On the pseudoclassical aspect, using the corresponding initial state ( p 0 , θ 0), we can acquire N t states after t iterations by the pseudoclassical map. The positions of these states are indicated in Fig. 1 as well. By comparison, we can observe that they can indeed wellcapture the skeleton of the quantum state. In addition to the skeleton, more information on the quantum state is encoded in the complex amplitudes of the pseudoclassical states. To determine if this information is su ffi -cient to forecast the expected value of a given observable, the angular momentum di ff usion behavior is considered as an example, quantified by 〈 ( p -p 0) 2 〉 , which is of particular interest in the quantum kicked rotor research. Fig. 2 compares the results by the quantum and the pseudoclassical dynamics. We may expect that in the limit of δ → 0, the agreement between them should be progressively enhanced, which is well corroborated. Note that despite its success, as shown in Fig. 1 and Fig. 2, owing to the issue of state proliferation, the implementation of the pseudoclassical theory would be prohibitively costly for a long evolution time. For instance, in the case discussed, the involved states would have reached up to 10 9 for t = 30. However, for a system that has the translational symmetry in θ as in cases C1 and C2, the state proliferation challenge can be overcome. As two examples of case C1 and C2, we consider r = 1, s = 3, and ω = 3 and r = 1, s = 4, and ω = 2, respectively. For the former, N = 3 and the pseudoclassical map reads as where A 1 = -√ 3 i 3 , A 2 = A 3 = 3 + √ 3 i 6 , and ( ˜ p , ˜ θ ) is generated from ( p , θ ) according to Eq. (6). This suggests that, after each step, a point will be mapped into three of the same weight but of two di ff erent phases. The results of the Husimi distribution of the quantum state at t = 0 to 3 and the pseudoclassical map are shown in Fig. 3. It can be observed that at t = 1, the initial quantum coherent state is mapped into three, and their centers perfectly overlap with the three resulting pseudoclassical states. At t = 2, of the nine expected pseudoclassical states, three pairs cancel each other so that only three survive. This is completely supported by quantum evolution. Finally, at t = 3, the three remaining pseudoclassical states merge into one rather than split into nine, successfully further predicting quantum evolution. For the quantum evolution, it can be straightforwardly shown that U 3 f = 1, suggesting that the number of involved coherent states must reduce to one after every three steps. For the latter case of r = 1, s = 4, and ω = 2, the pseudoclassical map is the same as Eq. (14) but with ˜ p = p + k sin(2 θ ) and ˜ θ = θ + ˜ p instead. As A 1 , 2 = 1 ± i 2 , a point will be mapped into two with the same weight but di ff erent phases. Fig. 4 compares the quantum and pseudoclassical dynamics, and good agreement between them is the same as in the previous case. Note that for this case, although U 4 f = 1, U 2 f /nequal 1. However, U ' 2 f = 1 such that ( UfU δ ) 2 = U ' 2 δ , which explains why after every two steps, the number of pseudoclassical states becomes one. In these two cases, as the state proliferation problem is well suppressed owing to translational symmetry, the pseudoclassical evolution can be conveniently conducted up to a much longer time than in the general case. For these two cases, Figs. 5(a) and 5(b) compare the time dependence of 〈 ( p -p 0) 2 〉 calculated with the quantum and the pseudoclassical dynamics over a wide time range. In particular, k is fixed, but perturbation δ is changed to observe how the quantum results depend on it. Indeed, as expected, as δ decreases, the quantum result tends to approach the pseudoclassical re- In addition, for the di ff usion time t di ff , which is empirically defined as the time when the quantum result deviates from the pseudoclassical result by 15% from below, it follows the scaling t di ff ∼ δ -2 [see Figs. 5(c) and 5(d)], the same as in the conventional semiclassical limit of the kicked rotor model if δ is recognized with the e ff ective Planck constant [16]. All these results consistently support the e ff ectiveness of our pseudoclassical theory.", "pages": [ 4, 5, 6 ] }, { "title": "4 Application to the PT -symmetric kicked rotor", "content": "Quantum mechanical Hamiltonians that are PT -symmetric but not Hermitian has recently been a frontier subject [1723]. A Hamiltonian H is regarded as PT -symmetric if [ H , PT ] = 0, where the parity operator, P , is a unitary operator that satisfies P 2 = 1 and the time-reversal operator, T , is an antiunitary operator that satisfies T 2 = ± 1. As a result, PT is an antiunitary operator as well. Surprisingly, as observed in some previous works and highlighted in Ref. [17], it is feasible for a PT -symmetric Hamiltonian to have a real spectrum, despite the fact that it can be non-Hermitian. Moreover, as the gain (or loss) parameter λ that controls the degree of non-Hermiticity changes, a spontaneous PT symmetry breaking may take place. In this section, we attempt to use our pseudoclassical theory to the PT -symmetric kicked rotor model whose Hamiltonian is [18] where λ ≥ 0 is the non-Hermitian parameter that controls the strength of the imaginary part of the kicking potential. For λ = 0, it reduces to the conventional Hermitian kicked rotor. When α ≡ /planckover2pi1 T / I = 4 π r s with r and s two coprime integers, similar to the conventional kicked rotor, the quasienergy spectrum is absolutely continuous and composed of s quasienergy bands, which is also named quantum resonance [18]. In the following, we will focus on the perturbed case where α = 4 π r s + δ , for which the Floquet operator can be written as It includes two parts: U NH = UfU NH δ with Uf = exp ( -i 2 π r s ˆ ν 2 ) and U NH δ = exp ( -i δ 2 ˆ ν 2 ) exp( -i k δ cos( ωθ )) exp( k λ δ sin( ωθ )). The di ff erence lies in the last term of U NH δ . For λ = 0, this term is identity, U NH δ reduces to U δ and U NH reduces to U . For λ /nequal 0 and in the limit of δ → 0, when the last term of U NH δ acts on a quantum state expressed as the superposition of multiple coherent states, only the component coherent state | pj , θ j 〉 that maximizes sin( ωθ ) (with θ j ) significantly contributes to the result; the contributions of other components can be ignored. In such a sense, one role that the last term of U NH δ plays is a 'selector'. Formally, the pseudoclassical counterpart of this role can be denoted as M s . It chooses the component state ( pj , θ j ) from others as the first step of the pseudoclassical dynamics. The last term of U NH δ also plays a key role in retrieving the pseudoclassical counterpart, denoted as M NH δ , of U NH δ . Based on the generalized canonical structure theory [20], in the limit of δ → 0, the motion of the center of a given coherent state U NH δ acts on is governed by the following equations: Thus, the pseudoclassical operation M NH δ thus represents the integration of these two functions up to a unit time. For λ = 0, it reduces to M δ given by Eq. (6). Formally, the pseudoclassical map for the PT -symmetric kicked rotor can be written as To test its e ff ectiveness, a general case of the PT -symmetric kicked rotor is simulated with r = 1, s = 4, and ω = 1, the same as in Figs. 1 and 2. Figs. 6 and 7 illustrate the results. Fig. 6 displays the Husimi distribution of the quantum state evolved from an initial coherent state. Owing to the gain (or loss) operation of the selecting operator exp( k λ δ sin θ ), only two component coherent states appear for t ≥ 2, in clear contrast to the conventional kicked rotor corresponding to λ = 0 (see Fig. 1 for comparison). Namely, the state proliferation problem in the latter is e ff ectively suppressed here by the selection operator. Meanwhile, we can see that the positions of the two component coherent states are well predicted by the pseudoclassical map. An interesting feature of the PT -symmetric kicked rotor is that it can generate the directed current [18, 19]. Here we study this property with α being slightly perturbed from the quantum resonance condition. Fig. 7 compares the expected values of the momentum, 〈 p 〉 , for quantum and pseudoclassical dynamics. It can be observed that as δ decreases, they do converge and 〈 p 〉 ∼ t , which implies that the directed current also exists in the deep quantum regime near quantum resonances and interestingly, it has a pseudoclassical explanation.", "pages": [ 6, 7, 8 ] }, { "title": "5 Summary", "content": "In this study, by considering the quantum e ff ect of the free rotation at quantum resonances, a generalized pseudoclassical theory is designed for the kicked rotor model. Its e ff ectiveness suggests that, even in the deep quantum regime, quantum dynamics may have a close connection to classical dynamics. With regard to this, one may wonder if it is imperative to extend the conventional quantum chaos study from the semiclassical regime [1] to the deep quantum regime. In this context, it depends on how general the pseudoclassical theory could be, which warrants further investigation, except the kicked top and the kicked rotor model, if the pseudoclassical theory can be extended to other Floquet systems. Extensive experiments have been conducted on the kicked rotor model with cold atoms [24] owing to its paradigmatic role in illustrating quantum chaos. It would be enticing to examine the e ff ects of the pseudoclassical dynamics near higher-order quantum resonances. To this end, it is necessary to adapt the pseudoclassical theory to the kicked particle model first, which is in progress. This work is supported by the National Natural Science Foundation of China (Grants No. 12075198, No. 12247106, and No. 12247101). Conflict of interest The authors declare that they have no conflict of interest.", "pages": [ 8 ] }, { "title": "Appendix A Derivation of Equation (8)", "content": "Regarding the eigenstates {| n 〉} of the operator ˆ ν , the coherent state | ˜ p , ˜ θ 〉 can be expressed as | ˜ p , ˜ θ 〉 = ∑ n cn | n 〉 . Applying the operator exp ( -i 2 π r s ˆ ν 2 ) to both sides, ( - 2 π r s ) | 〉 s k = 0 1 ∑ ( - 2 π r s ) ∑ mod ( n , s ) = k | 〉 Note that two coherent states separating in θ by φ can be related by the translation operator exp( -i ˆ νφ ), we have - exp i ν ˆ 2 p , ˜ ˜ θ = exp i k 2 cn n . (a1) by setting φ = 2 π lr / s , where l is an integer. Thereafter, multiplying both sides with exp( i 2 πλ lr / s ), where λ is an integer, 0 ≤ λ ≤ s -1, and taking summation over l from l = 0 to s -1, Finally, by replacing λ with k and substituting this equation into Eq. (a1), we get Eq. (8).", "pages": [ 8, 9 ] }, { "title": "Appendix B Translational symmetry of cases C1 and C2", "content": "To prevent the proliferation problem of the pseudoclassical dynamics, one way is to introduce the translational symmetry into the phase space so that two or more state points will be mapped into one by M f . This can be fulfilled by setting the proper integer value of ω . First, for case C1 where s is odd, based on the analytical results by Ref. [14], all s Gaussian sums are nonzero so that N = s and any two neighboring state points resulted by acting M f to a given state are separated in θ by 2 π/ s . Hence, to ensure the resultant state points overlap at the next step, we can set ω = s . It ensures that the number of states is up-bounded by s throughout. For case C2, where s is even, note that the Gaussian sum Gl can be rewritten as It is evident that the last term, 1 + exp ( -i 2 π r ( s 4 -l 2 )) , is zero for all odd l when mod( s , 4) = 0 and is zero for all even l when mod( s , 4) = 2. On the one hand, this implies that half of all s Gaussian sums are zero, whereas the other half are nonzero based on Ref. [14], so that N = s / 2. On the other hand, this suggests that of all s / 2 state points resulted by acting M f to a given state, any two neighboring points are separated in θ by 4 π/ s . Thus, similar to case C1, to ensure the evolving state points overlap at the following steps, we can set ω = N = s / 2. This guarantees that the number of states is up-bounded by N = s / 2 throughout.", "pages": [ 9 ] }, { "title": "Appendix C Commutation relation of Uf and U δ", "content": "In the representation of ˆ ν , the elements of Uf and U δ are, respectively, following which the matrix element for the commutator ( UfU δ -U δ Uf ) m , m ' is Note that exp ( -i k ωδ cos ( ωθ ) ) is periodic with the period 2 π/ω ; thus, the integration term in the above equation can be rewritten as Substituting it into Eq. (c7), For case C1 where s is odd, as ω = s , the delta function in Eq. (c9) suggests that m -m ' = ns ( n is an integer). Substituting it into Eq. (c7), it becomes because the di ff erence term in Eq. (c7) is However, for case C2 where s is even and ω = s 2 , the di ff erence term in Eq. (c7) does not equal zero in general, and thus, UfU δ -U δ Uf /nequal 0. However, if we split the Floquet operator U as U = U ' f U ' δ with U ' f = Uf exp( i π r ν ) and U ' δ = exp( -i π r ν ) U δ , as shown in the following, U ' f and U ' δ do commutate. The matrix element for U ' f and U ' δ is, respectively, and thus, Note that here the integration term is the same as Eq. (c7); however, the delta function suggests m -m ' = n s 2 instead ( n is an integer). Similarly, substituting it into Eq. (c13), we obtain because the di ff erence term in Eq. (c13) is for any even s .", "pages": [ 9, 10 ] } ]

[ { "title": "Prediction of the SYM-H Index Using a Bayesian Deep Learning Method with Uncertainty Quantification", "content": "Yasser Abduallah 1 , 2 , Khalid A. Alobaid 1 , 2 , 3 , Jason T. L. Wang 1 , 2 Haimin Wang 1 , 4 , 5 , Vania K. Jordanova 6 , Vasyl Yurchyshyn 5 Huseyin Cavus 7 , 8 , Ju Jing 1 , 4 , 5 1 Institute for Space Weather Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA 2 Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA 3 College of Applied Computer Sciences, King Saud University, Riyadh 11451, Saudi Arabia 4 Center for Solar-Terrestrial Research, New Jersey Institute of Technology, Newark, NJ 07102, USA 5 Big Bear Solar Observatory, New Jersey Institute of Technology, Big Bear City, CA 92314, USA 6 Space Science and Applications, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 7 Department of Physics, Canakkale Onsekiz Mart University, 17110 Canakkale, Turkey 8 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "We propose a novel deep learning framework, named SYMHnet, which employs a graph neural network and a bidirectional long short-term memory network to cooperatively learn patterns from solar wind and interplanetary magnetic field parameters for short-term forecasts of the SYM-H index based on 1-minute and 5-minute resolution data. SYMHnet takes, as input, the time series of the parameters' values provided by NASA's Space Science Data Coordinated Archive and predicts, as output, the SYM-H index value at time point t + w hours for a given time point t where w is 1 or 2. By incorporating Bayesian inference into the learning framework, SYMHnet can quantify both aleatoric (data) uncertainty and epistemic (model) uncertainty when predicting future SYM-H indices. Experimental results show that SYMHnet works well at quiet time and storm time, for both 1-minute and 5-minute resolution data. The results also show that SYMHnet generally performs better than related machine learning methods. For example, SYMHnet achieves a forecast skill score (FSS) of 0.343 compared to the FSS of 0.074 of a recent gradient boosting machine (GBM) method when predicting SYM-H indices (1 hour in advance) in a large storm (SYM-H = -393 nT) using 5-minute resolution data. When predicting the SYM-H indices (2 hours in advance) in the large storm, SYMHnet achieves an FSS of 0.553 compared to the FSS of 0.087 of the GBM method. In addition, SYMHnet can provide results for both data and model uncertainty quantification, whereas the related methods cannot.", "pages": [ 2 ] }, { "title": "Plain Language Summary", "content": "In the past several years, machine learning and its subfield, deep learning, have attracted considerable interest. Computer vision, natural language processing, and social network analysis make extensive use of machine learning algorithms. Recent applications of these algorithms include the prediction of solar flares and the forecasting of geomagnetic indices. In this paper, we propose an innovative machine learning method that utilizes a graph neural network and a bidirectional long short-term memory network to cooperatively learn patterns from solar wind and interplanetary magnetic field parameters to provide short-term predictions of the SYM-H index. In addition, we present techniques for quantifying both data and model uncertainties in the output of the proposed method.", "pages": [ 2 ] }, { "title": "1 Introduction", "content": "Geomagnetic activities and events are known to have a substantial impact on the Earth. They can damage and affect technological systems such as telecommunication networks, power transmission systems, and spacecraft (Ayala Solares et al., 2016; Jordanova et al., 2020). These activities are massive and scale on orders of magnitude (Newell et al., 2007). It may take a few days to recover from the damage, depending on its severity. These activities and events cannot be ignored regardless of whether they are in regions at high, medium, or low latitudes (Carter et al., 2016; Gaunt & Coetzee, 2007; Moldwin & Tsu, 2016; Tozzi et al., 2019; Viljanen et al., 2014). Therefore, several activity indices have been developed to measure the intensity of the geomagnetic effects. These indices characterize the magnitude of the disturbance over time. Modeling and forecasting these geomagnetic indices have become a crucial area of study in space weather research. Some indices, such as Kp, describe the overall level of geomagnetic activity while others, such as the disturbance storm time (Dst) index (Woodroffe et al., 2016), describe a specific area of geomagnetic activity. The Dst index has been used to classify a storm based on its intensity (Bala & Reiff, 2012; Gruet et al., 2018; Lazz'us et al., 2017; Lu et al., 2016; Xu et al., 2023). The storm is intense when Dst is less than -100 nT, moderate when Dst is between -100 nT and -50 nT, and weak when Dst is greater than -50 nT (Gruet et al., 2018; Nuraeni et al., 2022). Another important index is the sym- etric H-component index (SYM-H), which is used to represent the longitudinally symmetric disturbance of the intensity of the ring current during geomagnetic storms. The SYM-H index is the one-minute version of the DST index, obtained by data from more stations (Rangarajan, 1989; Siciliano et al., 2021; Vichare et al., 2019; Wanliss & Showalter, 2006). On the other hand, ASY-H (the asymmetric geomagnetic disturbance of the horizontal component) is quantified as the longitudinally asymmetric part of the geomagnetic disturbance field at low latitude to midlatitude. In addition, there are other indices that can be used to measure the activity of the storm as described in Mayaud (1980). A lot of efforts have been devoted to developing strategies to alleviate the geomagnetic effects on technologies and humans, but it is almost impossible to offer complete protection from the effects (Siciliano et al., 2021). Some of these strategies are to predict the occurrence and intensity of geomagnetic storms to offer some level of mitigation of their damaging effects. For example, Burton et al. (1975) established an empirical connection between interplanetary circumstances and Dst using a linear forecasting model. Temerin and Li (2002) developed an explicit model to predict Dst on the basis of solar wind data for the years 1995-1999, by finding functions and values of free parameters that minimize the root square error (RMS error) between their model and the measured Dst. Wang et al. (2003) used differential equation models to examine the effect of the dynamic pressure of the solar wind on the decay and injection of the ring current. Yurchyshyn et al. (2004) proposed that the hourly averaged magnitude of the Bz component of the magnetic field in interplanetary ejecta is correlated with the speed of the CME, which may open a way to predict the Dst index using CME parameters. Ayala Solares et al. (2016) performed predictions of global magnetic disturbance in near-Earth space in a case study for the Kp index using Nonlinear AutoRegressive with eXogenous (NARX) models. Due to the intrinsic complex response of the circumterrestrial environment to changes in the interplanetary medium, these simple models were unable to properly and fully depict the evolution of the solar wind-magnetosphere-ionosphere system (Consolini & Chang, 2001; Klimas et al., 1996; Siciliano et al., 2021). To surpass the limitations of simple models and acquire the complex response of the magnetosphere, researchers resorted to more advanced models such as artificial neural networks (ANNs). The use of ANNs focused on the prediction of the Dst and Kp indices. Gleisner et al. (1996) constructed the first Dst prediction model employing a time-delay ANN with solar wind parameters as input variables. Lazz'us et al. (2017) created a particle swarm optimization method to train ANN connection weights to improve the accuracy of the prediction of the Dst index. Bala and Reiff (2012) combined ANNs and physical models with solar wind and interplanetary magnetic field parameters such as velocity, interplanetary magnetic field (IMF) magnitude, and clock angle. Chandorkar et al. (2017) used Gaussian processes (GP) to build an autoregressive model to predict the Dst index 1 hour in advance based on the past solar wind velocity, the IMF component B z , and the values of the Dst index. This method generated a predictive distribution rather than a single prediction point. However, the mean values of the estimations are not as accurate as those generated by ANNs. Gruet et al. (2018) overcame the poor performance of GP and constructed a Dst index estimation model by merging GP with a long shortterm memory (LSTM) network to obtain more accurate results. More recently, Xu et al. (2023) developed a new GP regression model that performed better than related distance correlation learning methods (Lu et al., 2016) in forecasting the Dst index during intense geomagnetic storms. Rastatter et al. (2013) compared the effectiveness of 30 Dst forecast models and found that none of the models performed consistently the best for all events. Relatively few researchers have focused on the prediction of SYM-H. This happens probably because of the high temporal resolution of 1 minute for the SYM-H index, which gives rise to a more difficult problem in estimating SYM-H due to its highly oscillating nature (Siciliano et al., 2021). However, some SYM-H index prediction techniques have been reported in the literature. Cai et al. (2010) presented the first 5-minute average estimates of the SYM-H index throughout large storms between 1998 and 2006 using a NARX neural network with IMF and solar wind data. Bhaskar and Vichare (2019) predicted both the SYM-H and ASY-H indices for solar cycle 24 by employing the NARX neural network in a similar way. Both Bhaskar and Vichare (2019) and Cai et al. (2010) used the IMF magnitude ( B ), B y and B z components, as well as the density and velocity of the solar wind as input data for their models. Siciliano et al. (2021) provided a comprehensive examination of two well-known deep learning models, namely long short-term memory (LSTM) and a convolutional neural network (CNN), with an average temporal resolution of 5 minutes for the estimation of SYM-H index values (1 hour in advance). The authors used the IMF component B z , squared values of the magnitude of the IMF B and the B y component, measured at L1 by the ACE satellite in GSM coordinates. ColladoVillaverde et al. (2021) created neural network models for the SYM-H and ASY-H predictions by combining CNN and LSTM. The authors considered 42 geomagnetic storms between 1998 and 2018 for model training, validation, and testing purposes. Iong et al. (2022) developed a model using gradient boosting machines to predict the SYM-H index (1 and 2 hours in advance) with a temporal resolution of 5 minutes. In this paper, we present a new method, named SYMHnet, that utilizes cooperative learning of a graph neural network (GNN) and a bidirectional long short-term memory (BiLSTM) network with Bayesian inference to conduct short-term (1 or 2 hours in advance) predictions of the SYM-H index for solar cycles 23 and 24. We consider temporal resolutions of 1 minute and 5 minutes, respectively, for the SYM-H index. To our knowledge, this is the first time that 1-minute resolution data have been used to predict the SYM-H index. Furthermore, our method can quantify both model and data uncertainties when producing prediction results, whereas related machine learning methods cannot. The remainder of this paper is organized as follows. Section 2 describes the data, including the solar wind and IMF parameters, as well as geomagnetic storms, used in this study. Section 3 presents the methodology, explaining the SYMHnet framework, its architecture, and the uncertainty quantification algorithm. Section 4 evaluates the performance of SYMHnet on 1-minute and 5-minute resolution data. We also report the experimental results obtained by comparing SYMHnet with related machine learning methods on 5-minute resolution data. Section 5 presents a discussion and concludes the paper.", "pages": [ 2, 3, 4 ] }, { "title": "2 Database", "content": "In training and evaluating SYMHnet, we built a database that combines the solar wind and IMF parameters with the geomagnetic storms studied in this paper. This database contains 42 storms selected from the past two solar cycles (#23 and #24). The storms occurred between 1998 and 2018.", "pages": [ 4 ] }, { "title": "2.1 Solar Wind and IMF Parameters", "content": "We consider seven solar wind, IMF, and derived parameters: IMF magnitude ( B ), B y and B z components, flow speed, proton density, electric field and flow pressure. These parameters have been used in related studies (Bhaskar & Vichare, 2019; Cai et al., 2010; Denton et al., 2016; Iong et al., 2022). The parameters' values along with the SYM-H index values are collected from the NASA Space Science Data Coordinated Archive available at https://nssdc.gsfc.nasa.gov (King & Papitashvili, 2005). Data are collected with 1- and 5-minute resolutions.", "pages": [ 4 ] }, { "title": "2.2 Geomagnetic Storms", "content": "We work with the same storms as those considered in previous studies (ColladoVillaverde et al., 2021; Iong et al., 2022; Siciliano et al., 2021). Table 1 lists the storms used to train SYMHnet. Table 2 lists the storms used to validate SYMHnet. Table 3 lists the storms used to test SYMHnet. The training set, validation set, and test set are disjoint. Thus, SYMHnet can make predictions on storms that it has never seen during training. Note that each storm period listed in Tables 1, 2, and 3 contains both quiet time and storm time, as indicated by the maximum SYM-H and minimum SYM-H values in the period.", "pages": [ 5 ] }, { "title": "3 Methodology", "content": "Machine learning (ML) and its subfield, deep learning (DL) (Goodfellow et al., 2016), have been used extensively in the space weather community for predicting solar flares (Abduallah et al., 2021; Huang et al., 2018; Liu et al., 2019), flare precursors (Chen et al., 2019), coronal mass ejections (Alobaid et al., 2022; Liu et al., 2020), solar energetic particles (Abduallah et al., 2022; Laurenza et al., 2009; Lavasa et al., 2021; N'u˜nez, 2011; Stumpo et al., 2021), and geomagnetic indices (Amata et al., 2008; Bala & Reiff, 2012; Bhaskar & Vichare, 2019; Collado-Villaverde et al., 2021; Gruet et al., 2018; Lazz'us et al., 2017; Pallocchia et al., 2006; Siciliano et al., 2021). Different from the existing methods, SYMHnet combines a graph neural network (GNN) and a bidirectional long shortterm memory (BiLSTM) network to jointly learn patterns from input data. GNN learns the relationships among the parameter values in the input data, while BiLSTM captures the temporal dynamics of the input data. As our experimental results show later, this combined learning framework works well and generally performs better than related machine learning methods for SYM-H index forecasting.", "pages": [ 5, 6 ] }, { "title": "3.1 Parameter Graph", "content": "We construct an undirected unweighted fully connected graph (FCG) for the solar wind, the IMF and the derived parameters considered in this study, where each node represents a parameter and there is an edge between every two nodes. Because the parameter values are time series, we obtain a time series of parameter graphs where the topologies of the graphs are the same, but the node values vary as time goes on. For example, Figure 1 shows three parameter graphs constructed at time points t , t + 1, t + 2, respectively, with a resolution of 1 minute to predict the SYM-H index 1 hour in advance. In Figure 1, the leftmost graph at t contains the values of the seven parameters, represented by seven nodes or circles, at the time point t . The FCG symbol in the center indicates that this is a fully connected graph in which every two nodes are connected by an edge. (For simplicity, only a portion of the edges are shown in the figure.) Furthermore, the graph contains a node that represents the value of the SYM-H index at the time point t + 1 hour. During training, this SYM-H index value is used as the label for the graph. The GNN in SYMHnet will learn the relationships among the parameters' values and the relationships between the parameters' values and the label. If we want to predict the SYM-H index 2 hours in advance, then the label will be the SYMH index value at the time point t + 2 hours. The middle graph at t + 1 in Figure 1 contains the values of the seven parameters at the time point t + 1 minute. In addition, this graph contains the SYM-H index value at the time point ( t + 1 minute) + 1 hour, which is the label for this graph. If we want to predict the SYM-H index 2 hours in advance, then the label will be the SYM-H index value at the time point ( t + 1 minute) + 2 hours. The rightmost graph at t + 2 in Figure 1 contains the values of the seven parameters at the time point t + 2 minutes. Additionally, this graph contains the SYM-H index value at the time point ( t + 2 minutes) + 1 hour, which is the label for this graph. If we want to predict the SYM-H index 2 hours in advance, then the label will be the SYM-H index value at the time point ( t + 2 minutes) + 2 hours. During testing/prediction, given the values of the seven parameters at a time point t ' (without a label), SYMHnet will predict the label, which is the SYM-H index value at the time point t ' + 1 hour (for 1-hour ahead predictions) or the SYM-H index value at the time point t ' + 2 hours (for 2-hour ahead predictions), as detailed in Section 3.2.", "pages": [ 6, 7 ] }, { "title": "3.2 The SYMHnet Framework", "content": "Figure 2 illustrates the SYMHnet framework. During training, we feed the input data sample at each time point in turn to SYMHnet. The input data sample at the time point t consists of the parameter graph G t constructed at t and a sequence of m records X t -m +1 , X t -m +2 , . . . , X t where X i , t -m +1 ≤ i ≤ t , represents the record collected at the time point i . X i contains the seven values of the solar wind and IMF parameters along with the SYM-H index value at the time point i . Including previous SYM-H index values in the input to predict future SYM-H indices improves prediction accuracy (Iong et al., 2022). The number of records, m , in the input is set to 10 which was determined by our experiments. When m < 10, BiLSTM cannot effectively capture the temporal patterns in the data. When m > 10, it causes additional overhead for larger se- quence sizes without improving prediction accuracy. The label of the graph G t is used as the label of the input data sample at the time point t . The parameter graph G t is sent to SYMHnet's GNN component (Panagopoulos et al., 2021) while the sequence of m records, X t -m +1 , X t -m +2 , . . . , X t , is sent to SYMHnet's BiLSTM component (Abduallah et al., 2022). The GNN, illustrated in Figure 2(b), contains a graph convolutional layer followed by a rectified linear unit (ReLU), which is followed by another graph convolutional layer and ReLU. The BiLSTM network, illustrated in Figure 2(c), is composed of two LSTM layers (Hochreiter & Schmidhuber, 1997) with opposite directions when processing the data. This architecture allows the BiLSTM network to use one LSTM layer to read the sequence from the end to the beginning, denoted as forward, and the other LSTM layer to read the sequence from the beginning to the end, denoted as backward. GNN is good for learning the correlations between nodes (parameters) in a graph (Panagopoulos et al., 2021) while BiLSTM is suitable for learning the temporal patterns in time series (Abduallah et al., 2022; Siami-Namini et al., 2019). SYMHnet combines the learned parameter correlations and temporal patterns into a joint pattern, which is then passed to two dropout and dense layers. A dropout layer provides a mechanism to randomly drop a percentage of neurons to avoid over-fitting on the training data so that the SYMHnet model can generalize to unseen test data. It also enables the Monte Carlo (MC) sampling method described in Section 3.3 because the internal structure of the network is slightly different each time neurons are dropped (Gal & Ghahramani, 2016; Jiang et al., 2021). Each neuron in a dense layer connects to every neuron in the preceding layer (Goodfellow et al., 2016). The dense layer helps to change the dimensionality of the output of the preceding layer so that the SYMHnet model can easily define the relationship between the values of the data on which the model works. In this way, we better train our model, and the model learns things more effectively. Table 4 summarizes the details of the model architecture. During testing/prediction, we feed an unlabeled test data sample to SYMHnet where the test data sample is the same as the training data sample, except that the test data sample does not have a label. The trained SYMHnet model will predict the label based on the input test data sample. SYMHnet uses the MC dropout sampling method described in Section 3.3 to produce, for a test data sample, a predicted SYM-H index value accompanied by results of aleatoric uncertainty and epistemic uncertainty.", "pages": [ 7, 9 ] }, { "title": "3.3 Uncertainty Quantification", "content": "Quantification of uncertainty is essential for the reproducibility and validation of a model (Volodina & Challenor, 2021). Uncertainty quantification with deep learning has been used in computer vision (Kendall & Gal, 2017), space weather (Gruet et al., 2018), and solar physics (Jiang et al., 2021). There are two types of uncertainty: aleatoric and epistemic. Aleatoric uncertainty captures the inherent randomness of data, hence also referred to as data uncertainty. Epistemic uncertainty occurs due to the inexact weight calculations in a neural network and is also known as model uncertainty. In incorporating Bayesian inference into SYMHnet, our goal is to find the posterior distribution over the weights of the network, W , given the observed training data, X , and the labels Y , that is, P ( W | X,Y ). The posterior distribution is intractable (Jiang et al., 2021), and one has to approximate the weight distribution (Denker & LeCun, 1990). We use variational inference as suggested by Graves (2011) to learn the variational distribution on the weights of the network, q ( W ), by minimizing the Kullback-Leibler (KL) divergence of q ( W ) and P ( W | X,Y ). Training a network with dropout (Srivastava et al., 2014) is equivalent to a variational approximation on the network (Gal & Ghahramani, 2016). Furthermore, minimizing the loss function of cross-entropy (CE) (Goodfellow et al., 2016) can have the same effect as minimizing the KL divergence term. Minimizing CE loss in classification problems is equivalent to minimizing mean squared error (MSE) loss in regression problems (Hung et al., 2020; Kline & Berardi, 2005). Therefore, we use the MSE loss function and the root mean squared propagation (RMSProp) optimizer with a learning rate of 0.0002 to train SYMHnet. Table 5 summarizes the hyperparameters and their values used by SYMHnet. We use ˆ q ( W ) to represent the optimized weight distribution. During testing/prediction, SYMHnet uses the MC dropout sampling method (Gal & Ghahramani, 2016) to quantify uncertainty. Specifically, we process the test data K times to generate K MC samples where K is set to 100. We have experimented with different K values. Using a K value of less than 100 does not generate enough samples; the produced uncertainty ranges are too large to be useful. Using a K value of larger than 100 increases computation time, while the model performance remains the same. As a consequence, we set K to 100 to process the test data 100 times. Each time, a set of weights is randomly drawn from ˆ q ( W ). We obtain the mean and variance for the K samples. The mean is the anticipated SYM-H value. According to Jiang et al. (2021), we split the variance into aleatoric and epistemic uncertainties.", "pages": [ 10 ] }, { "title": "4.1 Performance Metrics", "content": "To assess the prediction accuracy of SYMHnet and compare it with related machine learning models, we adopt the following metrics: root mean square error (RMSE), forecast skill score (FSS) and R-squared (R 2 ). These metrics have been used in the forecasting of geomagnetic indices and are recommended in the literature (Camporeale, 2019; Iong et al., 2022; Liemohn et al., 2018). Our experiments were carried out by feeding time series data samples from the training storms in Table 1 (training set) to train a model. We then used the time series data samples from the validation storms in Table 2 (validation set) to validate the model and optimize its hyperparameters. Finally, we used the trained model to predict the SYM-H index values of the time series data samples from the test storms in Table 3 (test set). RMSE measures the difference between prediction and ground truth for each test data sample. It is calculated as follows: where n is the number of test data samples in a test storm in Table 3, and ˆ y i ( y i , respectively) represents the predicted SYM-H index value (observed SYM-H index value, respectively) at the time point i in the test storm. The smaller the RMSE, the more accurate the model. FSS is calculated using the prediction provided by the Burton equation (O'Brien & McPherron, 2000a) as a baseline and is defined as follows (Iong et al., 2022; Murphy, 1988): where y b i represents the prediction provided by the Burton equation at the time point i in the test storm. The FSS value between 0 and 1 indicates that the model is better than the baseline, while the negative FSS value indicates that the model is worse than the baseline (Iong et al., 2022). R 2 determines the amount of variance of the observed data explained by the predicted data. It is calculated as follows: where ¯ y is the mean of the observed SYM-H index values for the test data samples in the test storm. The larger the R 2 , the more accurate the model. For each metric, the mean and standard deviation of the metric values for all test storms in the test set (Table 3) are calculated and recorded.", "pages": [ 11 ] }, { "title": "4.2 Results Based on 1-Minute Resolution Data", "content": "In this section, we present experimental results based on the 1-minute resolution data in our database. First, we conducted an ablation study to analyze and assess the components of SYMHnet. Then we performed case studies on a moderately large storm (storm #36 with SYM-H = -137 nT) and a very large storm (storm #37 with SYMH = -393 nT) in the test set shown in Table 3 where both storms were previously investigated by Iong et al. (2022). It should be noted that the work of Iong et al. (2022) was based on 5-minute resolution data. To our knowledge, no previous method used 1minute resolution data to predict the SYM-H index.", "pages": [ 11 ] }, { "title": "4.2.1 Ablation Study with 1-Minute Resolution Data", "content": "We considered three variants of SYMHnet: SYMHnet-B, SYMHnet-G and SYMHnetBG. SYMHnet-B represents the subnetwork of SYMHnet with the BiLSTM component removed. SYMHnet-G represents the subnetwork of SYMHnet with the GNN component removed. SYMHnet-BG represents the subnetwork of SYMHnet with both the BiLSTM and GNN components removed. Thus, SYMHnet-BG simply contains the dense layers in SYMHnet, which amounts to a simple multilayer perceptron network. When conducting the ablation study, we turned off the uncertainty quantification mechanism. Table 6 presents the average values for RMSE, FSS, and R 2 (with standard deviations enclosed in parentheses) obtained by the four models: SYMHnet, SYMHnet-B, SYMHnet-G and SYMHnet-BG, based on the 1-minute resolution data in our database. The best metric values are highlighted in boldface. It can be seen from Table 6 that SYMHnet outperforms its three variants. SYMHnet-B is the second best among the four models, implying that a GNN is effective in solving time series regression problems (Bloemheuvel et al., 2022). SYHMnet-G, which contains a BiLSTM network but no GNN, does not perform well. This finding is consistent with those in Collado-Villaverde et al. (2021), who showed that LSTM performed worse than a combination of LSTM and CNN in SYMH forecasting. Finally, SYMHnet-BG is the worst among the four models. This happens because SYMHnet-BG loses the advantages offered by GNN and BiLSTM networks.", "pages": [ 12 ] }, { "title": "4.2.2 Case Studies with 1-Minute Resolution Data", "content": "Here we conducted case studies by using SYMHnet to predict the SYM-H index values in storms #36 and #37 given in Table 3 based on the 1-minute resolution data in our database. Additional case studies on other storms can be found in Appendix A. The period of storm #36 started on 18 January 2004 and ended on 27 January 2004, with a minimum SYM-H value of -137 nT and a maximum SYM-H value of 41 nT during the period. The period of storm #37 started on 4 November 2004 and ended on 14 November 2004, with a minimum SYM-H value of -393 nT and a maximum SYM-H value of 92 nT during the period. Figure 3 shows the predictions and measured error of the SYMHnet model in storm #36 and storm #37 respectively. In the figure, each point on a yellow dashed line represents the prediction made at the corresponding time x on the Xaxis. For 1-hour ahead (2-hour ahead, respectively) predictions, the point/prediction at time x is produced based on the solar wind/IMF parameters at time x - 1 hour ( x - 2 hours, respectively). There is a lag of 1 hour (for 1-hour ahead predictions) or 2 hours (for 2-hour ahead predictions) as in previous studies (Collado-Villaverde et al., 2021; Iong et al., 2022). It can be seen from Figure 3 that the SYMHnet model works well at both quiet time and storm time. The measured error ranges between -15 nT and 23 nT for storm #36 and between -50 nT and 34 nT for storm #37. The more intense the storm, the larger the measured error. 374 374 375 375 376 376 377 377 100 100 0 1h ahead prediction manuscript submitted to Space Weather 1h ahead prediction 2h ahead prediction Figure 3. Predictions for storm #36 (top) and storm #37 (bottom) made by the SYMHnet 0 Observed SYM-H Predicted SYM-H Epistemic Uncertainty Aleatoric Uncertainty is 5-minute rather than 1-minute. Furthermore, the label of the parameter graph G t , G t +5 , G t +10 is the SYM-H index value at time point t + w hour, ( t + 5 minutes) + w hour, ( t + 10 minutes) + w hour, respectively, for w -hour ahead predictions where w is 1 or 2. is 5-minute rather than 1-minute. Furthermore, the label of the parameter graph G t , G t +5 , G t +10 is the SYM-H index value at time point t + w hour, ( t + 5 minutes) + w hour, ( t + 10 minutes) + w hour, respectively, for w -hour ahead predictions where w is 1 or 2. Figure 4 presents uncertainty quantification results produced by SYMHnet in storm #36 and storm #37, respectively, based on the 1-minute resolution data in our database. In the figure, the red line represents the observed values of the SYM-H index, and the yellow dashed line represents the predicted values of the SYM-H index. The light-blue region shows the epistemic uncertainty (model uncertainty) and the light-gray region shows the aleatoric uncertainty (data uncertainty) of the predicted outcome. It can be seen in Figure 4 that the yellow dashed line (predicted values) is reasonably close to the red line (observed values), again demonstrating the good performance of SYMHnet. The lightblue region is tinier than the light-gray region, indicating that the model uncertainty is lower than the data uncertainty. This is due to the fact that the uncertainty in the predicted outcome is primarily caused by the noise in the input test data, not by the SYMHnet model.", "pages": [ 12, 13, 14 ] }, { "title": "4.3 Results Based on 5-Minute Resolution Data", "content": "SYMHnet can be easily modified to process 5-minute resolution data. As described in Section 3.2, the input data sample at the time point t is composed of the parameter graph G t and a sequence of m records. The difference is that the cadence of the m records here is 5-minute rather than 1-minute. Furthermore, the labels of the parameter graphs G t , G t +5 , G t +10 are the SYM-H index values at the time points t + w hour, ( t + 5 minutes) + w hour, ( t + 10 minutes) + w hour, respectively, for w -hour ahead predictions where w is 1 or 2. In the following, we present experimental results based on the 5-minute resolution data in our database. As in Section 4.2, we conducted an ablation study, this time using the 5-minute resolution data. We then performed case studies on storms #36 and #37. Finally, we compared SYMHnet with related machine learning methods, all of which utilized the 5-minute resolution data in our database. Since the related methods cannot quantify uncertainty, we turned off the uncertainty quantification mechanism in SYMHnet while conducting the comparative study.", "pages": [ 14 ] }, { "title": "4.3.1 Ablation Study with 5-Minute Resolution Data", "content": "Table 7 presents the average values for RMSE, FSS and R 2 (with standard deviations enclosed in parentheses) obtained by the four models: SYMHnet, SYMHnet-B, SYMHnet-G and SYMHnet-BG, based on the 5-minute resolution data in our database. The best metric values are highlighted in boldface. It can be seen from Table 7 that SYMHnet is again the best among the four models for the 5-minute resolution data, a finding consistent with that in Table 6 for the 1-minute resolution data.", "pages": [ 14 ] }, { "title": "4.3.2 Case Studies with 5-Minute Resolution Data", "content": "Figure 5 shows the predictions and measured error of SYMHnet in storms #36 and #37, respectively, and Figure 6 presents the uncertainty quantification results produced by SYMHnet in these storms respectively, based on the 5-minute resolution data in our database. Unlike Figures 3 and 4, in which both quiet time and storm time are shown, 412 412 413 413 414 414 415 415 416 416 417 417 100 100 0 1h ahead prediction 0 1h ahead prediction manuscript submitted to Space Weather 1h ahead prediction 2h ahead prediction 0 Observed SYM-H Predicted SYM-H Epistemic Uncertainty Aleatoric Uncertainty ory (LSTM) and a convolutional neural network (CNN), referred to as the LCNN method, to forecast the SYM-H index (1 and 2 hours in advance). Iong et al. (2022) utilized gradient boosting machines, referred to as the GBM method, to forecast the SYM-H index ory (LSTM) and a convolutional neural network (CNN), referred to as the LCNN method, to forecast the SYM-H index (1 and 2 hours in advance). Iong et al. (2022) utilized gradient boosting machines, referred to as the GBM method, to forecast the SYM-H index (also 1 and 2 hours in advance). Siciliano et al. (2021) compared LSTM and CNN for Figures 5 and 6 focus on the peak storm time. In Figure 5, the measured error ranges between -24 nT and 25 nT for storm #36 and between -52 nT and 36 nT for storm #37. These results indicate that SYMHnet can properly forecast the SYM-H index even in the most intense storm period. the prediction of the SYM-H index (only 1 hour in advance). Since the related methods cannot predict uncertainties, we turned off the uncertainty quantification compothe prediction of the SYM-H index (only 1 hour in advance). Since the related methods cannot predict uncertainties, we turned off the uncertainty quantification compoIn Figure 6, the red line represents the observed values of the SYM-H and the yellow dashed line represents the predicted values of the SYM-H. The light-blue area shows (also 1 and 2 hours in advance). Siciliano et al. (2021) compared LSTM and CNN for 0 the epistemic uncertainty (model uncertainty) and the light-gray area shows the aleatoric uncertainty (data uncertainty) of the predicted outcome. It can be seen from Figure 6 that the red line representing the observed SYM-H values is within the uncertainty interval, indicating SYMHnet's predicted values together with the uncertainty values well cover the observed values. The overall findings here are similar to those from the 1-minute resolution data shown in Figure 4.", "pages": [ 14, 15, 16 ] }, { "title": "4.3.3 Comparative Study with 5-Minute Resolution Data", "content": "Several researchers performed SYM-H forecasting using machine learning and the 5-minute resolution data. Collado-Villaverde et al. (2021) combined long short-term memory (LSTM) and a convolutional neural network (CNN), referred to as the LCNN method, to forecast the SYM-H index (1 and 2 hours in advance). Iong et al. (2022) utilized gradient boosting machines, referred to as the GBM method, to forecast the SYM-H index (also 1 and 2 hours in advance). Siciliano et al. (2021) compared LSTM and CNN for the prediction of the SYM-H index (only 1 hour in advance). Although the methods including ours use slightly different data samples, these methods are all developed to predict the SYM-H index values in the same set of storms. The purpose of this comparative study is to compare the prediction results/accuracies of, rather than specific models/data samples in, these methods. This comparison methodology has commonly been used in SYM-H forecasting (Collado-Villaverde et al., 2021; Iong et al., 2022; Siciliano et al., 2021). Since the related methods cannot predict uncertainties, we turned off the uncertainty quantification component in SYMHnet while carrying out the comparative study. The Burton equation (O'Brien & McPherron, 2000a), used as the baseline, is also included. The performance metric values of each method for each test storm in the test set (Table 3) are calculated. The best metric values are highlighted in boldface. Tables 8 and 9 compare the RMSE results of these methods for 1-hour and 2-hour ahead SYM-H predictions, respectively, based on the RMSE values available in the related studies (Collado-Villaverde et al., 2021; Iong et al., 2022; O'Brien & McPherron, 2000a; Siciliano et al., 2021). Tables 10 and 11 compare the FSS results of these methods for 1-hour and 2-hour ahead SYM-H predictions, respectively, based on the FSS values available in the related studies (Collado-Villaverde et al., 2021; Iong et al., 2022; Siciliano et al., 2021). Table 12 compares the R 2 results of these methods for 1-hour ahead and 2-hour ahead SYM-H predictions, respectively, on the same test storms. Iong et al. (2022) did not provide R 2 results, and hence the GBM method was excluded from Table 12. These tables show that SYMHnet performs better than the related methods for all except two test storms (#28 and/or #40), demonstrating the good performance and feasibility of our tool for SYM-H forecasting.", "pages": [ 16 ] }, { "title": "5 Discussion and Conclusion", "content": "Geomagnetic activities have a significant impact on Earth, which can cause damages to spacecraft, electrical power grids, and navigation systems. Geospace scientists use geomagnetic indices to measure and quantify the geomagnetic activities. The SYMH index provides information about the response and behavior of the Earth's magnetosphere during geomagnetic storms. Therefore, a lot of effort has been put into SYMH forecasting. Previous work mainly focused on 5-minute resolution data and skipped 1-minute resolution data. The higher temporal resolution of the 1-minute resolution data poses a more difficult challenge to forecast due to its highly oscillating character. This oscillating behavior could make the data more noisy to a machine learning model. As a consequence, the model requires more iterations during training with a larger number of neurons in order to learn more features and patterns hidden in the data. In our study, the SYMHnet model architectures for processing the 1-minute resolution data and 5-minute resolution data are the same, as shown in Figure 2. The con-", "pages": [ 16 ] }, { "title": "2-h ahead prediction (FSS)", "content": "figuration details and hyperparameter values of SYMHnet for processing the 5-minute resolution data are shown in Tables 4 and 5. When processing the 1-minute resolution data, the model is configured with a larger number of neurons in the dense layers, a higher percentage in the dropout layers, and a larger number of epochs during the training phase. This configuration is designed to combat the highly oscillating behavior of the 1-minute resolution data. Results from our experiments demonstrated the good performance of SYMHnet at both quiet time and storm time. These results were obtained from a database of 42 storms that occurred between 1998 and 2018 during the past two solar cycles (#23 and #24). As done in previous studies (Collado-Villaverde et al., 2021; Iong et al., 2022; Siciliano et al., 2021), 20 storms, listed in Table 1, were used for training, 5 storms, listed in Table 2, were used for validation, and 17 storms, listed in Table 3, were used for testing. Based on the tables, the 42 storms were distributed to 14 distinct years. To avoid bias in drawing a conclusion from the above experiments, we conducted an additional experiment using 14-fold cross validation where the data was divided into 14 partitions or folds. Each fold corresponds to one year in which at least one storm occurred. The sequential order of the data in each fold was maintained. In each run, one fold was used for testing and the other 13 folds together were used for training. Thus, the training set and test set are disjoint, and the trained model can predict unseen SYMH values in the test set. There were 14 folds and consequently 14 runs where the average performance metric values over the 14 runs were calculated. The results of the 14fold cross validation were consistent with those reported in the paper. These results indicate that the SYMHnet tool can be used to predict future SYM-H index values without knowing whether a storm is going to start. When the predicted SYM-H value is less than a threshold (e.g., -30 nT), the tool detects the occurrence of a storm. Thus, we conclude that the proposed SYMHnet is a viable machine learning method for short-term, 1 or 2-hour ahead forecasts of the SYM-H index for both 1- and 5-minute resolution data.", "pages": [ 19 ] }, { "title": "Acknowledgments", "content": "We appreciate the editor and anonymous referees for constructive comments and suggestions. We acknowledge the use of NASA/GSFC's Space Physics Data Facility's OMNIWeb and CDAWeb services, and OMNI data. This work was supported in part by U.S. NSF grants AGS-1927578, AGS-1954737, AGS-2149748, AGS-2228996, AGS-2300341 and OAC-2320147. Huseyin Cavus was supported by the Fulbright Visiting Scholar Program of the Turkish Fulbright Commission.", "pages": [ 20 ] }, { "title": "Appendix A Additional Case Studies with 1-Minute Resolution Data", "content": "Figure A1 shows the predictions and measured error of SYMHnet in storms #28, #31, #33, #40, and #42, respectively, and Figure A2 presents the uncertainty quantification results produced by SYMHnet in these storms, respectively, based on the 1minute resolution data in our database. The period of storm #28 started on 9 January 1999 and ended on 18 January 1999, with a minimum SYM-H value of -111 nT and a maximum SYM-H value of 9 nT. The period of storm #31 started on 2 April 2000 and ended on 12 April 2000, with a minimum SYM-H value of -315 nT and a maximum SYMH value of 16 nT. The period of storm #33 stared on 26 March 2001 and ended on 4 April 2001, with a minimum SYM-H value of -434 nT and a maximum SYM-H value of 109 nT. The period of storm #40 started on 26 June 2013 and ended on 4 July 2013, with a minimum SYM-H value of -110 nT and a maximum SYM-H value of 19 nT. The period of storm #42 started on 22 August 2018 and ended on 3 September 2018, with a minimum SYM-H value of -205 nT and a maximum SYM-H value of 26 nT. In Figure A1, the measured error ranges between -46 nT and 7 nT for storm #28, between -58 nT and 2 nT for storm #31, between -69 nT and 32 nT for storm #33, between -12 nT and 4 nT for storm #40, and between -26 nT and 7 nT for storm #42. Generally, the more intense the storm, the larger the measured error. In Figure A2, we see that SYMHnet's predicted values together with the uncertainty values well cover the observed values, a finding consistent with that in Figure 4.", "pages": [ 21 ] }, { "title": "References", "content": "Abduallah, Y., Jordanova, V. K., Liu, H., Li, Q., Wang, J. T. L., & Wang, H. (2022). Predicting solar energetic particles using SDO/HMI vector magnetic data products and a bidirectional LSTM network. The Astrophysical Journal Supplement Series , 260 (1), 16. https://doi.org/10.3847/1538-4365/ ac5f56 Abduallah, Y., Wang, J. T. L., Nie, Y., Liu, C., & Wang, H. (2021). DeepSun: Machine-learning-as-a-service for solar flare prediction. Research in Astronomy and Astrophysics , 21 (7), 160. https://doi.org/10.1088/1674-4527/21/7/ 160 Alobaid, K. A., Abduallah, Y., Wang, J. T. L., Wang, H., Jiang, H., Xu, Y., . . . Jing, J. (2022). Predicting CME arrival time through data integration and ensemble learning. Frontiers in Astronomy and Space Sciences , 9 , 1013345. https://doi.org/10.3389/fspas.2022.1013345 Amata, E., Pallocchia, G., Consolini, G., Marcucci, M. F., & Bertello, I. (2008). Comparison between three algorithms for Dst predictions over the 2003-2005 period. Journal of Atmospheric and Solar-Terrestrial Physics , 70 (2-4), 496502. https://doi.org/10.1016/j.jastp.2007.08.041 Ayala Solares, J. R., Wei, H.-L., Boynton, R. J., Walker, S. N., & Billings, S. A. (2016). Modeling and prediction of global magnetic disturbance in nearEarth space: A case study for Kp index using NARX models. Space Weather , 14 (10), 899-916. https://doi.org/10.1002/2016SW001463 Bala, R., & Reiff, P. (2012). Improvements in short-term forecasting of geomagnetic activity. Space Weather , 10 (6). https://doi.org/10.1029/2012SW000779 Bhaskar, A., & Vichare, G. (2019). Forecasting of SYMH and ASYH indices for geomagnetic storms of solar cycle 24 including St. Patrick's day, 2015 storm using NARX neural network. Journal of Space Weather and Space Climate , 9 , A12. https://doi.org/10.1051/swsc/2019007 Bloemheuvel, S., van den Hoogen, J., Jozinovi'c, D., Michelini, A., & Atzmueller, M. (2022). Graph neural networks for multivariate time series regression with application to seismic data. International Journal of Data Science and Analytics . https://doi.org/10.1007/s41060-022-00349-6 Burton, R. K., McPherron, R. L., & Russell, C. T. (1975). An empirical relationship between interplanetary conditions and Dst. Journal of Geophysical Research (1896-1977) , 80 (31), 4204-4214. https://doi.org/10.1029/ JA080i031p04204 Cai, L., Ma, S. Y., & Zhou, Y. L. (2010). Prediction of SYM-H index during large storms by NARX neural network from IMF and solar wind data. Annales Geophysicae , 28 (2), 381-393. https://doi.org/10.5194/angeo-28-381-2010 Camporeale, E. (2019). The challenge of machine learning in space weather: Nowcasting and forecasting. Space Weather , 17 (8), 1166-1207. https://doi.org/ 10.1029/2018SW002061 Carter, B. A., Yizengaw, E., Pradipta, R., Weygand, J. M., Piersanti, M., Pulkkinen, A., . . . Zhang, K. (2016). Geomagnetically induced currents around the world during the 17 March 2015 storm. Journal of Geophysical Research: Space Physics , 121 (10), 10,496-10,507. https://doi.org/10.1002/ 2016JA023344 Chandorkar, M., Camporeale, E., & Wing, S. (2017). Probabilistic forecasting of the disturbance storm time index: An autoregressive Gaussian process approach. Space Weather , 15 (8), 1004-1019. https://doi.org/10.1002/2017SW001627 Chen, Y., Manchester, W. B., Hero, A. O., Toth, G., DuFumier, B., Zhou, T., . . . Gombosi, T. I. (2019). Identifying solar flare precursors using time series of SDO/HMI images and SHARP parameters. Space Weather , 17 (10), 1404-1426. https://doi.org/10.1029/2019SW002214 Collado-Villaverde, A., Mu˜noz, P., & Cid, C. (2021). Deep neural networks with convolutional and LSTM layers for SYM-H and ASY-H forecasting. Space Weather , 19 (6), e02748. https://doi.org/10.1029/2021SW002748 Consolini, G., & Chang, T. S. (2001). Magnetic field topology and criticality in geotail dynamics: Relevance to substorm phenomena. Space Science Reviews , 95 , 309-321. https://doi.org/10.1023/A:1005252807049 Denker, J. S., & LeCun, Y. (1990). Transforming neural-net output levels to probability distributions. In Proceedings of the 3rd International Conference on Neural Information Processing Systems (p. 853-859). Denton, M. H., Henderson, M. G., Jordanova, V. K., Thomsen, M. F., Borovsky, Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In Proceedings of the 33rd International Conference on Machine Learning (pp. 1050-1059). https://doi.org/10 .5555/3045390.3045502 Gaunt, C. T., & Coetzee, G. (2007). Transformer failures in regions incorrectly considered to have low GIC-risk. In 2007 IEEE Lausanne Power Tech (p. 807812). https://doi.org/10.1109/PCT.2007.4538419 Gleisner, H., Lundstedt, H., & Wintoft, P. (1996). Predicting geomagnetic storms from solar-wind data using time-delay neural networks. Annales Geophysicae , 14 , 679-686. Goodfellow, I. J., Bengio, Y., & Courville, A. C. (2016). Deep Learning . MIT Press. Graves, A. (2011). Practical variational inference for neural networks. In J. ShaweTaylor, R. Zemel, P. Bartlett, F. Pereira, & K. Q. Weinberger (Eds.), Advances in Neural Information Processing Systems (Vol. 24). Curran Associates, Inc. Gruet, M. A., Chandorkar, M., Sicard, A., & Camporeale, E. (2018). Multiplehour-ahead forecast of the Dst index using a combination of long short-term memory neural network and Gaussian process. Space Weather , 16 (11), 18821896. https://doi.org/10.1029/2018SW001898 Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural Computation , 9 , 1735-80. https://doi.org/10.1162/neco.1997.9.8.1735 Huang, X., Wang, H., Xu, L., Liu, J., Li, R., & Dai, X. (2018). Deep learning based solar flare forecasting model. I. Results for line-of-sight magnetograms. The Astrophysical Journal , 856 (1), 7. https://doi.org/10.3847/1538-4357/ aaae00 Hung, C.-C., Chen, Y.-J., Guo, S. J., & Hsu, F.-C. (2020). Predicting the price movement from candlestick charts: a CNN-based approach. International Journal of Ad Hoc and Ubiquitous Computing , 34 (2), 111-120. https://doi.org/10.1504/IJAHUC.2020.107821 Iong, D., Chen, Y., Toth, G., Zou, S., Pulkkinen, T., Ren, J., . . . Gombosi, T. (2022). New findings from explainable SYM-H forecasting using gradient boosting machines. Space Weather , 20 (8), e2021SW002928. https:// doi.org/10.1029/2021SW002928 Jiang, H., Jing, J., Wang, J., Liu, C., Li, Q., Xu, Y., . . . Wang, H. (2021). Tracing H α fibrils through Bayesian deep learning. The Astrophysical Journal Supplement Series , 256 (1), 20. https://doi.org/10.3847/1538-4365/ac14b7 Jordanova, V. K., Ilie, R., & Chen, M. W. (2020). Ring Current Investigations: The Quest for Space Weather Prediction . Elsevier. https://doi.org/10.1016/ C2017-0-03448-1 Kendall, A., & Gal, Y. (2017). What uncertainties do we need in Bayesian deep learning for computer vision? In I. Guyon et al. (Eds.), Advances in Neural Information Processing Systems (Vol. 30). Curran Associates, Inc. King, J. H., & Papitashvili, N. E. (2005). Solar wind spatial scales in and com- parisons of hourly Wind and ACE plasma and magnetic field data. Journal of Geophysical Research: Space Physics , 110 (A2). https://doi.org/10.1029/ 2004JA010649 Klimas, A. J., Vassiliadis, D., Baker, D. N., & Roberts, D. A. (1996). The organized nonlinear dynamics of the magnetosphere. Journal of Geophysical Research: Space Physics , 101 (A6), 13089-13113. https://doi.org/10.1029/ 96JA00563 Kline, M., & Berardi, L. (2005). Revisiting squared-error and cross-entropy functions for training neural network classifiers. Neural Comput. Appl. , 14 (4), 310-318. https://doi.org/10.1007/s00521-005-0467-y Laurenza, M., Cliver, E. W., Hewitt, J., Storini, M., Ling, A. G., Balch, C. C., & Kaiser, M. L. (2009). A technique for short-term warning of solar energetic particle events based on flare location, flare size, and evidence of particle escape. Space Weather , 7 (4), S04008. https://doi.org/10.1029/ 2007SW000379 Lavasa, E., Giannopoulos, G., Papaioannou, A., Anastasiadis, A., Daglis, I. A., Aran, A., . . . Sanahuja, B. (2021). Assessing the predictability of solar energetic particles with the use of machine learning techniques. Solar Physics , 296 (7), 107. https://doi.org/10.1007/s11207-021-01837-x Lazz'us, J. A., Vega, P., Rojas, P., & Salfate, I. (2017). Forecasting the Dst index using a swarm-optimized neural network. Space Weather , 15 (8), 1068-1089. https://doi.org/10.1002/2017SW001608 Liemohn, M. W., McCollough, J. P., Jordanova, V. K., Ngwira, C. M., Mor- ley, S. K., Cid, C., . . . Vasile, R. (2018). Model evaluation guidelines for geomagnetic index predictions. Space Weather , 16 (12), 2079-2102. https://doi.org/10.1029/2018SW002067 Liu, H., Liu, C., Wang, J. T. L., & Wang, H. (2019). Predicting solar flares using a long short-term memory network. The Astrophysical Journal , 877 (2), 121. https://doi.org/10.3847/1538-4357/ab1b3c Liu, H., Liu, C., Wang, J. T. L., & Wang, H. (2020). Predicting coronal mass ejections using SDO/HMI vector magnetic data products and recurrent neural networks. The Astrophysical Journal , 890 (1), 12. https://doi.org/10.3847/ 1538-4357/ab6850 Lu, J., Peng, Y., Wang, M., Gu, S., & Zhao, M. (2016). Support vector machine combined with distance correlation learning for Dst forecasting during intense geomagnetic storms. Planetary and Space Science , 120 , 48-55. https://doi.org/10.1016/j.pss.2015.11.004 Mayaud, P. N. (1980). What is a geomagnetic index? In Derivation, Meaning, and Use of Geomagnetic Indices (p. 2-4). American Geophysical Union (AGU). https://doi.org/10.1002/9781118663837.ch2 Moldwin, M. B., & Tsu, J. S. (2016). Stormtime equatorial electrojet groundinduced currents. In Ionospheric Space Weather (p. 33-40). American Geophysical Union (AGU). https://doi.org/10.1002/9781118929216.ch3 Murphy, A. H. (1988). Skill scores based on the mean square error and their relationships to the correlation coefficient. Monthly Weather Review , 116 (12), 2417. https://doi.org/10.1175/1520-0493(1988)116<2417: SSBOTM>2.0.CO;2 Newell, P. T., Sotirelis, T., Liou, K., Meng, C.-I., & Rich, F. J. (2007). A nearly universal solar wind-magnetosphere coupling function inferred from 10 magnetospheric state variables. Journal of Geophysical Research: Space Physics , 112 (A1). https://doi.org/10.1029/2006JA012015 Nuraeni, F., Ruhimat, M., Aris, M. A., Ratnasari, E. A., & Purnomo, C. (2022). Development of 24 hours Dst index prediction from solar wind data and IMF Bz using NARX. Journal of Physics: Conference Series , 2214 (1), 012024. https://dx.doi.org/10.1088/1742-6596/2214/1/012024 N'u˜nez, M. (2011). Predicting solar energetic proton events (E > 10 MeV). Space Weather , 9 (7). https://doi.org/10.1029/2010SW000640 O'Brien, T. P., & McPherron, R. L. (2000a). An empirical phase space analysis of ring current dynamics: Solar wind control of injection and decay. Journal of Geophysical Research: Space Physics , 105 (A4), 7707-7719. https://doi.org/ 10.1029/1998JA000437 Pallocchia, G., Amata, E., Consolini, G., Marcucci, M. F., & Bertello, I. (2006). Geomagnetic D st index forecast based on IMF data only. Annales Geophysicae , 24 (3), 989-999. https://doi.org/10.5194/angeo-24-989-2006 Panagopoulos, G., Nikolentzos, G., & Vazirgiannis, M. (2021). Transfer graph neural networks for pandemic forecasting. In Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence (pp. 4838-4845). https://doi.org/10.1609/aaai.v35i6.16616 Rangarajan, G. K. (1989). Indices of geomagnetic activity. Geomatik , 3 , 323-384. Rastatter, L., Kuznetsova, M. M., Glocer, A., Welling, D., Meng, X., Raeder, J., . . . Gannon, J. (2013). Geospace environment modeling 2008-2009 challenge: D st index. Space Weather , 11 (4), 187-205. https://doi.org/10.1002/ swe.20036 Siami-Namini, S., Tavakoli, N., & Namin, A. S. (2019). The performance of LSTM and BiLSTM in forecasting time series. In IEEE International Conference on Big Data (p. 3285-3292). https://doi.org/10.1109/ BigData47090.2019.9005997 Siciliano, F., Consolini, G., Tozzi, R., Gentili, M., Giannattasio, F., & De Michelis, Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research , 15 (56), 1929-1958. http://jmlr.org/papers/v15/srivastava14a.html Stumpo, M., Benella, S., Laurenza, M., Alberti, T., Consolini, G., & Marcucci, M. F. (2021). Open issues in statistical forecasting of solar proton events: A machine learning perspective. Space Weather , 19 (10), e2021SW002794. https://doi.org/10.1029/2021SW002794 Temerin, M., & Li, X. (2002). A new model for the prediction of Dst on the basis of the solar wind. Journal of Geophysical Research: Space Physics , 107 (A12), SMP 31-1-SMP 31-8. https://doi.org/10.1029/2001JA007532 Tozzi, R., De Michelis, P., Coco, I., & Giannattasio, F. (2019). A preliminary risk assessment of geomagnetically induced currents over the Italian territory. Space Weather , 17 (1), 46-58. https://doi.org/10.1029/2018SW002065 Vichare, G., Thomas, N., Shiokawa, K., Bhaskar, A., & Sinha, A. K. (2019). Spatial gradients in geomagnetic storm time currents observed by Swarm multispacecraft mission. Journal of Geophysical Research (Space Physics) , 124 (2), 982-995. https://doi.org/10.1029/2018JA025692 Viljanen, A., Pirjola, R., Pr'acser, E., Katkalov, J., & Wik, M. (2014). Geomagnetically induced currents in Europe - Modelled occurrence in a continent-wide power grid. J. Space Weather Space Clim. , 4 , A09. https://doi.org/ 10.1051/swsc/2014006 Volodina, V., & Challenor, P. (2021). The importance of uncertainty quantification in model reproducibility. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 379 (2197), 20200071. https://doi.org/10.1098/rsta.2020.0071 Wang, C. B., Chao, J. K., & Lin, C.-H. (2003). Influence of the solar wind dynamic pressure on the decay and injection of the ring current. Journal of Geophysical Research: Space Physics , 108 (A9). https://doi.org/10.1029/", "pages": [ 24, 25, 26, 27 ] }, { "title": "2003JA009851", "content": "Wanliss, J. A., & Showalter, K. M. (2006). High-resolution global storm index: Dst versus SYM-H. Journal of Geophysical Research (Space Physics) , 111 (A2), A02202. https://doi.org/10.1029/2005JA011034 Woodroffe, J. R., Morley, S. K., Jordanova, V. K., Henderson, M. G., Cowee, M. M., & Gjerloev, J. G. (2016). The latitudinal variation of geoelectromagnetic disturbances during large (Dst ≤ -100 nT) geomagnetic storms. Space Weather , 14 (9), 668-681. https://doi.org/10.1002/2016SW001376 Xu, W., Zhu, Y., Zhu, L., Lu, J., Wei, G., Wang, M., & Peng, Y. (2023). A class of Bayesian machine learning model for forecasting Dst during intense geomagnetic storms. Advances in Space Research , 72 (9), 3882-3889. https://doi.org/10.1016/j.asr.2023.07.009 Yurchyshyn, V., Wang, H., & Abramenko, V. (2004). Correlation between speeds of coronal mass ejections and the intensity of geomagnetic storms. Space Weather , 2 (2), S02001. https://doi.org/10.1029/2003SW000020", "pages": [ 28 ] } ]

[ { "title": "Mark Taylor", "content": "H. H. Wills Physics Laboratory, Tyndall Avenue, University of Bristol, UK; [email protected] Abstract. TOPCAT is a desktop GUI tool for working with tabular data such as source catalogues. Among other capabilities it provides a rich set of visualisation options suitable for interactive exploration of large datasets. The latest release introduces a Corner Plot window which displays a grid of linked scatter-plot-like and histogram-like plots for all pair and single combinations from a supplied list of coordinates.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "TOPCAT 1 (Taylor 2005) is an established desktop GUI application for working with tabular data. It has many capabilities including manipulation of data and metadata, Virtual Observatory access, and crossmatching, but one of its particular strengths is to enable interactive exploratory analysis of large tables. Such tables may have millions of rows and hundreds of columns. Exploration of such bulky and high-dimensional data is facilitated in TOPCAT by provision of a number of highly configurable and responsive interactive visualisation windows, focussed on presentation of point clouds over a wide range of densities. Features include scatter plots in 1, 2 and 3 dimensions, the ability to label points with shapes, vectors, error bars and text, extensive options for colouring points by density or additional coordinates, and the ability to plot arbitrary combinations of table columns using a powerful expression language. The most recent release, v4.9, introduces a new 'Corner' plot window, which presents a grid of scatter-plot-like and histogram-like plots for all pair and single combinations of a supplied list of coordinates. This provides a simultaneous view of all the 2-d and 1-d projections of a high-dimensional dataset, which can be a valuable aid for identifying interesting features in complex data. An example is shown in Figure 1. This plot type is also available as the command plot2corner in STILTS (Taylor 2006), the command-line counterpart of TOPCAT. This type of visualisation is not novel; it has been used since the 1980s (Cleveland 1993) under the names 'Scatter Plot Matrix' , 'SPLOM' , 'Pairs Plot' and 'Corner Plot' , and graphics packages producing them exist for instance in Python, R and IDL. Its adoption here benefits from the performance, interactive features and ease of use provided by the existing application GUI, and gives TOPCAT users a new tool for exploring high-dimensional data.", "pages": [ 1 ] }, { "title": "2. Features", "content": "The Corner Plot shares its features with the other TOPCAT visualisation windows: Ease of use: coordinates can be added, deleted and reordered with a few mouse clicks Interactivity: each panel can be panned and zoomed in one or two dimensions easily using the mouse, and the other panels in the same row / column adjust to keep the axes aligned Linked Views: graphical selections and point highlighting on one panel are immediately visible in the other panels (and in other windows) Overplotting: multiple datasets, selections and plot types can appear in each panel Calculations: coordinates can be existing columns or algebraic expressions based on them Scalability: multipanel displays for millions of rows are quite responsive even on modest hardware Flexibility: many plotting and shading options are available alongside simple scatter plots and histograms: transparency, weighted density maps, contours, KDEs, text labels, etc Configurability: interactive controls can specify lower / upper / full panel matrix, logarithmic or inverted axes, axis labels, grid line drawing, colour map adjustments, etc The linked views, especially in combination with interactive graphical region selection (blob drawing), make it very easy to identify regions, populations or points that are apparent in one coordinate plane and see where they appear in the other displayed planes. To facilitate setting up plots with many panels, an extra 'Fill' control is provided which allows easy selection of coordinates, including an option to use all pair di ff erences (or ratios) from a selection of columns, for instance all pairwise colours from a list of magnitudes (or fluxes)", "pages": [ 3 ] }, { "title": "3. Implementation", "content": "Assembling multiple plot panels in this way is not conceptually di ffi cult, and the TOPCAT codebase already provided most of the data handling infrastructure required, as well as some support for multi-panel plots. However, implementing a plot type with multiple panels derived from a single set of input coordinates broke some of the assumptions in the existing plotting framework. This meant that significant code restructuring was required, especially for the graphical and command-line user interfaces, so implementation was quite time-consuming.", "pages": [ 3 ] }, { "title": "4. Future Work", "content": "The new corner plot is working quite well, but not perfectly. In particular the axis labelling developed for other plot types can become overcrowded and imperfectly aligned. Some performance improvements could be made by concurrent plotting of multiple panels and eliminating some duplicated work. Usage in the field may come up with other bugs, complaints or suggestions. If this plot type proves popular, improvements may be made in future, in line with user feedback. Other possibilities for multi-panel plots may also be investigated. Acknowledgments. This work was funded by the UK's Science and Technology Facilities Council (STFC).", "pages": [ 4 ] }, { "title": "References", "content": "Clementini, G., et al. 2016, A&A, 595, A133. ASP Conf. Ser., 29 1609.04269 Cleveland, W. S. 1993, Visualizing Data (Summit, NJ: Hobart Press) Taylor, M. B. 2005, in ADASS XIV, edited by P. Shopbell, M. Britton, & R. Ebert, vol. 347 of -2006, in ADASS XV, edited by C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, vol. 351 of ASP Conf. Ser., 666", "pages": [ 4 ] } ]

[ { "title": "Bifurcation mechanism of quasihalo orbit from Lissajous Orbit", "content": "Mingpei Lin a, ∗ , Hayato Chiba a a Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "This paper presents a general analytical method to describe the center manifolds of collinear libration points in the Restricted Three-body Problem (RTBP). It is wellknown that these center manifolds include Lissajous orbits, halo orbits, and quasihalo orbits. Previous studies have traditionally tackled these orbits separately by iteratively constructing high-order series solutions using the Lindstedt-Poincar'e method. Instead of relying on resonance between their frequencies, this study identifies that halo and quasihalo orbits arise due to intricate coupling interactions between in-plane and out-of-plane motions. To characterize this coupling e ff ect, a novel concept, coupling coe ffi cient η , is introduced in the RTBP, incorporating the coupling term η ∆ x in the z -direction dynamics equation, where ∆ represents a formal power series concerning the amplitudes. Subsequently, a uniform series solution for these orbits is constructed up to a specified order using the LindstedtPoincar'e method. For any given paired in-plane and out-of-plane amplitudes, the coupling coe ffi cient η is determined by the bifurcation equation ∆ = 0. When η = 0, the proposed solution describes Lissajous orbits around libration points. As η transitions from zero to non-zero values, the solution describes quasihalo orbits, which bifurcate from Lissajous orbits. Particularly, halo orbits bifurcate from planar Lyapunov orbits if the out-of-plane amplitude is zero. The proposed method provides a unified framework for understanding these intricate orbital behaviors in the RTBP. Keywords: Restricted three-body problem, Center manifold, Lissajous orbit, Quasihalo orbit, Coupling coe ffi cient", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The Restricted Three-body Problem (RTBP) serves as a fundamental dynamical model to study the motion of asteroids or satellites under the gravitational influence of two primaries [1, 2, 3]. The RTBP has five equilibrium points called the Lagrange points Li ( i = 1, 2, . . . , 5), among which three so-called collinear libration points are unstable, and the remaining two are triangular libration points. The linear behavior of three collinear libration points is of the type center × center × saddle. The center manifold of three collinear libration points includes Lissajous orbits around libration points, halo orbits, and quasihalo orbits around halo orbits [4]. These libration point orbits play a key role in astronomy and many space science missions [5, 6, 7, 8]. Numerical methods (numerical integration, optimization technology) and analytical methods (parameterization method, norm form, Lindstedt-Poincar'e method) are often used to study the dynamics in the center manifolds of collinear libration points. In most cases, numerical methods still require accurate analytical initial values. On the other hand, high-order analytical solutions explicitly and accurately describe the dynamics around libration points, which is very useful for the preliminary space mission design. Farquhar initially proposed the third-order solution for small-amplitude Lissajous orbits and introduced the concept of 'halo orbit' [9, 10]. Richardson [11] presented a third-order analytical solution of halo orbits around collinear libration points in the RTBP. He employed the LindstedtPoincar'e method to construct this solution by introducing a correction term to modify the out-of-plane frequency. By means of the Lindstedt-Poincar'e method, Jorba and Masdemont [12, 13] obtained high-order series solutions of Lissajous orbits and halo orbits around collinear libration points with the help of ad hoc algebraic manipulators. Celletti and Luo et al. [14, 15] conducted an analytical study of the Lissajous and halo orbits around collinear points in RTBP for arbitrary mass ratios, utilizing a canonical transformation procedure. Paez and Guzzo [16] provided a semi-analytical construction of halo orbits and halo tubes in the elliptic RTBP by implementing a nonlinear Floquet-Birkho ff resonant normal form. However, these analytical methods cannot yield high-order series solutions of quasihalo orbits. G'omez et al. [17] took a halo orbit with a constant amplitude as a backbone (periodic-varying equilibrium point), and further employed Floquet transformation and Lindstedt-Poincar'e method to construct quasihalo orbits. This method necessitates specifying the magnitude of the reference halo orbit and using Floquet transformation, and does not directly provide a complete analytical expression for quasihalo orbits.'. So far, there is no semi-analytical method to uniformly describe Lissajous orbits, halo orbits and quasihalo orbits in the center manifolds, although some numerical methods, such as norm form [12] and multiple Poincar'e section [18], can provide a comprehensive description for the center manifolds of RTBP. References [11, 12, 13] successfully gave the third or higher-order series solutions of the Lissajous and halo orbits, yet they failed to construct a high-order series solution of the quasihalo orbits. Since the discovery of halo orbits in 1969, the astrodynamics community has traditionally attributed their origin to bifurcations from planar Lyapunov orbits when the in-plane and out-of-plane frequencies are in a 1:1 resonance. In this paper, we also contend that halo orbits indeed arise from bifurcations of planar Lyapunov orbits. However, We employ a new mechanism, coupling-induced bifurcation, to explain the generation of halo and quasihalo orbits. Consider a two-degree-of-freedom nonlinear dynamical system similar to the part of the center manifolds in the RTBP, but in this system, the degrees of freedom are decoupled, i.e., [ x , y ] T = [ f ( x ) , f ( y ) ] T . It is evident that bifurcations of periodic orbits do not occur even under frequency resonance. Drawing a distinction between this hypothetical dynamical system and the RTBP, it can be inferred that the true trigger for the emergence of halo orbits through planar Lyapunov orbit bifurcation is the nonlinear coupling e ff ect between the two degrees of freedom, rather than a resonance between the two frequencies. Considering the nonlinear coupling of in-plane and out-of-plane motions, this paper introduce the concept of a coupling coe ffi cient η into the RTBP for the first time. Additionally, a new correction term η ∆ x is incorporated into the RTBP equation to construct a unified semi-analytical solution for the center manifolds of collinear libration points in the RTBP. Here, ∆ is a power series concerning in-plane and out-of-plane amplitudes. When the coupling coe ffi cient η is zero, the correction item is inactive, and the series solution describes Lissajous orbits around libration points. Otherwise, the series solution describes quasihalo orbits, of course, including halo orbits. In this way, the center manifolds of collinear libration points in the RTBP are uniformly described with a high-order series solution. The contribution of this paper lies in the novel proposition of the couplinginduced bifurcation mechanism to explain the generation of quasihalo orbits. This marks the first complete approximate analytical solution for quasihalo orbits, providing a comprehensive analytical solution for the entire central manifold, including Lissajous orbits, halo orbits, and quasihalo orbits. Additionally, the practical convergence domain of quasihalo orbits is presented for the first time. Lastly, the proposed method transforms the dynamic orbit bifurcation problem of vector fields into a static bifurcation problem for the solutions of the bifurcation equation. This method is applicable for general analytical bifurcation analysis in dynamical systems. The remainder of this paper is organized as follows. Section 2 introduces the dynamical model of the RTBP. In section 3, a general analytical construction method for the center manifolds of collinear libration points in the RTBP is presented. Results and discussions are provided in Section 4. Finally, Section 5 makes some concluding remarks.", "pages": [ 2, 3, 4 ] }, { "title": "2. Dynamical model", "content": "This section introduces the dynamical model of the classical RTBP. This model serves as a good approximation for the motion of an infinitesimal particle (asteroids or spacecraft) under the gravitational attraction of two primaries. The attraction of the particle on the primaries is neglected so the two primaries rotating around their common center of mass in a Kepler orbit. Here, our focus lies on the circular orbit, i.e., a Kepler orbit with zero eccentricity. The motion of the particle is usually described in a synodic coordinate system. In this system, the origin is positioned at the centroid of the two primaries. The X -axis points from the smaller primary to larger primary, the Z -axis is perpendicular to the plane of the Kepler orbit and positive in the direction of the angular momentum, and the Y -axis completes a right-hand triad. Consequently, the position vector of the smaller primary and larger primary is ( -µ , 0, 0) and (1 -µ , 0, 0), respectively. Then, the di ff erential equation governing the motion of the particle in the synodic coordinate system is expressed as follows [19]: with where µ = m 2 / ( m 1 + m 2) is the mass parameter of the system. m 1 and m 2 are the mass of the smaller primary and larger primary, respectively. r 1 and r 2 are the distance from the particle to the smaller primary and larger primary, respectively. This model has a Jacobi integral As mentioned in the Introduction section, the RTBP has five equilibrium points. In this paper, we focus on the three collinear libration points L 1, L 2, and L 3. Let γ i ( i = 1 , 2 , 3) denote the distance from Li to the closet primary. This distance is determined by the unique positive root of the Euler quantic equation [4, 19], When focusing on the selected libration point Li , a coordinate transformation is performed to move the origin of coordinate system to the libration point Li , Then, the dynamical model (1) of the RTBP can be reformulated as where In order to construct a high-order series solution for the center manifolds in the RTBP in the following section, the motion equation (7) is expanded in power series using the Legendre polynomials [12, 20], where ρ = x 2 + y 2 + z 2 , Pn is Legendre polynomials, and the constant coe ffi cients cn ( µ ) are depend on the system parameters µ , The left-hand linear terms form the foundational components of the series solution, while the right-hand nonlinear ones are crucial for constructing halo and quasihalo orbits. They are well-known as bifurcations from large-amplitude planner Lyapunov orbit.", "pages": [ 4, 5, 6 ] }, { "title": "3. Analytical construction of center manifolds in RTBP", "content": "In this section, a semi-analytical solution for the center manifolds of collinear libration points in the RTBP is constructed using the Lindstedt-Poincar'e method. Lindstedt-Poincar'e method is an iterative computational technology that begins from the fundamental first-order solution of the system. Subsequently, it continuously adjusts the relationship between the frequency and amplitude to obtain a higher-order series solution by iterating the known low-order solution step by step.", "pages": [ 6 ] }, { "title": "3.1. Form of the analytical solution", "content": "First, the first-order solution for the center manifolds of collinear libration points in the RTBP should be found. It is well known that the center manifolds include Lissajous orbits, halo orbits, and quasihalo orbits around halo orbits. For Lissajous orbits, the first-order solution naturally arises by solving the linear part of (9), The solution of (11) is where α and β represent the in-plane and out-of-plane amplitudes, respectively. φ 1 and φ 2 denote the corresponding phases. As stated in the Introduction section, halo orbits bifurcate from the planar Lyapunov periodic orbits without requiring the in-plane and out-of-plane frequencies to be equal. This phenomenon is primarily due to the nonlinear coupling of in-plane and out-of-plane motions in RTBP. Considering the coupling between in-plane and out-of-plane motions, a natural concept of coupling coe ffi cient η can be defined to characterize the degree of coupling between the linear motion in plane and out of plane. Consequently, the basic first-order solution can be reformulated as To derive the solution (14) for (11), it is necessary to introduce a correction term and rewrite it as where d 00 = c 2 -ω 0 2 /nequal 0 is a constant correction factor and η d 00 = 0. It is evident that for linear equation (15), the only trivial solution is η = 0, indicating the absence of coupling between the in-plane and out-of-plane motions. Halo / quasihalo orbits only appears when the coupling coe ffi cient η is not-zero in higher-order solutions. Therefore, we extend (15) to higher-order case by introducing a higherorder correction term to the third equation of (9) as follows: where the higher-order term η ∆ x is the product of the coupling coe ffi cient η nonlinear correction factor ∆ , and x , satisfying ∆ = dij α i β j and η ∆ = the process, non-trivial solutions of η are determined from the condition , the ∑ 0. During ∆ = where θ 1 = ω t + ϕ 1, θ 2 = ν t + ϕ 2. Considering the nonlinear terms, the frequencies should also be expanded in the power series of α and β , Moreover, we have the constraint which provides the implicit relationship between η and α and β , i.e., η = η ( α, β ). Equations (17), (18), and (19) together constitute a comprehensive analytical description of the central manifolds around the collinear libration in the RTBP. Remark 1 . In fact, the above correction is not unique, as long as the correction term can represent the coupling between in-plane and out-of-plane motions, such as adding η ∆ y to the third equation or η ∆ z to the first (second) equation of (16). Remark 2 . When η = 0, the solution (17) represents Lissajous orbits. Specifically, if β = 0, it corresponds to planar Lyapunov orbits; if α = 0, it corresponds to vertical Lyapunov orbits. When η /nequal 0, the solution (17) represents quasihalo orbit. Particularly, if η > 0 ( η < 0) and β = 0, it results in north (south) halo orbits. No solutions exist if α = 0, which will be demonstrated in the next section. In sum up, the high-order series solution (17) uniformly describes the center manifolds of collinear libration points in the RTBP. Remark 3 . In (17) and (18), i and j ∈ N , k and m ∈ Z . Due to the symmetry of the RTBP, x ( t ) and z ( t ) are formulated as a cosine series, and y ( t ) as a sine series. p and q have the same parity as i and j , respectively. Besides, due to the symmetry of sine and cosine functions, it can be assumed that m ≥ 0, and n ≥ 0 when m = 0. The series of ω and v only include even items. These properties are useful for conserving computational storage and time.", "pages": [ 6, 7, 8 ] }, { "title": "3.2. Solving for undetermined coe ffi cients", "content": "Now, our goal is to compute the coe ffi cients xijkm , yijkm , zijkm , ω i j , vij in (17) and (18) up to a finite order n . The Lindstedt-Poincar'e method is utilized to calculate these coe ffi cients following an iterative scheme from the linear solution. Compared to the solution of the linear part (14), we can determine x 1010 = 1, y 1010 = κ , z 1010 = η , z 0101 = 1, ω 00 = ω 0, v 00 = v 0. By substituting this linear solution into (16), the coe ffi cients of the second-order solution can be derived. Similarly, when the coe ffi cients up to order n -1 are obtained, i.e., x ( t ), y ( t ), and z ( t ) are determined up to order n -1, ω and v are determined up to order n -2. Substituting them into the right side of (16), we can obtain three power series up to order n , denoted by p , q , and r . Here, what we are interested in are those n -order terms. Without losing generality, the n -order terms of p , q , and r are denoted by pijkm , qijkm , and rijkm ( i + j = n ) respectively. - - Next, we analyze the composition of n -order terms on the left side of (16). According to (17), the derivatives of variable x can be expressed as Similarly, we can obtain the derivatives of y and z . Let f g denote the first-order derivative term, where f represents the power series of the frequencies ( ω and v ), and g represents the coordinate variables ( x , y , and z ). ( i j ) f and ( i j ) g denote their corresponding order. Then, the f g satisfying ( i j ) f + ( i j ) g = n constitutes the n -order terms we require. When ( i j ) f is 0 or n -1, the corresponding ( i j ) g is n or 1 and f g is an unknown term that needs to be solved. When ( i j ) f = 1 , 2 , ..., n -2, f g is a known term that needs to be moved to the right-hand side of (16). Table 1 summarizes the unknown and known terms of the first derivatives and the product of ∆ and x , where δ i j denotes the Kronecker function. For the second derivatives of x , y , and z , it can be similarly summarized as shown in Table 2. Then we move all the known terms to the right-hand side of (16), add them to pijkm , qijkm , and rijkm , and re-denote them as ¯ pijkm , ¯ qijkm and ¯ rijkm . Besides, we need to address the calculation of the unknown term Ω i -1 j and N i j -1. In fact, they are composed of the unknown term 2 ω 00 ω i -1 j (2 v 00 vij -1) and the remaining known terms, i.e., Ω i -1 j = 2 ω 00 ω i -1 j + ∑ i 1 , j 1 , i 2 , j 2 C 1 -δ i 1 i 2 δ j 1 j 2 2 ω i 1 j 1 ω i 2 j 2 , N i j -1 = a combination. Similar results can be obtained for the second derivatives of y and z . In summary, the linear equation of n-order unknown coe ffi cients is yielded 2 v 00 vij -1 + ∑ i 1 , j 1 , i 2 , j 2 C 1 -δ i 1 i 2 δ j 1 j 2 2 vi 1 j 1 vi 2 j 2 , where i 1 + j 1 + i 2 + j 2 = n -1 and C k n denotes where /pi1 km = k ω 00 + mv 00, d 00 = ( c 2 -ω 2 00 ) η , and When ( k , m ) /nequal (1 , 0) , ( k , m ) /nequal (0 , 1), (21) becomes the regular linear equations (23). The coe ffi cients xijkm , yijkm , and zijkm can be solved immediately. When ( k , m ) = (1 , 0), xijkm , yijkm , and zijkm are couple. Thus, xijkm can be set zero, zijkm = η xijkm = 0. In this case, (21) is turned to linear equations (24). Then, yijkm and ω i -1 j are solved by the first two equations of (24). di -1 j is obtained from di -1 j = ¯ rijkm η C 1 -δ i 1 i 2 δ j 1 j 2 2 ω i 1 j 1 ω i 2 j 2 . When ( k , m ) = (0 , 1), the coe ffi cient of zijkm is zero, and therefore zijkm is set zero. In this case, (21) simplifies into linear equations (25). Then, xijkm and yijkm are solved by the first two equations of (25). vij -1 is obtained from -2 v 00 vij -1 = ¯ rijkm + d 00 xijpq + C 1 -δ i 1 i 2 δ j 1 j 2 2 vi 1 j 1 vi 2 j 2 .", "pages": [ 9, 10, 11 ] }, { "title": "4. Results", "content": "In this section, the third-order analytical solution for central manifolds around the collinear libration points in the RTBP with arbitrary system parameter µ is derived. Moreover, the construction of the series solution up to a certain order n is implemented for the given system parameter µ , such as the Sun-Earth system ( µ = 3.040423398444176e-6) or Earth-Moon system ( µ = 1.215058191870689e-2), utilizing the C ++ 17 programming language.", "pages": [ 11 ] }, { "title": "4.1. Third-order analytical solution", "content": "It is well-known that halo / quasihalo orbits in the RTBP first appear in the third-order series solution. To achieve a comprehensive description of the center manifolds, their third-order analytical solution is derived using the analytical construction method described in Section 3, as follows: x = α cos θ + a + a + ( a + ( a + ( a + ( a + ( a + a y = κα sin θ + b + ( b + ( b + ( b + ( b + ( b + b 24 31 37 ηαβ sin( θ 4 η 3 + b + b 3 η 311 315 317 319 η 2 η 2 η ηβ 3 + b + b 316) αβ + b 318) αβ sin θ 2 + b ηαβ cos( θ β 25 27 31 36 38 2 + a 4 η 3 η 3 + a + a + a 2 η 310 312 314 η 2 η ηβ 3 + a 311) αβ + a 313) αβ cos θ 1 2 + ( b 1 2 η η ) α 32 38 312 η ) α - + a 21 θ 2) + b + b 33) α 2 β sin θ 2 2 2 β sin (2 θ sin( θ sin( θ 320 ηβ 3 1 1 ( 1 2 a 21 + θ 2) + a cos 2 θ 2 β 28 32 37 39 η η ) α η ) α + a 33) α 2 2 β cos(2 θ β cos(2 θ 2 2 cos( θ cos( θ 315 + b ηβ 3 22 ) η 2 26 α 2 ηαβ cos( θ cos 3 θ 1 1 1 1 + θ 2) - + 2 θ 2) - 2 θ 2) cos 3 θ 2 η 25 3 2 ) α β 2 sin θ + ( b 1 + 2 θ 2) - 2 2 , sin 2 θ sin 2 θ 1 + ( b 39 - 2 θ 2) sin 3 θ 2 , with the frequencies and the coupling correction term η θ 2) + ( b 3 ( + ( a 23 1 + a - 34 1 2 34 + b η + b 4 η + b 310 η ) α 313 2 η 35 2 24 θ 2) 3 + a 23 ) α 35 2 η ) α ηαβ sin( θ 2 η β sin (2 θ + b 314) αβ + b 36) α 1 cos 2 θ 2 β cos θ 1 + θ 2) 3 + θ 2) 2 sin θ sin 3 θ θ 2) 1 2 3 + + a + a 2 η 1 22 1 2 1 1 where aij , bij , dij , eij , and li are constant as provided in Appendix A. Equation (28) establishes an explicit relationship between η and α and β , i.e., η = η ( α, β ). It obviously has a trivial solution η = 0 for any values of α and β . In this case, the third-order solution (26) describes Lissajous orbits. With the increment of α and β , other non-zero real solutions will bifurcate if we have In this case, solution (26) describes quasihalo orbits. To find the critical condition for bifurcation, we let η = 0 in (29) and obtain It is not di ffi cult to calculate and verify that l 1 > 0 , l 2 < 0 , l 3 < 0 , l 4 > 0 and l 5 < 0 for the three collinear libration points with all system parameter µ ∈ (0 , 0 . 5]. Hence, (30) is a hyperbolic equation where bifurcation occurs and suitable nonzero real solutions exist for (29) when ∆ ( α, β ) as illustrated in Fig. 1. Particularly, if the amplitude β is set to zero, the third-order solution (26) describes halo orbits. It is seen from (30) that the minimum permissible value of α is given as (Without loss of generality, only the case of α > 0 is considered in the following) When α > α min and β = 0, by solving (29) we can find two solutions of η , The solution branch corresponds to northern halo orbits (Class I) for η > 0 and southern halo orbits (Class II) for η < 0. These results align with the classical outcomes of the third-order analytical solution of halo orbits presented in [11]. To sum up, quasihalo orbits bifurcate from Lissajous orbits when η is a solution of (29). In particular, halo orbits bifurcate from planar Lyapunov periodic orbits when η is a solution of (29) with β = 0. Remark 4 . The third-order solution (26) serves as an initial approximation to the center manifolds of around the collinear libration points in the RTBP. A higher-order series solution is required for real space missions. In such case, the bifurcation equation is no longer a hyperbola. Figure 1 just provides a basic outline of the feasible region for η , and in fact, its real feasible region is more intricate than depicted in Fig. 1. A higher-order numerical feasible region of η will be presented in the following section.", "pages": [ 11, 12, 13, 14 ] }, { "title": "4.2. Numerical results", "content": "The semi-analytical computation of center manifolds up to a certain order n for the given system parameter µ is implemented to verify the accuracy of the proposed method. Appendix B shows the coe ffi cients of expansion for center manifolds of L 1 in Sun-Earth system, up to order 3. The third-order analytical solution in Subsection 4.1 illustrates that a feasible region of coupling coe ffi cient η of center manifolds is bounded by a hyperbola. Similar to the process in (30) for the third-order solution, higher-order ∆ ( α, β ) can be obtained during the computation of higher-order semi-analytical solutions. For each given paired amplitudes ( α, β ), non-trivial values of η can be determined from ∆ ( α, β ) = 0. Then, The feasible region of η of center manifolds around L 1 in the Earth-Sun system with di ff erent orders is presented in Fig. 2, where only the range of α ∈ [0 , 0 . 35], β ∈ [0 , 0 . 4] and η ∈ (0 , 3 . 0] is considered due to the symmetry of η and the divergence of (17) for large amplitudes. The colorbar represents the number of solutions N ( η ) of ∆ ( α, β ) = 0 with the range η ∈ (0 , 3 . 0]. It can be seen that N has only two possible values N = 0 or N = 1, distributed on both sides of a hyperbola boundary when the order n = 3. This result is consistent with the analytical solution in Subsection 4.1. As the order increases, parts of the region with N = 0 and N = 1 are replaced by the region with N ( η ) > 1. This means that, with increasing amplitudes α and β , more than one quasihalo orbit bifurcates from a Lissajous orbit. Remark 5 . The polynomials ∆ ( α, β ) may encompass all the information about local bifurcation around collinear libration points in the RTBP. It is reasonable to believe that with further increases in order, the number of solutions of η will also increase. Furthermore, it is observed that the number of solutions for η multiplies (due to only η ∈ (0 , 3 . 0] being considered, some solutions for η are omitted here). This is a typical period-doubling bifurcation, indicating that chaos naturally occurs in the center manifolds of collinear libration points in the RTBP. However, the high-order series solution (17) cannot describe this phenomenon due to its divergence. Then, a contour map of η values with an order n = 19 is computed as shown in Fig. 3. The feasible region corresponds to the bottom-right part of Fig. 2. Each contour map describes the distribution of one solution of η . Empty regions indicate no solution for η . Six contour maps represent a maximum of six solutions of η at the 19-th order. Figure 3 shows that the feasible region is largest when there is only one non-zero solution for η (top left). Subsequently, as the number of nonzero η solution increases, the feasible region becomes small. When the amplitude α and β are small, there are no non-zero η , corresponding to the empty region in the top-left part of Fig. 3. In this case, the higher-order series (17) only describes planar and vertical Lyapunov orbits and Lissajous orbits. Figure 4 shows a typical plot of planar Lyapunov orbits, vertical Lyapunov orbits and Lissajous orbits in the synodic coordinate system. Upon increasing α and β to the region with one solution of η , Fig. 3 shows that η values change from zero to non-zeros but they are very small, which means the week coupling of two degrees of freedom (in-plane and out-of-plane motions). In this case, quasihalo orbits closely resemble the corresponding Lissajous orbits, with the amplitudes identical to quasihalo orbits but η values being zeros. This observation is evident from Fig. 5, where η = 0.08180669069. As the coupling e ff ect ( η ¿ 0) of the in-plane motion acts on the out-of-plane motion, a planesymmetric Lissajous orbit (linear part in z -direction: z 1 = β cos θ 2) becomes an approximately oblique-upward symmetric quasihalo orbit (linear part in z -direction: z 1 = β cos θ 2 + ηα cos θ 1). With further increment in α and β , one η value rapidly grows larger and more solutions for η appear. Figure 7 shows two quasihalo orbits with the same amplitudes. It can be seen that the quasihalo orbit with the smaller value of η exhibits week coupling and remains similar to Lissajous orbits. The larger value of η leads to a significant coupling e ff ect on the motion in z-directions, resulting in a typical quasihalo orbit, as shown in Fig. 7b. In this case, in-plane and out-of-plane motions are strong coupling. As α and β increase to the region shown in the bottom-left of Fig.3, four solutions for η can be found from ∆ = 0, i.e., N ( η ) = 4. Figure 8 shows four quasihalo orbits corresponding to these four solutions for the given α and β . It can be seen they have some strange bendings, likely due to the corresponding α and β not being within the practical region of convergence. Now, we analyze the practical convergence domain of the proposed analytical solution by comparing it with numerical solutions. Firstly, an initial condition is obtained for given amplitudes α and β from the analytical solution (17). Subsequently, numerical integration of the dynamical equations is performed over a normalized time length T = π . The accuracy of the analytical solution is determined by comparing the Euclidean norm of di ff erence in position vectors at final time between the analytical and numerically integrated solutions. Performing the same procedure for each pair of amplitudes ( α , β ) within a given range yields the practical convergence domain of the solution with a specified order. Figure 6 shows the domain of practical convergence of the proposed analytical solution up to order 35 for Lissajous orbits ( η = 0) and Quasihalo orbits ( η /nequal 0) around L 1 of the Earth-Sun system. Here Fig. 6(a) is similar the result in [12]. This is because, when η is zero, no bifurcation occurs, and the analytical solution is identical to the solution of Lissajous orbits presented in [12]. However, when η is non-zero, coupling e ff ects between di ff erent degrees of freedom lead to the bifurcation, specifically the generation of quasihalo orbits. Figure 6(b), for the first time, provides the actual convergence domain of the approximate analytical solution for quasihalo orbits generated by the first bifurcation from Lissajous orbits. It can be observed that the practical convergence domain for quasihalo orbits is significantly smaller than the practical convergence domain for the Lissajous orbit. This is due to the coupling in the plane direction causing a large actual amplitude in the z-direction, even when the amplitude β is small. It is known that center manifolds around collinear libration points in the RTBP are four-dimensional, making direct graphical display challenging. As a byproduct of the proposed analytical method for constructing center manifolds, the dynamics inside center manifolds can be globally described in a two-dimensional Poincar'e section by fixing z = 0 with z > 0 and the Jacobian integral C = C 0 in synodic coordinate system. First, arbitrary phase angles φ 1 and φ 2 and arbitrary initial time t0 from (17) is chosen. For the chosen C 0, paired amplitudes ( α , β ) and corresponding initial states ( x 0 y 0 z 0) can be computed. Then, starting from each initial state, every points ( xi , yi ) is plotted in the two-dimensional Poincar'e section when zi = 0 within a specified time interval according to (17). Figure 9 shows two Poincar'e sections with two di ff erent Jacobian integrals, defining a closed region. The boundary of the region is a planar Lyapunov orbit with the paired amplitudes ( α max, 0) and a fixed point on y -axis represents a vertical Lyapunov orbit with the paired amplitudes (0, β max), where α max and β max are maximum in-plane and out-of-plane amplitudes for the selected C 0, respectively. Other circles inside this region respond to Lissajous orbits with the paired amplitudes ( α , β ) where α ¡ α max and β < β max. The bifurcation occurs when the increase in C results in a non-zero solution η /nequal 0 for the equation ∆ = ∑ 0 ≤ i + j ≤ n dij α i β j = 0. This visual computing not only shows the emergence of well-known halo orbits from planar Lyapunov orbits but also illustrates the generation of quasi-periodic orbits from Lissajous orbits. Remark 6 . The computed results presented in Fig. 9 are similar to those found in previous studies [4, 12]. However, these results were computed through numerical continue methods with high computational complexity or displayed in normal coordinate via a complicated normal form. In contrast, the Poincar'e sections of center manifolds can be directly derived from the high-order series solution (17) in this paper.", "pages": [ 14, 15, 17, 18, 21 ] }, { "title": "5. Conclusions", "content": "Understanding the coupling between in-plane and out-of-plane motions and its connection to the bifurcation of halo / quasihalo orbits is crucial for unraveling the mechanism governing their generation and obtaining their unified an analytical solution. This paper introduced a novel concept of coupling coe ffi cients into the RTBP for the first time, incorporated a new correction term into the RTBP equation to characterize this coupling e ff ect and successfully derive a unified ana- lytical solution for the center manifolds of collinear libration points in the RTBP. When the bifurcation equation ∆ = 0 has no real solutions, the series solution (17) describes planar Lyapunov orbits, vertical Lyapunov orbits and Lissajous orbits. When the equation ∆ = 0 has only zero solutions, the bifurcation occurs. When the equation ∆ = 0 yields non-zero real solution, the series solution (17) further describes both halo orbits and quasihalo orbits, with the latter being constructed analytically for the first time. Remarkably, larger amplitudes lead to multiple solutions of the equation ∆ = 0 with the higher-order series solution, indicating the discovery of a multiple bifurcation, which can be explicitly calculated. Although the proposed analytical method is aimed at constructing center manifolds around collinear libration points in the RTBP, it can be extended to invariant manifolds in the RTBP and even general multi-degree-of-freedom dynamical systems. Besides, the coupling e ffi cient η and the equation ∆ = 0 serve as conditions for bifurcation. Thus, this method can also be referred to as an analytical bifurcation calculation method.", "pages": [ 21, 22 ] } ]

[ { "title": "ABSTRACT", "content": "The equilibrium resulting in a recombining plasma with arbitrary metallicity 𝑍 , heated by a mean radiation field 𝐸 as well as by sound waves dissipation due to thermal conduction, dynamic and bulk viscosity is analyzed. Generally, the heating by acoustic waves dissipation induces drastic changes in the range of temperature where the thermochemical equilibrium may exist. An additional equilibrium state appears which is characterized by a lower ionization and higher gas pressure than the equilibrium resulting when the wave dissipation is neglected. The above effects are sensibly to the values of the gas parameters as well as the wavelength and intensity of the acoustic waves. Implications in the interstellar gas, in particular, in the high velocity clouds are outlined. Keywords : Recombining astrophysical plasma, intellestelar medium (ISM), high velocity clouds (HvCs), metallicity. mean radiation field.", "pages": [ 1 ] }, { "title": "HEATING BY DISSIPATION OF SOUND WAVES IN THE INTERSTELLAR GAS", "content": "Miguel H. Ibañez Sanchez 1 , Sandra M. Conde C 2 ., and Pedro L. Contreras E. 2,3 Address: 1 Centro de Física Fundamental, Universidad de Los Andes, Mérida, 5101 Venezuela, 2 Formally at Centro de Física Fundamental, Universidad de Los Andes, Mérida, 5101 Venezuela. 3 Departmento de Física, Universidad de Los Andes, Mérida, 5101 Venezuela, Corresponding author email: [email protected]", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "It has been well established (Landau et al.,1987; Mihalas et al.,1984; Stix, 1992) that when the gas dynamic equations are linearized, assuming small disturbances, a rotational and a potential mode result which are independent each other. If viscosity is accounted for, it produces damping on the potential mode and if thermal conduction is taken into account an additional damped thermal mode appears (Landau et al.,1987). The above two dissipative effects have been invoked as one of the mechanisms of heat input in different astrophysical plasmas, particularly, in the solar, and more generally in stellar atmospheric plasmas (Bird,1964; Narain et al.,1990; Stein et al,1972; Stein et al.,1974), in the interstellar medium (ISM) (Spitzer,1978; Spitzer,1982, Spitzer,1990), and more recently in the intracluster gas (Fabian et al.,2003; Fabian et al.,2005; Ruszkowski et al.,2004; Ferland et al.;2009; Ibañez et al.,2005; Fajardo et al.,2021). Most of the works on this subject does not consider bulk viscosity dissipation, none the less the corresponding dissipation can be larger than both the dynamic viscosity and the thermal conduction dissipation, when chemical precesses are present (including ionization and recombination precesses) (Ibañez et al.,1993; Ibañez et al.,2019). In addition to the quantitative changes on the acoustic dissipation, the bulk viscosity also introduces important qualitative changes, in particular due to the fact that the bulk viscosity coefficient is dispersive (depends on the wave frequency), as it will be shown later on. The present work is aimed to analyze how much a thermochemical chemical equilibrium is modified if heating by sound wave dissipation is taken into account in an homogeneous recombining plasma able to cool down by the well known cooling function for plasmas with arbitrary metallicity 𝑍 , ionized and heated by a mean radiation field 𝐸 as well as by dissipation of the acoustic waves. Implications in the structure of the interstellar medium (ISM) and in high velocity clouds (HvCs) will be outlined. Non-homogeneity effects will be neglected and only three dissipative mechanisms will be considered, i.e. the thermal conduction 𝜅 , the dynamic viscosity 𝜂 , and the bulk viscosity 𝜁 . Molecular formation which appears once the hydrogen molecule H2 forms (in gas phase or adbortion on cool grains), in particular the formation of the CO molecule (which is a much more stronger coolant than the corresponding atomic cooling), as well as the strong cooling by solid grains and their important opacity effects will not be taken into account at the present approximation. These additional effects completely change the radiative heating and cooling of the interstellar gas. These complications as well as magnetohydrodynamic effects will be carried out elsewhere (Fajardo et al.,2021).", "pages": [ 1 ] }, { "title": "BASIC EQUATIONS", "content": "For a fluid where a chemical reaction of the form ∑ 𝑏 𝑖 𝐶 𝑖 𝑖 = 0 is proceeds, the equations of gas dynamic can be written in the form where 𝑑 𝑑 𝑡 / is the convective derivative, 𝑋(𝜌,𝑇,𝜉) is the net rate function, ℒ(𝜌, 𝑇, 𝜉) the net rate of cooling per unit mass and time, which are functions of the degree of ionization 𝜉 (parameter determining the advance of the the reaction), the density 𝜌, and the temperature T . 𝜅 is the thermal conduction coefficient, 𝜂 is the dynamic viscosity, and 𝜁 the bulk viscosity (complex in this case). Furthermore an ideal gas state equation (5) has been assumed. 𝑅 is the universal gas constant, and the coefficients A, B and 𝜇 are defined as follows where 𝑥 𝑖 0 is initial concentration, 𝛾 𝑖 and 𝜖 𝑖 0 are the specific heat ration and zero point energy of the i-esime gas component, respectively, 𝑘 𝐵 is the Boltzmann constant. Additionally, for a plasma with a degree of ionization 𝜉, the thermal conduction becomes where ln Λ 𝑠 (𝜌, 𝑇, 𝜉) is the relation between the Debye screening, and the impact particles parameter (Braginskii, 1965; Spitzer 1962). The dynamic viscosity following (Braginskii, 1965; Spitzer 1962) takes the form On the other hand, the complex bulk viscosity 𝜁 becomes where 𝜏 is the chemical relaxation time and the sound velocities 𝑐 ∞ 2 and 𝑐 0 2 are defined as Here 𝜉 0 is the value of the chemical parameter at chemical equilibrium and ( 𝜕𝜉 0 𝜕𝜌 ) . In this work the c.g.s. unit system is used.", "pages": [ 2, 3 ] }, { "title": "LINEAR WAVES", "content": "If the velocity field of sound disturbances with wave number 𝑘 and frequency 𝜔 is assumed to be 𝑣 = 𝑣 𝑥 = 𝑣 1 cos(𝑘𝑥 𝜔𝑡) , and 𝑣 𝑦 = 𝑣 𝑧 = 0 as a first approximation, the time averaged in a volume 𝑉 of energy dissipation by the bulk viscosity 𝜁 of sound waves becomes as can be realized easily, the real part 𝜁 𝑅 of the bulk viscosity 𝜁 is On the other hand, we have that where ( 𝜕𝜉 𝜕𝜌 ) 𝑒𝑞 is calculated at equilibrium. This result will allow to calculate the bulk viscosity coefficients for reacting gases, as it will be shown later on. Therefore, the heat input due to dynamic as well as bulk viscosity (Landau et al.,1987), and the thermal conduction becomes where and 𝑐 𝑃 being the specific heat at constant pressure.", "pages": [ 3 ] }, { "title": "THERMAL EQUILIBRIUM OF A PHOTOIONIZED GAS", "content": "Commonly the heat/loss function for a low density plasma at equilibrium temperatures (neglecting molecular formation as quoted out in the introduction) in the range 30 𝐾 < 𝑇 < 3 × 10 4 𝐾 can be written as where Λ (𝜌, 𝑇, 𝜉) is the cooling rate and Γ 0 (𝜌, 𝑇, 𝜉) is the heating input (different from wave dissipation) per unit volume and time, respectively. So, at thermal equilibrium, ℒ (𝜌, 𝑇, 𝜉) = 0 , i.e. the cooling rate becomes For an optically thin hydrogen plasma with metallicity 𝑍 heated and ionized by a background radiation field of mean photon energy E , and an ionization rate 𝜍 , the net rate function 𝑋(𝜌, 𝑇, 𝜉) and the cooling and heating rates rate per unit volume and time are respectively given by (Corbelli et al., 1995), and ] where 𝜙 denotes the number of secondary electrons, 𝐸 ℎ the heat released per photoionization (Shull et al., 1985), 𝛬 𝐻𝑍 , 𝛬 𝑒𝑍 , 𝛬 𝑒𝐻 and 𝛬 𝑒𝐻 + respectively are the cooling efficiencies by collisions of neutral hydrogen-ions and metal atoms (Launay et. al., 1977; Dalgamo et al., 1972), electrons-ions and metal atoms (Dalgamo et al., 1972), 𝐿𝑦𝛼 emission by neutral hydrogen (Spitzer, 1978), and hydrogen recombination, on the spot approximation (Seaton, 1959). From (Corbelli et al., 1995), follows that at ionization equilibrium state 𝑋(𝜌, 𝑇, 𝜉) = 0 , therefore On the other hand, the damping scale length 𝑙 𝑑 is given by where equation (18) holds as far as the damping scale-length is larger than the sound wave length 𝜆 = 2 𝜋 𝑘 / i.e. where = 𝑣 1 𝑐 0 / (notice that 𝜖 does not have units); so, the dimensions of the region heated by sound waves dissipation is larger than the sound wavelength and the above approximation holds.", "pages": [ 3, 4 ] }, { "title": "Interstellar gas (ISM)", "content": "At the present subsection the results obtained previously will be applied to a gas with characteristic values of the parameters representative of the interstellar medium, i.e 𝑁 0 𝜌 = 1, 𝑍 = 1, 𝑎𝑛𝑑 𝐸 = 10 2 𝑒𝑉, and in the range of temperature where the hydrogen ionization-recombination takes place. Figure 1 is a plot of the ionization 𝜉 (𝑎), the pressure 𝑝 (𝑘 𝐵 𝜍) / (𝑏) , the heat input Γ 0 (𝑐) , and the dissipative coefficients (𝛾 - 1)𝜅 𝑐 𝑃 / (dash line in d), 4/3 𝜂 (point line in d), 𝜁 𝑅 𝑎𝑛𝑑 𝛾 𝑑 (thick line in d) without heat input by sound waves dissipation, (𝜖 𝑘 = 0 𝑐𝑚 =1 ) , and for a plasma at equilibrium 𝑖. 𝑒., 𝑋(𝜌, 𝑇, 𝜉) = 0, 𝑎𝑛𝑑 ℒ (𝜌, 𝑇, 𝜉) = 0 . For the above particular values of density, metallicity and mean photon energy, the thermal equilibrium may only exist for 𝑇 < 2.254 × 10 4 𝐾 , the nature of the equilibrium for this particular case has been analyzed elsewhere approximation (Ibañez, 2009). As it can be seen in Fig. 1(d), the dynamic viscosity term (point line), and the thermal conduction term (dash line) are very small respect to the bulk viscosity 𝜁 𝑅 (thick line) which determines the total dissipation 𝛾 𝑑 , that is undistinguished from 𝜁 𝑅 at the scale of Fig. 1(d). When one considers sound waves dissipation the above results can drastically change depending on the value of the sound wavelength, more exactly on the value of 𝜖𝑘 . In fact, for the above values of parameters of the gas, sound waves with 𝜖𝑘 ≲ 10 -18 𝑐𝑚 -1 only produce small changes on variables the of interest, in particular at high temperatures 𝑇 > 10 3 𝐾 as it is apparent in Figure 2, which are plots as those on Fig. 1 but for 𝜖 𝑘 = 10 -18 𝑐𝑚 -1 . This is due to the fact that the heat input Γ 0 by dissipation of sounds shown by the dash line in Fig. 2(c), at this wavelengths, is less than one order of magnitude than the radiative heating Γ 𝑤 seen in the thick line of Fig. 2(c). These two values of ionization and pressure produce two values of the heating rates Γ 0 (𝜌, 𝑇, 𝜉) 𝑎𝑛𝑑 Γ 𝜔 (𝜌, 𝑇, 𝜉) Fig. 3(c) as well as two value in the dissipation coefficients 𝜅, 𝜂, 𝑎𝑛𝑑 𝜁 𝑅 in Fig. 3(d) in the above interval of temperature. From the physical point of view, the above results imply that when the heat input by acoustic waves in a recombining gas becomes important, the gas can be at equilibrium in two different phases: one at high ionization and low pressure, and the another one at low ionization and high pressure for the same value of the density. The effects shown in Fig. 3 shift towards higher values of the temperature when 𝜖𝑘 increases as can be seen in Figure 4 which is as Fig. 3 but for 𝜖𝑘 = 10 -16 𝑐𝑚 -1 . For this particular value of 𝜖𝑘 , the ionization Fig. 4(a), pressure Fig. 4(b), both heating rates Fig. 4(c), as well as, the dissipative terms Fig. 4(d) show a bifurcation point at a temperature 𝑇 = 1.14 × 10 4 𝐾 . A low ionization branch and high pressure branch appear in the range of temperature 1.14 × 10 4 < 𝑇 < 1.268 × 10 4 𝐾 . For these equilibrium states the heating Γ 0 (𝜌, 𝑇, 𝜉) < Γ 𝜔 (𝜌, 𝑇, 𝜉) . Contrary to the high ionization (and pressure) branch were the radiation heating is higher than the wave heating in Figs. 4(a), 4(b), and 4(c). For this case, the thermal conduction dissipation terms dominates over the viscosity terms. The low ionization branch corresponds to a total dissipative branch 𝛾 𝑑 higher than that corresponding to higher ionization shown in Fig. 4(d). When the value of 𝜖𝑘 increases the interval of temperature where this equilibrium may exist decreases in such a way that the equilibrium can not exist for 𝜖𝑘 > 5 × 10 -16 𝑐𝑚 -1 . Figure 5 shows the plot of the the variables under discussion have been plotted 𝜖𝑘 = 5 × 10 -16 𝑐𝑚 -1 . For this particular value, the ionization and pressure becomes one-valued function of temperature (as in the case 𝜖𝑘 = 10 -18 𝑐𝑚 -1 ) but the gas pressure shows a minimum value at 𝑇 = 1.849 × 10 4 𝐾 seen in Fig. 5(b). At this limiting value the acoustic dissipation Γ 𝜔 (𝜌, 𝑇, 𝜉) becomes higher than the radiation heating Γ 0 (𝜌, 𝑇, 𝜉) as it is apparent from Fig. 5(c). Note that in this case, the dissipation is due mainly to thermal conduction, as for the previous 𝜖𝑘 value, please see Fig. 5(d). Figure 6 with the damping length scale 𝑙 𝑑 is obtained for the following four values of the magnitude 𝜖𝑘, 𝑖. 𝑒., 𝜖𝑘 = 10 -18 𝑐𝑚 -1 (𝑎) , 𝜖𝑘 = 10 -17 𝑐𝑚 -1 (𝑏) , 𝜖𝑘 = 10 -16 𝑐𝑚 -1 (𝑐) , and 𝜖𝑘 = 5 × 10 -16 𝑐𝑚 -1 (𝑑) . The temperature dependence of the damping length scale 𝑙 𝑑 (Eq. 24) is shown in the Figs 6(a), 6(b), 6(c) and 6(d), respectively. Figure 7 shows the plots for the corresponding ratios 𝜆 (2 𝜋 𝑙 𝑑 ) / . For large wavelengths when the sound wave heat input is much more smaller than the radiative heating seen in fig 6(a), regions of the order of parsecs up to kiloparsecs can exist in thermochemical equilibrium in a wide range of temperature. When the wave number decreases this range of temperature decreases as well as the dimensions of the regions where such an equilibrium may exist, and the two equilibria become apparent. Regions of high pressure are larger than those at lower pressure for 𝜖𝑘 = 10 -17 𝑐𝑚 -1 Fig. 6(b) but the opposite occurs for 𝜖𝑘 = 10 -16 𝑐𝑚 -1 in Fig. 6(c). Close to the limiting value were the equilibrium may exist only small regions, of the order of 10 -3 𝑝𝑐 and in a narrow interval of temperature can exist. As it is apparent from Figure 7 the present approximation holds, 𝜆 𝑙 𝑑 / < 1 in the range of temperature where the recombination-ionization becomes important.", "pages": [ 4, 5, 6, 7, 8, 9, 10 ] }, { "title": "High velocity clouds at high galactic latitude (HvCs)", "content": "Observations suggest that for the hydrogen gas at large galactic latitude, and in the halos of other galaxies the metallicity ranges between 0.03 ≲ 𝑍 ≲ 0.3 $0.03 and a the mean photon energy between 0.3 ≲ 𝐸 ≲ 2 𝑘𝑒𝑉 (Collins et al.,2004; Maller et al.,2004; Miller et al.,2015; Wakker et al., 2001; Wakker, 2004; Tripp et al.,2003; Olano, 2008; Lockman et al., 2008, Ibañez et al.,2011, Conde, 2010) Thus equilibrium states may exist for 𝜖𝑘 smaller than a threshold value (𝜖𝑘) 𝑡ℎ𝑟 which depends on the exact values of 𝑍 𝑎𝑛𝑑 𝐸 (see Figure 8). For 𝜖𝑘 > (𝜖𝑘) 𝑡ℎ𝑟 equilibrium states can not exit. Therefore, the hydrogen gas into High velocity Clouds (HvCs) can be in thermochemical equilibrium as far as the HvCs have dimensions 𝑙 > 𝑙 𝑡ℎ𝑟 , where 𝑙 𝑡ℎ𝑟 = 3.24 × 10 -19 (𝜖𝑘) 𝑡ℎ𝑟 𝑝𝑐 the above range of values of metallicity and mean photon energy. Strictly speaking this threshold value has to be taken as a limiting value, for real situations one should expect thermochemical equilibrium for gas clouds with dimensions 𝑙 ≫ 𝑙 𝑡ℎ𝑟 . This conclusion can also be applied to the intracluster clouds, however this important case will be analyzed elsewhere. Additionally, depending on the particular values of 𝑍 𝑎𝑛𝑑 𝐸 , temperature gaps appear where the equilibrium can not exist. Say for instance, for 𝑍 = 0.3 𝑎𝑛𝑑 𝐸 = 0.3 𝑘𝑒𝑉 the gap is between temperature 2.405 × 10 4 < 𝑇 < 8.149 × 10 5 𝐾 , and for 𝑍 = 0.03 between 2.297 × 10 4 < 𝑇 < 2.255 × 10 5 𝐾 , for this particular value of 𝑍 only one equilibrium exists just at 𝑇 = 2.255 × 10 5 𝐾 . From the above results one may conclude that HvCs at thermal equilibrium may only exist in well defined of ranges of temperature, (depending on 𝐸, 𝑍 𝑎𝑛𝑑 𝑁 0 𝜌 ) and with dimensions 𝑙 ≫ 𝑙 𝑡ℎ𝑟 . .", "pages": [ 11, 12 ] }, { "title": "SUMMARY", "content": "Generally, the dissipation of acoustic waves by thermal conduction, dynamic viscosity. as well as, bulk viscosity can introduce drastic quantitative and qualitative changes in the equilibrium of the interstellar gas. Depending on the particular values of the metallicity, the mean radiation energy input, and the particle density: Strictly speaking, in any plasma, in particular, in a very complex interstellar medium (ISM) a (discrete as well as continuos, depending on the origin of the waves) spectral distribution for sound waves should be present. However, determining for which particular 𝜖𝑘 values of such spectrum the equilibrium (𝑋(𝜌, 𝑇, 𝜉) = 0 𝑎𝑛𝑑 ℒ (𝜌, 𝑇, 𝜉) =0) is modified by the waves dissipation is an important issue. We emphasize that this work is essentially restricted to a linear approximation, in which the wave dissipation is a second order phenomena (Landau et al.,1987), for this, the results depend on 𝜖𝑘 which allows one to get a \"first\" information on the dependence on the wave amplitude. On the other hand, turbulence is by itself a non-linear phenomena, a problem out the scope of the paper. Certainly, turbulence should be present in the different \"regions\" of the ISM, which is a typical (very complex) plasma where many relaxation time scales go into play. Models beyond the qualitative ones (Kolmogorov-like) (Braun et al.,2012) do not exist, and the understanding of turbulent plasmas under realistic physical conditions is an open question.", "pages": [ 12, 13 ] }, { "title": "Declaration of competing interest", "content": "The authors declare that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.", "pages": [ 13 ] }, { "title": "REFERENCES", "content": "Bird, G. A. 1964. A gas-dynamic model of the outer solar atmosphere. The Astrophysical Journal. 139(2):684-689. DOI: http://dx.doi.org/10.1086/147794 Braginskii, SI. 1965. Transport processes in a plasma. Reviews of Plasma Physics. 1:205-217. 30. Braun H. and Schmidt W., 2012. A semi-analytic model of the turbulent multi-phase interstellar medium, Mon. Not. R. Astron. Soc. 42:1838, https://doi.org/10.1111/j.1365-2966.2011.19889.x Corbelli, E., and Ferrara, A. 1995. Instabilities in photoionized interstellar gas. The Astrophysical Journal. 447(2):708-720. Collins, J. A.; Shull, J. M.; and Giroux M. L. 2004. Highly Ionized High-Velocity Clouds toward PKS 2155-304 and Markarian 509. The Astrophysical Journal 605(1):216 DOI: 10.1086/382269 Conde C. Sandra M. 2010. Calentamiento de plasmas por ondas hidrodinamicas. M. Sc. Thesis, University of Los Andes, Center for Fundamental Physics. Dalgarno A. and .McCray R. A. 1972. Heating and ionozation of HI regions, Ann. Rev. Astron. Astroph., 10, 375 -426. Fabian, AC., Sanders, JS., Allen, SW., Crawford, CS., Iwasawa, K., Johnstone, RM., Schmidt, RW. and Taylor, GB. 2003. A deep Chandra observation of the Perseus cluster: shocks and ripples. Monthly Notices of the Royal Astronomical Society. 344(3):L43-L47. DOI: https://doi.org/10.1046/j.1365-8711.2003.06902.x Fabian, AC., Reynolds, CS., Taylor, GB. and Dunn, RJH. 2005. Monthly Notices of the Royal Astronomical Society. 363(3):891-896. DOI: https://doi.org/10.1111/j.1365-2966.2005.09484.x Fajardo M. Jorge J.; Ibañez S. Miguel H, and Contreras E. Pedro L. 2021. An eigenvalue analysis of damping in optical thin plasmas.Rev. Mex. Fis. 67 061502 1-13 DOI: https://doi.org/10.31349/RevMexFis.67.061502 Ferland, GJ., Fabian, AC., Hatch, NA., Johnstone, RM., Porter, RL., Van Hoof, PAM. and Williams, RJR. 2009. Collisional heating as the origin of filament emission in galaxy clusters. Monthly Notices of the Royal Astronomical Society. 392(4):1475-1502. DOI: https://doi.org/10.1111/j.1365-2966.2008.14153.x Ibáñez, SMH. 2009. The stability of optically thin reacting plasmas: Effects of the bulk viscosity. The Astrophysical Journal. 695(1):479-487. DOI: https://doi.org/10.1088/0004-637X/695/1/479 Spitzer, L. 1990. Theories of the hot interstellar gas. Annual Review of Astronomy and Astrophysics. 28:71-101. DOI: https://doi.org/10.1146/annurev.aa.28.090190.000443 Spitzer, L. 1962. Physics of Fully Ionized Gases. Interscience, John Wiley and Sons Inc., New York. pp61. Stix, T. H. 1992. Waves in plasmas. Springer, Berlin. Tripp, T.M., Wakker, B.P., Jenkins, E.B., Bowers, C.W., Danks, A.C., Green, R.F., Heap, S.R., Joseph, C.L., Kaiser, M.E., Linsky, J.L., and Woodgate, B.E. 2003. Complex C: A Low-Metallicity, High-Velocity Cloud Plunging into the Milky Way. The Astronomical Journal, 125:3122-3144 Wakker, B. P. 2001. Distamces and metallicities of high- and intermediate- velocity clouds Astrophysical J. Supplement, 136:463-535 Wakker, B. P. 2004. High velocity clouds and the local group. IAU Symposium Series. 217:2", "pages": [ 13, 15 ] } ]

[ { "title": "Constructing 'Wavefunctions' for One-Body and Two-Body Gravitational Orbits in Classical Mechanics", "content": "Jixin Chen* Nanoscale & Quantum Phenomena Institute, Department of Chemistry and Biochemistry, Ohio University, Athens Ohio 45701", "pages": [ 1 ] }, { "title": "Abstract", "content": "The circular orbits and elliptical orbits of moving objects in a gravitational field are essential information in astronomy. There have been many methods developed in the literature and textbooks to describe these orbits. In this report, I propose to use the vis -viva equation to construct a complex function to store the state of a moving object in elliptical orbits such that one can calculate its near future numerically. This state function is constructed by splitting its momentum into real and imaginary parts with one perpendicular to the radius of the mass center and the other parallel. The wavefunctions of electrons of hydrogen atoms in quantum mechanics inspire this idea. The equations are derived for onebody problems. Two-body problems can be constructed with subsets of one-body problems with the same center of mass, but different effective mass pinned there, significantly different from existing methods and providing the same results.", "pages": [ 1 ] }, { "title": "Keywords", "content": "Classical mechanics, elliptic orbit, complex state functions, planet and satellite.", "pages": [ 1 ] }, { "title": "Introduction", "content": "Calculating the elliptic orbits of moving objects in a star system under gravitational forces is important to humankind with the greatest breakthrough many of us may agree to attribute to Isaac Newton's classical mechanics. The gravitational system is a similar condition to the electrons orbiting atomic cores sharing the same mathematical equation in forces being both dependent on one over square the distance between the objects. In the atomic system, the motions of electrons are described by wavefunctions proposed by Erwin Schrödinger which are sets of complex numbers. I was wondering if we can construct a 'wavefunction' for classical mechanics , particularly the motions of objects with gravitational forces with elliptical orbits, which I am giving a try in this report, and you can see it might have some advantages of doing so.", "pages": [ 1 ] }, { "title": "Orbits of one-body problems", "content": "The classical mechanics of planet orbits such as the Earth orbiting the Sun or satellites orbiting the Earth have been solved analytically e.g. with the vis -viva equation as reported in textbooks and literature. 1 -6 Due to the Sun 's massive mass relative to that of the Earth, this problem can be approximated as a onebody problem. The center of mass is very close to the center of the large object thus we can assume the large object does not move. The small object (mass m ) has potential energy due to the gravitational pull from the large object (mass M ) assuming obeying Newton's law of gravity when setting the potential energy at infinity to be zero as the reference point: where G is the gravitational constant and r is the distance between the centers of these two objects. The kinetic energy of the small object is: The total energy of the small object at any given condition at distance r is the sum of the potential energy and its kinetic energy: If E tot > 0, there is a range of angles of v for the object to hit the surface of the larger object which is a sphere in our example, and another range to be parabolic or hyperbolic escape orbits. If E tot < 0, the small object is caught by the gravitational field of the larger object whose orbit will be determined by the angle of v among which we are interested in the circular orbit and elliptic orbits. Under a special condition when the centrifugal force balances the gravitational force in the opposite direction, with a symmetry argument, we can see that there is only one angle and value of v for a circular orbit when half of its potential energy from infinite far is lost and the angle of the velocity is perpendicular to the radius of the orbit: We can compute the equilibrium radius ( R E) of this circular orbit to any object with a conserved energy E tot < 0, regardless of its current position: Under the special case when a stable circular orbit is formed, the angular momentum of the small object is, This is the maximum angular momentum of the small object with total energy E tot because the velocity at r = R E is perpendicular to the radius, the line connecting the small and the large object, 𝑣 ⊥ = 𝑣 ( Fig. 1A ). All the rest cases of E tot < 0, the small object follows an elliptic orbit because at R E when the values of the velocity are the same as their circular sibling, there is an angle deviate from that of the circular orbit. We can set a parameter 0 <= a <= 1 to describe the angular momentum of the small object, such that the angular momentum when it passes R E is: When a = 1, the object follows the circular orbit; when 0 < a < 1, the object follows elliptic orbits; and when a = 0 which should not exist, the small object free falls towards the center of the large object. This factor aligns the opposite trend with the eccentricity of the ellipse that has been widely used in the literature and textbooks to describe this elliptical orbit problem. 1 Compared to eccentricity, the angular momentum factor is more consistent with the idea in quantum mechanics that the angular momentum is quantized. We have learned from Ke pler's 2 nd law and Newton's law s of motion that the small object conserves this angular momentum , i.e. at any possible distance, At any given moment, if we assign the angles of the elliptic orbits with respect to the radius r as θ ( Fig. 1A ), we can see that using the results from the vis -viva equation, Thus, we conclude that at radius r with the angular momentum factor √𝑎 and total energy E tot, Thus, use the large object as the reference of frame, if we were able to measure the small object at any given time in space given its distance to the center of the large object, the mass of the large object, the velocity of the small object, and the angle with respect to the radius which defines the ecliptic plane, we can predict its entire orbit from this point of time into the future. We can combine the two parts of momentum into a single complex number to describe the state of the small object which naturally build in the angle of the orbits respected to the concentered rings: We can see that this wavefunction is a solution to 1 2𝑚 𝜓𝜓 ∗ = 𝐸 𝑘 = 𝐸 tot -𝐸 𝑉 for the orbits with conservation of angular momentum and conservation of total energy as the boundary conditions. Both Newton's 2 nd law and Equation 15 can be used to numerically simulate the motion ( Fig. 1B , 1C ). However, when using constant time step and the correlation ∆𝜃 = 𝑣 ⊥ ∆𝑡 𝑟 to calculate the rotation angles, the propagated error is larger than Newton's method because the step size near perigee is large. This reduced accuracy near perigee is general for both methods in simulating elliptical orbits with large eccentricity. The error is significantly reduced if take constant rotation angle step instead of constant time step in the simulation, within a range that the sampling density at apogee is not too small. In addition, the sign in Equation 15 is tuned manually for the two sides of the long semiaxis now which is somewhat inconvenient. When leaving apogee, the next step radius is approximated to, When leaving perigee, the next step radius is approximated to, Optimizing these two approximation equations could give a better simulation accuracy.", "pages": [ 2, 3, 4 ] }, { "title": "Orbits of two-body problems", "content": "For the two-body problem, a typical theory will set the center of mass (CoM) stationary over space and time as the reference point ( Fig. 2A ), 1,6 -8 where Thus, according to Newton's 3 rd law, the conservation of momentum gives the correlation between the motion of the two bodies ( Fig. 2B ): In both Newtonian mechanics and Lagrangian mechanics, the reduced mass is, which is used to calculate the relative motion between the two bodies. However, it is difficult to construct a rotation system with respect to the CoM using this reduced mass. We shall split the two bodies and create separated effective masses for the two subsystems in order to use the results of the one-body system we have constructed in the last section ( Fig. 2C ). For body m 1, the effective mass pinned at the CoM is, such that the force m 1 feels from this effective mass pinned at CoM equals the force it feels from m 2 all the time due to synchronized motion. And for mass m 2, the effective mass pinned at CoM is, We split the two-body problem into two one-body problems with the same pinned center but different effective masses. Setting the potential energy at infinite far to be zero, the potential energy is also split to be integration of two forces, which is consistent with integrating the single force over the distance ( Equation 1 ). Thus, with an initial angular momentum, the rotational motion between the two bodies can be split into two different one-body rotations with respect to the center of mass at distance r , e.g. for m 1, where R E is the equivalent circular orbit radius and 0 <= a 1 <= 1 is its angular momentum factor. The same set of equations can be derived for m 2, just switch the subscription 1 and 2. We can also derive from Equations 20 , 26 , and 27 that a 2 = a 1, which makes sense since both orbits have the same shape. The conservation of energy can be expressed as a Hamiltonian, Splitting a two-body problem into two separated one-body problems makes it easy to judge if the orbit is circular ( a = 1), ellipse ( a < 1), or nonstable ( E tot >= 0). Fig. 3 shows two simulations using Newton's laws of motion and the wavefunction method demonstrating consistency between these two methods. From Einstein's point of view, c onsidering it takes time for one subset to update the information to the CoM with a limit at the speed of light, the two subsets will have CoMs at different locations and the rotation could be affected to induce precessions, while there is no such problem for a one-body system. Because all real gravitational systems are at least two-body problems, this method of splitting the system into subsets could be beneficial. The effective masses at the CoM in the two-body problem remain unchanged over time. Many-body problems cannot be solved by simply putting an effective mass at the CoM for each particle because the other particles have an overall drag force that is perpendicular to its radius of motion to the CoM. In addition, although the overall angular momentum remains the same for the system but the distribution among particles changes over time. It is challenging to construct a wavefunction without calculating the overall force and its direction.", "pages": [ 4, 5, 6, 7 ] }, { "title": "Conclusion", "content": "In summary, constructing a wavefunction to a classical mechanic problem of elliptical orbits in a gravitational system yields surprising simplicity of solving the problem. This could be a new way to calculate complicated planets and satellites orbits in a different angle of view from the established methods. The basic idea is to construct a one-body problem with a pinned center and a force between the center and the object following the gravitational laws. Then for two-body and many-body problems, subsets of one-body problems can be constructed with different pinned centers that have different effective masses even if the centers overlap in different subsets, e.g. the center of mass which is often chosen as the reference frame. I believe this is a new idea that has not been reported before to the best of knowledge of mine and I hope you agree that it is interesting to think this way.", "pages": [ 7 ] }, { "title": "Author information", "content": "*Corresponding author. Email: [email protected] ORCID: 0000-0001-7381-0918", "pages": [ 7 ] }, { "title": "Supporting information available", "content": "MATLAB source code available at https://github.com/nkchenjx/TwoBodyProblem", "pages": [ 7 ] }, { "title": "Acknowledgment", "content": "Chen thanks his family for their support.", "pages": [ 7 ] }, { "title": "Author Declarations", "content": "The authors have no conflicts of interest to disclose.", "pages": [ 7 ] }, { "title": "Data Availability", "content": "All data is included in the main text and the supporting information.", "pages": [ 7 ] } ]

[ { "title": "Noname manuscript No.", "content": "(will be inserted by the editor)", "pages": [ 1 ] }, { "title": "Spin-Orbit Synchronization and Singular Perturbation Theory", "content": "Clodoaldo Ragazzo · Lucas Ruiz dos Santos Received: date / Accepted: date Abstract In this study, we formulate a set of differential equations for a binary system to describe the secular-tidal evolution of orbital elements, rotational dynamics, and deformation (flattening), under the assumption that one body remains spherical while the other is slightly aspherical throughout the analysis. By applying singular perturbation theory, we analyze the dynamics of both the original and secular equations. Our findings indicate that the secular equations serve as a robust approximation for the entire system, often representing a slow-fast dynamical system. Additionally, we explore the geometric aspects of spin-orbit resonance capture, interpreting it as a manifestation of relaxation oscillations within singularly perturbed systems. Keywords Deformable body · tidal evolution · averaging · spin-orbit resonance · singular perturbation", "pages": [ 1 ] }, { "title": "Preamble", "content": "This work is dedicated to the memory of Prof. Jorge Sotomayor, a teacher and friend. Unlike typical mathematical publications, this paper contains no theorems. Instead, it focuses on applications of methods in Ordinary Differential Equations (ODE), a field where, as CR heard from Prof. J. K. Hale, 'techniques such as averaging, normal forms, and challenges like the N-body Instituto de Matem'atica e Estat'ıstica, Universidade de S˜ao Paulo, 05508-090 S˜ao Paulo, SP, Brazil E-mail: [email protected] L.S. Ruiz (ORCID 0000-0002-5705-5278) Instituto de Matem'atica e Computa¸c˜ao, Universidade Federal de Itajub'a, 37500-903 Itajub'a, MG, Brazil E-mail: [email protected] problem, Hilbert's XVI problem, and the Lorenz equation, become crucial in research, overshadowing the established general theory.' CR had the honor of collaborating with Prof. Sotomayor for nearly two decades at the Instituto de Matem'atica e Estat'ıstica da Universidade de S˜ao Paulo, where our daily interactions were enriched by his humorous insights on life. More than just a brilliant mathematician, he was vivacious, joyful, and optimistic. He often shared a belief that 'for a mathematical field to flourish, it must engage with other sciences or mathematical areas'. Prof. Sotomayor's work in ODEs, a discipline rooted in Isaac Newton's efforts to solve physical and geometrical problems, significantly advanced both the theoretical aspects of ODEs through his studies on bifurcations and their practical applications, notably in differential geometry's lines of curvature. His students and friends hope that his legacy endures: to approach ODE with joy and happiness.", "pages": [ 1, 2 ] }, { "title": "1 Introduction", "content": "The foundations of differential equations trace back to Newton's pioneering work in mechanics and differential calculus. Newton grounded the law of gravitation mathematically and solved the equations for the motion of two bodies. However, the Newtonian model primarily considers celestial bodies as point masses, a simplification that has its limitations given that celestial entities have finite dimensions. Planets and substantial satellites exhibit a near-spherical shape. Despite being relatively minuscule compared to their respective diameters, the deformations induced by spin and tidal forces have a considerable impact, instigating significant alterations in both rotation rates and orbits. It is worth noting that all the major satellites within our solar system, including the Moon, operate in a 1:1 spin-orbit resonance (see, e.g., Murray and Dermott (2000)), they complete a single rotation on their axis for every orbit around the planet. Mercury, however, maintains a 3:2 spin-orbit resonance, undergoing three rotations on its axis for every two revolutions around the Sun. Furthermore, a majority of these celestial entities follows elliptical orbits characterized by low eccentricity. Deciphering how this dynamic state was attained, along with determining the associated time scales, holds substantial significance in the scientific realm. The goal of this study is to introduce equations to describe the perturbative impact of deformations on the motion of two spherical bodies influenced by gravitational interaction. Subsequently, we demonstrate that in certain limiting scenarios, which bear physical relevance, these equations can be analyzed using the mathematical apparatus of singular perturbations. The earliest and most basic deformation model accounting for energy dissipation was put forth by George Darwin Darwin (1879), son of the renowned biologist Charles Darwin. Darwin built upon previous studies Thomson (1863) concerning the deformation of an elastic, homogeneous, incompressible sphere, extending the results to address a body constituted of a homogeneous, incompressible, viscous fluid. Subsequent to Darwin, a significant advancement came with the introduction of Love numbers Love (1911). When the tidal force is decomposed in time via its Fourier components and in space through spherical-harmonic components, the Love number for a specific harmonic frequency and sphericalharmonic mode is a scalar that correlates the amplitude of the tidal force to the deformation's amplitude. Essentially, Love numbers act as functions within the frequency space, offering a phenomenological approach to elucidate forcedeformation relationships. Estimates of Love numbers can be derived from observational data. Over the past 70 years, there has been a prolific output of scientific literature focusing on the tidal effects on the motion of celestial bodies. While it is challenging to encompass the breadth of these studies, we will mention a few we are particularly acquainted with. Kaula Kaula (1964) evaluated the rate of change of the orbital elements using Love numbers for each harmonic mode (see Bou'e and Efroimsky (2019) and Efroimsky (2012) for further insights on the work of Kaula). Numerous other scholars have investigated equations accounting for deformations averaged over orbital motion. Some important works in this area are: Goldreich (1966), Singer (1968), Alexander (1973), and Mignard (1979) (low-viscosity scenarios); and Makarov and Efroimsky (2013), Ferraz-Mello (2013), Correia et al. (2014), Ferraz-Mello (2015b), and Bou'e et al. (2016), Folonier et al. (2018), Ferraz-Mello (2019), Ferraz-Mello et al. (2020), Ferraz-Mello (2021) (low and high-viscosity scenarios). In this paper, for simplicity while maintaining physical relevance, we make the following assumptions: The foundational equations for the orbit and rotation of the extended body are standard. Various equations exist in the literature detailing the deformation of extended bodies. We utilize the equations provided in Ragazzo and Ruiz (2017), without the term accounting for the inertia of deformations Correia et al. (2018). The reduced and averaged equations we introduce here are not novel. Excluding centrifugal deformations, they match those in Correia and Valente (2022). Our analysis parallels the approach in Correia et al. (2014), Section 5. The primary contributions of this paper include: We adopt the geometric method set out by Fenichel Fenichel (1971), Fenichel (1974), Fenichel (1977), Fenichel (1979), and Krupa and Szmolyan (2001b) without fully verifying all the assumptions. A comprehensive mathematical analysis of the equations presented may necessitate extensive research. The paper is structured as follows: In Section 2, we outline the core equations of the system. We assess the magnitude of various terms and introduce a parameter representing the minor nature of the deformations. In Section 3, we examine the limit when deformations approach zero, averaging them over orbital motion. This leads to equations with 'passive deformations' that do not influence the orbit. In Section 4, we suggest that for minor deformations, the primary equations possess an attracting invariant manifold matching the deformations from Section 3. This manifold's existence depends on the body's rheology. As the body becomes more viscous, the manifold becomes less attractive 1 . Given the enhanced spin-orbit coupling at high viscosity, assessing the credibility of our calculations and assumptions in this section presents a compelling mathematical challenge. In Section 5, we average the orbital and spin equations based on the preceding section's invariant manifold. Section 6 reveals that the averaged equations exhibit a slow-fast split. The fast variable is the body's spin, while the slower variables are orbital eccentricity and the semi-major axis. In Section 7, we delineate a condition for the folding of the slow manifold and provide a numerical illustration of its geometry. We also present a geometric interpretation of the dynamics within this manifold, emphasizing rapid spin transitions as instances of 'relaxation jumps' Mishchenko (2013), Krupa and Szmolyan (2001b). Section 8 concludes the paper, recapping the pivotal mathematical queries regarding the simplification of the initial equations and the dynamics of the reduced equations. This paper was written concurrently with a companion paper Ragazzo and Ruiz (2024), which has a more physics-oriented content. The focus of Ragazzo and Ruiz (2024) is on the implications for dynamics of using rheological models more complex than the one employed here.", "pages": [ 2, 3, 4 ] }, { "title": "2 The fundamental equations.", "content": "Let m 0 and m represent the masses of two celestial bodies, which could be a planet and a star, or a planet and a satellite, etc. The body with mass m 0 is treated as a point mass, while the body with mass m is always a small deformation of a spherical body with a moment of inertia I · . We assume that the deformations do not alter the volume of the body, implying that I · remains constant, a result attributed to Darwin Rochester and Smylie (1974). Often, we will refer to the bodies simply as the point mass and the body. For convenience, we write the deviatoric part of the moment of inertia matrix I in non-dimensional form: where 1 is the identity and b is a symmetric and traceless matrix. We denote matrices and vectors in bold face. The matrix b is termed the deformation matrix. Consider an orthonormal frame { e 1 , e 2 , e 3 } . We assume that the vector x , from the center of mass of the body to the point mass, lies in the plane spanned by { e 1 , e 2 } . The angular velocity of the body, ω , is perpendicular to the orbital plane, represented as ω = ω e 3 . The deformation matrix is given by: Under the given assumptions, Newton's equation for the relative position is expressed as: where it is assumed that in the region occupied by the body, the gravitational field of the point mass is accurately represented by its quadrupolar approximation. The spin angular momentum of the body is denoted by ℓ s = ℓ s e 3 , with the index s representing spin, and is defined as: In the context of the quadrupolar approximation, Euler's equation for the variation of ℓ s is: For a rigid body, a specific frame exists, known as the body frame, in which the body remains stationary and its angular momentum with respect to this frame is zero. Similarly, for a deformable body, there is an equivalent frame, called the Tisserand frame, where the body's angular momentum is null. The orientation of the Tisserand frame K := { e T 1 , e T 2 , e T 3 } with respect to the inertial frame κ := { e 1 , e 2 , e 3 } is given by and by definition, the rate of change of the angle ϕ is given by: To complete the set of equations (2.3) and (2.5), we require additional equations for the deformation matrices. These equations were derived within the Lagrangian formalism and utilizing what was termed the 'Association Principle,' as detailed in Ragazzo and Ruiz (2015), Ragazzo and Ruiz (2017) (see, also Gevorgyan et al. (2020) addressing the treatment of Andrade rheology, Ragazzo et al. (2022) extending to bodies with permanent deformation, and Gevorgyan (2021) and Gevorgyan et al. (2023) exploring the relations with the rheology of layered bodies). To maintain simplicity in mathematical expressions, we consider only the basic rheology of 'Kelvin-Voigt' combined with self-gravity here. The exploration of more generalized rheologies, which might introduce new time scales to the problem, is reserved for a companion paper Ragazzo and Ruiz (2024). The Tisserand frame of the body is the natural frame to present the equations for deformations. In this frame, the deformation matrix and the position vector are denoted by capital letters as follows: The governing equation for B is:", "pages": [ 4, 5, 6 ] }, { "title": "where:", "content": "where X ⊗ X is a matrix with entries ( X ⊗ X ) ij = X i X j . To determine the Love number function associated with the deformation equation (2.9), we consider a simple harmonic force term of the form where ̂ F is a complex amplitude matrix, and σ ∈ R is the constant forcing frequency. Assuming a solution of the form B ( t ) = ̂ B e σt , we derive the relationship between the complex amplitudes as where C ( σ ) is the complex compliance and The complex Love number k 2 ( σ ), commonly defined differently (see, e.g., Ragazzo and Ruiz (2017)), is proportional to the complex compliance C ( σ ) as outlined in Mathews et al. (2002) (paragraph 21): where the number k · := 3 G I · R 5 1 γ + α denotes the secular Love number, representing the value of k 2 ( σ ) for static forces ( σ = 0). In the case of a fluid body, the elastic modulus α is zero, and The body is held together solely by self-gravity. For a homogeneous fluid body of any density, k f = 3 / 2. As discussed in Ragazzo (2020), this represents the maximum possible value of k f when the density of the body increases towards the center. Given that for any non-null elastic rigidity α > 0, k f > k · , we conclude that for any stably stratified body, Historical note. Darwin was the pioneer in deriving equation (2.13), while examining tides on a homogeneous body composed of viscous fluid. In page 13 of Darwin (1879), Darwin stated: 'Thus we see that the tides of the viscous sphere are the equilibrium tides of a fluid sphere as cos ϵ : 1, and that there is a retardation time ϵ σ '. In his paper, ν denotes fluid viscosity, and tan ϵ = 19 2 ν gRρ σ , where g represents surface gravity, and ρ is the mass per unit volume of the body. Given that for a homogeneous fluid body k · = k f = 3 / 2, Darwin's statement can be reformulated as Utilizing the relationships for a homogeneous spherical body, I · = 2 5 mR 2 , g = Gm R 2 , and ρ = m/ 4 πR 3 3 , where m is the mass and R is the radius of the fluid body, and from the relations k · = k f = 3 2 = 3 I · G R 5 1 γ and τ = η γ = 19 2 ν gRρ , we deduce which aligns with a relation in (Correia et al., 2018, Eq. (39)). The theory developed by Darwin Darwin (1879), Darwin (1880) has predominantly been applied in the frequency domain. Influenced by Darwin's work, Ferraz-Mello Ferraz-Mello (2013) formulated an equation for the motion of the surface of the body under tidal forcing in the time domain. When α = 0, the model in Correia et al. (2014) with τ e = 0, the model in FerrazMello (2013), and equation (2.9) are all equivalent (our τ corresponds to the τ in Correia et al. (2014), which is equal to the parameter '1 /γ ' used in FerrazMello (2013)). See Correia et al. (2014), paragraph above equation (90), and Ferraz-Mello (2015a) for the equivalence between the models in Ferraz-Mello (2013) and Correia et al. (2014).", "pages": [ 7, 8 ] }, { "title": "3 Zero deformation limit.", "content": "In numerous celestial mechanics problems, bodies maintain near-spherical shapes at all times, which can be reformulated as Given that equation (2.9) for B is linear, ∥ B ∥ is small if, and only if, ∥ F ∥ is small. The relative motion between two nearly spherical bodies approximates Keplerian motion. Let a , n , and e represent the semi-major axis, the mean motion (period/(2 π )), and the eccentricity of the Keplerian ellipses, respectively. The magnitude of the force terms in the deformation equation (2.9) is proportional to the following characteristic frequencies: The forces on the right-hand side of equation (2.9) are counteracted by the body's self-gravity and possibly elastic rigidity α ≥ 0. The static deformations are then given by where we used k · := 3 G I · R 5 1 γ + α . The order of magnitudes in equation (3.19) and inequality (2.15) imply This indicates that the region in phase space defined by the following inequalities: adheres to the small deformation hypothesis.", "pages": [ 8, 9 ] }, { "title": "3.1 The Zero Deformation Limit", "content": "Define the compliance ϵ d , where d denotes deformation, as follows: We then express and substitute into equations (2.3), (2.4), (2.5), and (2.9) to yield τ ˙ ˜ B + ˜ B = F where τ is defined in (2.12) and ˜ b = R ( ϕ ) ˜ BR -1 ( ϕ ). The zero deformation limit is defined by: In the zero deformation limit, equation (3.24) simplifies to: In this scenario, the body spin, ω , remains constant and x follows a Keplerian ellipse. To describe the Keplerian orbits, we change from variables ( x , ˙ x ) to ℓ ∈ R (orbital angular momentum), A (the Laplace vector), and f (the true anomaly), defined as: where The Laplace vector is normalized such that ∥ A ∥ = e is the orbital eccentricity and it points towards the periapsis, where ∥ x ∥ is minimized. The three vectors constitute an orthonormal basis, expressed in terms of the inertial frame basis vectors as Here, ϖ denotes the longitude of the periapsis, the angle between e A and e 1 . The orbit is represented by where R is the rotation matrix about the axis e 3 , as given in equation (2.6), and r ( t ) = ∥ x ( t ) ∥ .", "pages": [ 9, 10 ] }, { "title": "3.2 Passive deformations.", "content": "The equations at the zero deformation limit (3.26) in the new variables become (see, e.g., Murray and Dermott (2000) for details): where C and S are given in equation (2.10). In order to write S in a convenient way, we define the matrices with Y -2 = Y 2 , where the overline represents complex conjugation. These matrices have a simple transformation rule with respect to rotations about the axis e 3 , namely Using the tidal-force matrix in equation (2.10) can be written as In the basis { Y -2 , Y 0 , Y 2 } that implies In equation (3.38), the variables r , f , and ϕ = ωt are dependent on t . To solve the equation τ ˙ ˜ B + ˜ B = C + S , we do a harmonic analysis of the tidal force in equation (3.38) using: where M denotes the mean anomaly, ˙ M = n , and X n ' ,m k ( e ) is termed the Hansen coefficient. Equations (3.38) and (3.39) imply: where U k, -1 = U k, 1 = 0 and The symmetry property X n ' , -m -k = X n ' ,m k implies The centrifugal force in equation (2.10) can be represented as To obtain the almost periodic solution of the deformation equation solving for each Fourier mode separately suffices. An alternative approach involves using the variation of constants formula: Here, the definitions of the Love number k 2 and the secular Love number k · from equation (2.13) are used as well as the definitions of ζ c and ζ T from equation (3.21). Given that this formula indicates that the almost periodic solution of the tide equation is a time-averaged tidal force with an exponential weight decaying towards the past, characterized by time τ . Note that when τ > 0 is nearly zero, integration by parts of the right-hand side of equation (3.45) yields This represents the usual time delay approximation with corrections of the order of τ 2 . The limit case of τ →∞ also presents interest. Here, we can interpret the averaging in equation (3.45) as approximately the ordinary averaging", "pages": [ 10, 11, 12, 13 ] }, { "title": "4 Deformation Manifold.", "content": "The function t → B d provides a solution to the deformation equation (3.44) only when ϵ d = 0. To analyze the case where ϵ d > 0, we introduce new deformation variables δ B : and using these variables we write equation (3.24) τ δ B + δ B = ( ϵ d ) . For ϵ d = 0, equation (3.26) possesses the invariant manifold: The variables δ B are transversal to Σ 0 , and all associated eigenvalues equal -1 /τ < 0. Given this, a theorem by Fenichel (Fenichel, 1971, Theorem 3) suggests that for sufficiently small ϵ d , there is an invariant manifold represented as a graph: Additionally, Σ ϵ d approximates Σ 0 to order ϵ d , as visualized in Figure 1. The vector field on Σ ϵ d , considering corrections of order ϵ d , is derived from equations (4.48) by ignoring the variables δ B and setting ˜ B = B d in the equations for ˙ x and ℓ . Thus, the equation on Σ ϵ d is: where, b d = R ( ϕ ) B d R -1 ( ϕ ). The Fenichel theorem requires a specific condition concerning the eigenvalues of the linear equation: they must be sufficiently distant from the imaginary axis, depending on the flow on Σ 0 , which is fulfilled in this case since they are constant. When n and ω are neither small, to ensure the validity of the averaging, nor excessively large, which would violate inequalities (3.21) and result in large deformations, the approximation of Σ 0 by Σ ϵ d remains accurate. Under these conditions, changes in the Keplerian elements and spin are gradual, allowing the body ample time to adjust. The body maintains an average shape consistent with its secular Love number; for α = 0, it remains in hydrostatic equilibrium, countering centrifugal forces and slow tides. An intriguing scenario arises when either τn ≫ 1 or τω ≫ 1. Here, the body lacks the time to relax amid orbital and spin modifications, causing the deformation to retain a memory of a past initial state. In such situations, Fenichel's theorem is not applicable. If τ ≫ 1 and the initial condition is ˜ B = ˜ B · , the solution to the homogeneous equation τ ˙ ˜ B + ˜ B = 0 decays slowly as In Ragazzo et al. (2022), in a situation similar to this one, we added a permanent deformation ˜ B · to B d and continued. Adopting the same approach here is feasible, even without a mathematical basis. However, we must separate the orbital motion's averaging into two components: one for terms with B d and another for terms with ˜ B · . The averaging of terms associated with ˜ B · would resemble the averaging in rigid body problems. Here, we will not introduce the permanent deformation to keep the following analysis as simple as possible. Later in this paper, we'll explore situations where τn is large, assuming that, despite its size, Fenichel's conditions remain met. This assumption war- ants further mathematical scrutiny, potentially through multi-timescale system theories.", "pages": [ 13, 14, 15 ] }, { "title": "5 Orbital Averaging", "content": "We average equation (4.51) with respect to orbital motion. We set the scaling parameter ϵ d to 1. Equations (4.51) and (3.45) then become: Using variables ℓ , A , and f defined in equations (3.27) and (3.28), equation (5.53) transforms to: The terms requiring averaging are: where ⟨ h ⟩ = 1 2 π ∫ 2 π 0 h ( M ) dM represents the average over the mean anomaly. The total angular momentum is conserved and given by: The averaged result yields: The term E 1 : where we used, from equation (2.13), that k 2 ( -σ ) is the complex conjugate of k 2 ( σ ), represented as k 2 ( σ ). We write E 1 as The terms ( -5 2 E 2 + E 3 ) : The calculation of these terms resembles that of E 1 . The analysis was extended and performed using the software 'Mathematica'. We will skip the detailed steps. The outcomes are: (5.59) and where we used equations (3.30). The term E 4 : Detailed steps are omitted as before. The outcomes are: and The term ⟨ b d 33 ⟩ : where we used that X -3 , 0 0 = (1 -e 2 ) -3 / 2 Laskar and Bou'e (2010) 2 . For the Kepler problem, the following relations hold: Assuming ℓ > 0, we can use G ( m 0 + m ) = n 2 a 3 to write: Using the above relations, further calculations yield: Given that A = e ( cos ϖ e 1 +sin ϖ e 2 ) = e e A and ˙ e A = ˙ ϖ e H , we deduce: Thus, the final averaged equations are: (5.71)", "pages": [ 15, 16, 17, 18 ] }, { "title": "5.1 Computation of Hansen coefficients", "content": "The Hansen coefficients depend solely on the eccentricity. Following Cherniack (1972), we express, for n < 0 and m ≥ 0, Thus, to compute any series ( r a ) n e imf , one can employ series multiplication of the fundamental series of a/r and e if . This multiplication can be efficiently executed with an algebraic manipulator. For the computation of the series for a r and e if , one can refer to Murray and Dermott (2000) Section 2.5: where J k ( x ) denotes the Bessel function. The series for J k ( x ) converges absolutely for all values of x . Up to second order in eccentricity and with e iM = z the fundamental series are: These expressions and equation (5.72) imply X n,m k = O ( e | m -k | ).", "pages": [ 19 ] }, { "title": "5.2 The equations in Correia and Valente (2022)", "content": "Some relations between the Hansen coefficients presented in Correia and Valente (2022), equations (158) and (159), are: One can use these equations to simplify equations (5.71). After such simplifications, the equation governing the eccentricity is: For further simplification, one can apply ℓ = µ √ G ( m + m 0 ) a (1 -e 2 ), yielding: This result corresponds to equation (129) in Correia and Valente (2022). Our expression for the variation of the longitude of the periapsis, ˙ ϖ , differs from equation (130) in Correia and Valente (2022) due to the neglect of centrifugal deformation in the cited work.", "pages": [ 20 ] }, { "title": "6 Averaged Equations: A Geometrical Approach", "content": "In the following two sections, we analyze equation (5.71) from a geometric perspective using singular perturbation theory. The longitude of the periapsis, ϖ , is absent from the equation for ˙ e in (5.71). Therefore, the dynamics of the state variables e, ℓ, and ℓ s can be analyzed independently of ϖ . The conservation of total angular momentum, ℓ T = ℓ + ℓ s , implies that it is sufficient to observe the dynamics of e and ℓ s . While the dynamics unfolds within two-dimensional surfaces, on the level sets of angular momentum, analyzing the equations within a three-dimensional phase space proves more insightful. This approach facilitates a comprehensive understanding of the global dynamics and the impact of varying angular momentum. After some investigation, we selected ( ω, e, a ) as the phase-space variables, with n = √ G ( m + m 0 ) a 3 being a derived quantity. The differential equation for a = ℓ 2 µc (1 -e 2 ) is obtained from the equations for ˙ ℓ and ˙ e . Henceforth we use the approximation Equations (5.71) and the identity 3 c mn 2 a 3 = 3 m 0 m + m 0 imply Conservation of angular momentum ℓ T = √ µca √ 1 -e 2 + I · ω implies This suggests the following nondimensionalization of a : is defined as the radius of the circular orbit for two point masses, m 0 and m , possessing an orbital angular momentum of ℓ = ℓ T . Let be the angular frequency of the circular orbit of radius a · . Kepler's third law implies, n 2 a 3 = G ( m + m 0 ) = n 2 · a 3 · and so Conservation of angular momentum, as expressed in equation (6.79), implies where For the Mercury-Sun system, where m 0 is the mass of the Sun, ϵ = 6 . 8 × 10 -10 , and for the Earth-Moon system, where m 0 is the mass of the Moon, ϵ = 0 . 0036. Although ϵ appears to be very small for all problems of interest, in this section, we will conduct a geometric analysis with an arbitrary value of ϵ to elucidate the global properties of the equations. Using the above definitions equations (6.78) can be written in nondimensional form as where µ , c , A 0 , A 2 , and A 4 are given in equation (5.71). 6.1 Estimate of the Rate of Spin Variations. In a time scale where the unit of time corresponds to one radian of orbital motion, the spin angular velocity is ω/n , and, from equation (6.78), the rate of change of spin is where From equation (2.13) and we can express For a fixed pair ( e, n ), ˙ ω n 2 = V ( τn, ω n , e ) defines a differential equation for ω n . We aim to estimate two typical quantities associated with V : its maximum and the time constant near a stable equilibrium, as depicted in Figure 2. The maximum value of the function σ → | σ | 1+ σ 2 is 1 2 . Hence, Applying Parseval's identity, we get Based on Laskar and Bou'e (2010), X -6 , 0 0 = 3 e 4 8 +3 e 2 +1 (1 -e 2 ) 9 / 2 leading to The right side of this inequality increases with e , with values: 1 / 2 for e = 0, approximately 1 . 6 for e = 0 . 4, approximately 3 . 3 for e = 0 . 5, and approximately 8 for e = 0 . 6. Since m 0 m + m 0 ≤ 1, we deduce It is worth noting that ζ T , defined in equation (3.21), is a small quantity. For sufficiently large values of τn , the stable equilibria of ω n are close to semi-integers k 2 , with k = 1 , 2 , . . . , and for these values, the dominant term in the sum of V is the k th -term Correia et al. (2014). Thus, equation (6.87) yields the time constant for an equilibrium ω n ≈ k 2 . Note that V max is independent of the characteristic time of the rheology τ , whereas the time constant τ k has a linear dependency. A maximum rate speed V max proportional to ζ T k · will be observed during spin jumps. The prefactor 10 in equation (6.91) varies with the eccentricity e . 6.2 Equilibria, Linearization and the Invariant Subspace of Zero Eccentricity. Using the expresions for the Hansen coefficients in Section 5.1 we can compute the expansion of the right-hand side of equation (6.85) up to first order in eccentricity: These equations imply that the plane e = 0 is invariant. The only equilibria of equations (6.85) are on the plane e = 0, as shown in the next paragraph, and are given by the curve The equilibria of (6.85) satisfiy A 0 = 0 and 1 -e 2 2 A 2 + A 4 = 0. Equation (5.76) shows that these equations imply ̸ Wenotice, from (6.86), that x Im k 2 ( x ) < 0 ∀ x = 0 and hence (6.95) holds if and only each term of the sum is zero. The Hansen coefficients have the following properties: ∀ k = 0, X -3 , 0 k ( e ) = 0 ⇐⇒ e = 0 and ∀ k = 2, X -3 , 2 k ( e ) = 0 ⇐⇒ e = 0. This implies that e = 0 is a necessary condition for the existence of an equilibrium. ̸ ̸ Conservation of angular momentum implies that the orbits of the vector field (6.93) in the plane where e = 0 are parameterized by angular momentum. Equation (6.83) shows that the representation of these orbits in the plane (˜ a, ω n ) is given by the graphs as illustrated in Figure 3. In the significant case where ϵ ≈ 0, equation (6.83) suggests that 0 ≈ ϵ ω n = ˜ a 3 2 (1 -˜ a 1 2 √ 1 -e 2 ). Up to first order in eccentricity, we have ˜ a = 1 and n = n · . Using this approximation, the function ˙ e e in equation (6.93) is expressed as where n = n · = constant and ˜ c is a positive, although small, constant. The graph of ˙ e e ˜ c as a function of ω n for various values of τn is depicted in Figure 4. This figure illustrates that ˙ e e changes sign near the plane e = 0. Consequently, a solution with an initial eccentricity close to zero, yet sufficiently distant from the stable equilibrium at ω n = 1, may experience an increase in eccentricity.  The next step in understanding the dynamics of equation (6.85) involves linearization about the equilibria. It is evident from Figure 3 that the equilibria can be parameterized by their ˜ a coordinate. Thus, an equilibrium is represented by ( ω, ˜ a ) = ( ω e , ˜ a e ), where, according to equation (6.83), ˜ a e is the solution to The special equilibrium ˜ a e = 9 16 , corresponding to the bifurcation value ϵ = 27 256 , marked by the black dot in Figure 3, represents a threshold of stability: an equilibrium with ˜ a e < 9 16 is unstable, while an equilibrium with ˜ a e > 9 16 is stable. Given that 0 < ϵ < 27 256 ≈ 0 . 1, a perturbative calculation reveals that the largest root of this equation (stable equilibrium) satisfies This approximation remains accurate up to ϵ = 0 . 05. At equilibrium, the orbit is circular. If ℓ e = ℓ T -I · n e denotes the orbital angular momentum at equilibrium, then a e = ℓ 2 e µc . Since a e = ˜ a e a · and a · = ℓ 2 T µc , we obtain Thus, ˜ a e represents the square of the ratio of orbital angular momentum to total angular momentum at equilibrium. For the Mercury-Sun system, where m 0 is the mass of the Sun, ϵ = 6 . 8 × 10 -10 and ˜ a e ≈ 1. For the Earth-Moon system, where m 0 is the mass of the Moon, ϵ = 0 . 0036 and ˜ a e = 0 . 993. It appears that in most problems of interest, ˜ a e ≈ 1. The linearization of equation (6.85) at ( ω, ˜ a, e ) = ( ω e , ˜ a e , 0) is derived easily from equation (6.93): where ˜ n e = n 0 1 ˜ a 3 / 2 e . Each equilibrium has: one eigenvalue equal to zero, associated with the conservation of angular momentum; one negative eigenvalue λ e = -7˜ n e ϵ ˜ a 2 e (˜ n 2 e τ 2 +1) ( k · N ˜ n e τ ˜ a 3 e ) , with an eigenvector tangent to the eccentricity axis; and one eigenvalue λ 0 = -2˜ n e (˜ a 2 e -3 ϵ ) ˜ a 2 e ( k · N ˜ n e τ ˜ a 3 e ) , with an eigenvector in the plane e = 0 and tangent to the surface of constant angular momentum. As expected, λ 0 = 0 in the critical case where ϵ = 27 256 and ˜ a e = 9 16 , λ 0 > 0 if ˜ a e < 9 16 , and λ 0 < 0 if ˜ a e > 9 16 . Consider a solution to equation (6.93) that satisfies lim t →∞ ( e ( t ) , ˜ a ( t ) ) = ( 0 , ˜ a e ) , and let δ a ( t ) = ˜ a ( t ) -˜ a e . At a certain time ˜ t , this solution is sufficiently close to (0 , ˜ a e ) for the linear approximation to be valid. Since δ a ( t ) = e λ 0 ( t -˜ t ) δ a ( ˜ t ) and e ( t ) = e λ e ( t -˜ t ) e ( ˜ t ), we conclude that near the equilibrium, where Regardless of the value of the constant factor in equation (6.102), which in Figure 5 we assume to be one, the orbit's geometry near the equilibrium is controlled by the ratio λ 0 λ e . In Figure 5 LEFT, we illustrate how the orbit changes as λ 0 λ e varies, with the ratio λ 0 λ e = 1 being a critical value. For λ 0 λ e > 1, the orbit approaches the equilibrium along the e -axis, and for 0 < λ 0 λ e < 1, the orbit approaches the equilibrium along the δ a axis. In Figure 5 RIGHT, we demonstrate how to determine the special value of ˜ a e , corresponding to λ 0 λ e = 1, as a function of the parameter τ ˜ n e . The maximal value of this special ˜ a e is 169 225 ≈ 0 . 75, achieved when τ = 0. It appears that in most problems of interest, ϵ is very small, ˜ a e ≈ 1, and λ 0 /λ e ≫ 1, indicating that solutions approach the stable equilibrium along the e -axis, namely the weak-stable manifold of the equilibrium. 6.3 Slow-fast systems and singular perturbation theory For ϵ ≈ 0 equation (6.93) has the form of a slow-fast system: with x = ω ∈ R as the fast variable and y = ( e, ˜ a ) ∈ R 2 as the slow variables Fenichel (1979). Given an initial condition in the state space { ω, e, ˜ a } , the value of ω varies while ( e, ˜ a ) stays nearly constant until the state reaches the slow manifold where A 0 is given in equation (5.71). ̸ When ( x, y 0 ) is not close to Σ s (0), the fast dynamics is governed by the layer problem, ˙ x = f ( x, y 0 , 0). Here, the fast dynamics corresponds to the fast spin variation with fixed e and ˜ a . The spin decreases on points above Σ s (0) and decreases on points under Σ s (0), see Figure 10. Close to the slow manifold Σ s (0), the dynamics is approximated by the reduced problem, where the fast variable is given by an implicit function, solution of f ( Φ ( y ) , y, 0) = 0, and the slow variable solves the differential equation on Σ s (0), ˙ y = g ( Φ ( y ) , y, 0). The implicit function theorem ensures that Φ is locally determined at ( x 0 , y 0 ) ∈ Σ s if ∂ x f ( x 0 , y 0 , 0) = 0. In this case, Σ s (0) is called normally hyperbolic at ( x 0 , y 0 ). The results from geometric singular perturbation theory Fenichel (1979) state that if the system (6.104) has a normally hyperbolic slow manifold S 0 , for each small ϵ > 0 exists an invariant manifold S ϵ diffeomorphic to S 0 which is stable (unstable) if ∂ x f < 0 ( ∂ x f > 0) on S 0 . We will denote by Σ s ( ϵ ) the union of the hyperbolic components of perturbed slow manifold in (6.105). The dynamics across the entire phase space can be elucidated by examining the geometry of the slow manifold (6.105). Within the first octant B 1 := { ω > 0 , e > 0 , a > 0 } , Σ s (0) possesses a single connected component that splits B 1 into two regions. The conservation of angular momentum reduces the analysis to a two-dimensional problem. A diagram illustrating the local behavior of orbits near the stable equilibrium is presented in Figure 5 LEFT. A global illustration of the flow on a level set of angular momentum is shown in Figure 6.", "pages": [ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 ] }, { "title": "7 Spin-Orbit Resonances", "content": "In this section, we assume that the ratio ω n is at most on the order of tens, so that Under this condition, equation (6.83), i.e., ˜ a 3 2 (1 -˜ a 1 2 √ 1 -e 2 ) = ϵ ω n , yields two solutions for ˜ a . The first solution is ˜ a = ( ϵ ω n ) 2 / 3 + O ( ϵ ). This solution closely approximates the surface of constant angular momentum in a region that includes the unstable equilibrium ˜ a e ≈ 0. This approximation is depicted in Figure 3 by the nearly vertical red dot-dashed line near ˜ a e ≈ 0. We will not focus on this region. The second solution is ˜ a = 1 1 -e 2 + O ( ϵ ), which is of primary interest. This solution approximates the surface of constant angular momentum in a region containing the stable equilibrium ˜ a e ≈ 1. This approximation is depicted in Figure 3 by the nearly vertical red dot-dashed line near ˜ a e ≈ 1. Disregarding the error of order ϵ , we have ˜ a e = 1, a e = a · , and In the subsequent analysis we use these approximations. The geometry of the slow manifold Σ s (0) plays a crucial role in the capture into spin-orbit resonance, particularly where Σ s (0) is not normally hyperbolic. The slow manifold becomes non-normally hyperbolic at points where the projection map from Σ s (0) to the { a, e } plane is singular. These generic singular points of the projection are known as folds and collectively form the 'fold curves'. In Figure 7, the fold curves are depicted in blue on the slow manifold Σ s (0), which is represented as an orange surface. Although the fold curves themselves are smooth, their projection onto the { a, e } plane includes singular points termed 'cusps', at which a moving point on the projection reverses direction. A cusp point on a fold curve occurs where the tangent to the curve becomes parallel to the ω -axis. The flow dynamics near a fold are extensively described in the literature Krupa and Szmolyan (2001a). We illustrate the so called phenomenon of capture into spin-orbit resonance by a concrete example presented in Correia et al. (2014) and Correia et al. (2018). We use the parameters of the exoplanet HD80606b and its hosting star, namely m 0 = 2008 . 9 · 10 30 kg, m = 7 . 746 · 10 28 kg, I · = 8 . 1527 · 10 40 kg m 2 . The initial conditions are chosen as a = 0 . 455au, e = 0 . 9330 and ω = 4 π rad / day and hence ϵ = 1 . 35 · 10 -8 . The parameters of the rheology are k · = 0 . 5 and τ = 10 -2 year. In Figures 8 (top panels), the red curve represents a trajectory of the fundamental equations, given in Section 2, which was obtained by means of numerical integration. The numerically computed trajectory has consecutive transitions between stable branches of the perturbed slow manifold Σ s ( ϵ ). This trajectory shows a slow decrease of the eccentricity towards e = 0 while the spin-orbit ratio has fast transitions between integers and half-integers with final value ω/n = 1. The stable branches of Σ s ( ϵ ) are quite flat (parallel to the ( e, a )-plane) near the planes ω n = k 2 , k ∈ Z . These results are detailed in Figure 14 from Correia et al. (2018). We can observe in Figure 8 the full agreement between the solution of the fundamental equations and the fast-slow-geometric analysis of the averaged equations. The projection of the fold curves to the plane ( a, e ) are shown in Figure 8 DOWN-RIGHT. Each curve contains a cusp singularity and is labeled by an integer or half-integer. A point initially over ( a, e ) can be attracted to a resonance ω n = k 2 only if it is inside a dashed curve that intersects the curve labeled by k 2 ; see caption of Figure 8 for further information.", "pages": [ 29, 30 ] }, { "title": "7.1 Spin-Orbit Resonances Requires Large Relaxation Times τ .", "content": "The approximation ˜ a = (1 -e 2 ) -1 and equation (6.82) imply n = n · (1 -e 2 ) 3 / 2 . The imaginary part of the Love number (6.86) can then be written as where For e = 0, the slow manifold lacks any fold points for any value of τ > 0, as illustrated in Figure 3. Consider a fixed value e 1 > 0 for e . Equations (6.87) and (7.108) imply the existence of at least j -1 fold points in the region { 0 < e < e 1 , 0 < ω n < C } , where C > 0 represents a positive constant, if and only if ̸ has j zeroes for ω n ∈ (0 , C ). Given that X -3 , 2 k (0) = 1 and X -3 , 2 k ( e 1 ) = O ( e 1 ), function (7.110) can be expressed as ( ω/n -1) ˜ ϵ +( ω/n -1) 2 + O ( e 2 1 ). For 0 < ω n < C and a fixed ˜ ϵ > 0, this function exhibits a single zero near ω n = 1 if e 1 > 0 is sufficiently small. Furthermore, for a fixed e 1 > 0 and ˜ ϵ = 0, function (7.110) presents poles for every ω n = k 2 , k ∈ Z , thereby ensuring at least one zero in each interval ( k, k + 1 2 ), where k is any half-integer. A continuity argument suggests that if ˜ ϵ is sufficiently close to zero (implying τ is sufficiently large), then for any fixed e 1 , function (7.110) will have zeroes near j/ 2, for j = 1 , 2 , . . . . This analysis indicates that, particularly for small e 1 > 0, the condition ˜ ϵ ≪ 1 (equivalently, τ ≫ 1) is a necessary condition for the creation of folds in the slow manifold Σ s (0). For the Earth-Moon system, where m 0 is the mass of the Moon, ϵ = 0 . 0036 and n -1 · = 7 . 6 days, a value τ > 76 days gives ˜ ϵ < 0 . 0025. For the MercurySun system, where m 0 is the mass of the Sun, ϵ = 6 . 8 × 10 -10 and n -1 · = 13 days, a value τ > 130 days gives ˜ ϵ < 0 . 0025. In the case of the parameters chosen for HD80606b, ˜ ϵ ≈ 1 . 28 · 10 -5 . For ˜ ϵ ≪ 1 and close to a resonance ω/n = j/ 2, j ∈ { 2 , 3 , . . . } , Σ s (0) can be approximately computed as a power series in ϵ . If we substitute into the equation and solve the resulting equation for the coefficient of ˜ ϵ and ˜ ϵ 2 we obtain, We emphasize that the functions Φ ( j/ 2 , e, ˜ ϵ ) represent the O ( ϵ 0 ) approximations of the slow invariant manifold Σ s ( ϵ ). These functions determine the dynamics of the reduced system, serving as the initial step in comprehending the flow on Σ s ( ϵ ). Further exploration of this flow constitutes a subject for future work. Figure 9 illustrates the approximation of Σ s (0) on some resonances. We end this section with a topological description of the slow-fast dynamics of equation (6.85). In Figure 10 we present a sketch of flow lines for ϵ = 0 (LEFT panel) and ϵ > 0 small (RIGHT panel). Explanations are given in the Figure caption. The orientation of the fast flow lines was previously examined in Section 6.1. The orientation of the slow flow lines is determined by the monotonic decrease in eccentricity on Σ s (0). This is a consequence of the same argument employed to determine the equilibria, as presented in equation (6.95).", "pages": [ 31, 32, 33 ] }, { "title": "8 Conclusion", "content": "In this paper, we presented a set of equations for the evolution of the orbital elements in the gravitational two-body problem under the influence of tides. These equations, previously obtained by other authors, were derived here through a two-step procedure. Initially, we used the fact that tidal deformations are very small to demonstrate the existence of an invariant manifold, which we have termed the deformation manifold. Although our arguments are mathematically sound, they lack the appropriate quantifiers. The second step involves averaging the equations on the deformation manifold. This step is contingent upon the first, leading to uncertainties about whether the averaged equations are mathematically coherent with the large values of τn used in Section 7. In the physics literature, employing large values of τn in the averaged equations has been common practice. Analyzing the averaged equations mathematically presents a significant challenge due to the analytical complexity of the vector field, defined by infinite sums of Hansen coefficients, which are themselves infinite series in powers of eccentricity. Given the scientific significance of this problem, it warrants investigation from a mathematical perspective. The geometric theory of singular perturbation, potentially incorporating multiple time scales as suggested in our companion paper Ragazzo and Ruiz (2024), appears to be a suitable mathematical framework to address this challenge.", "pages": [ 34, 35 ] }, { "title": "Acknowledgements", "content": "C.R. is partially supported by by FAPESP grant 2016/25053-8. L.R.S. is supported in part by FAPEMIG (Funda¸c˜ao de Amparo 'a Pesquisa no Estado de Minas Gerais) under Grants No. RED-00133-21 and APQ-02153-23.", "pages": [ 35 ] }, { "title": "Conflict of interest", "content": "On behalf of all authors, the corresponding author states that there is no conflict of interest.", "pages": [ 35 ] }, { "title": "References", "content": "ME Alexander. The weak friction approximation and tidal evolution in close binary systems. Astrophysics and Space Science , 23:459-510, 1973. Gwenael Bou'e and Michael Efroimsky. Tidal evolution of the Keplerian elements. Celestial Mechanics and Dynamical Astronomy , 131:1-46, 2019. Gwenael Bou'e, Alexandre CM Correia, and Jacques Laskar. Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology. Celestial Mechanics and Dynamical Astronomy , 126(1-3):31-60, 2016. JR Cherniack. Computation of Hansen coefficients. SAO Special Report , 346, 1972. ACM Correia, C Ragazzo, and LS Ruiz. The effects of deformation inertia (kinetic energy) in the orbital and spin evolution of close-in bodies. Celestial Mechanics and Dynamical Astronomy , 130(8):51, 2018. Alexandre CM Correia and Ema FS Valente. Tidal evolution for any rheological model using a vectorial approach expressed in hansen coefficients. Celestial Mechanics and Dynamical Astronomy , 134(3):24, 2022. Alexandre CM Correia, Gwenael Bou'e, Jacques Laskar, and Adri'an Rodr'ıguez. Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology. Astronomy & Astrophysics , 571:A50, 2014. George Howard Darwin. I. On the bodily tides of viscous and semi-elastic spheroids, and on the ocean tides upon a yielding nucleus. Philosophical Transactions of the Royal Society of London , 170:1-35, 1879. GH Darwin. On the secular changes in the elements of the orbit of a satellite revolving about a planet distorted by tides. Nature , 21(532):235-237, 1880. Michael Efroimsky. Bodily tides near spin-orbit resonances. Celestial Mechanics and Dynamical Astronomy , 112(3):283-330, 2012. Yeva Gevorgyan, Isamu Matsuyama, and Clodoaldo Ragazzo. Equivalence between simple multilayered and homogeneous laboratory-based rheological models in planetary science. Monthly Notices of the Royal Astronomical Society , 523(2):1822-1831, 2023. William M Kaula. Tidal dissipation by solid friction and the resulting orbital evolution. Reviews of geophysics , 2(4):661-685, 1964. C Ragazzo and LS Ruiz. Tidal evolution and spin-orbit dynamics: The critical role of rheology. To appear , 2024.", "pages": [ 35, 36, 37 ] } ]

[ { "title": "ABSTRACT", "content": "A. V. Getling, & L. L. Kitchatinov, eds. doi:10.1017/xxxxx", "pages": [ 1 ] }, { "title": "Vindya Vashishth 1", "content": "1 Department of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi 221005 India email: [email protected] Abstract. The Sun and solar-type stars exhibit irregular cyclic variations in their magnetic activity over long time scales. To understand this irregularity, we employed the flux transport dynamo models to investigate the behavior of one solar mass star at various rotation rates. To achieve this, we have utilized a mean-field hydrodynamic model to specify differential rotation and meridional circulation, and we have incorporated stochastic fluctuations in the Babcock-Leighton source of the poloidal field to capture inherent fluctuations in the stellar convection. Our simulations successfully demonstrated consistency with the observational data, revealing that rapidly rotating stars exhibit highly irregular cycles with strong magnetic fields and no Maunder-like grand minima. On the other hand, slow rotators produce smoother cycles with weaker magnetic fields, long-term amplitude modulation, and occasional extended grand minima. We observed that the frequency and duration of grand minima increase with the decreasing rotation rate. These results can be understood as the tendency of a less supercritical dynamo in slow rotators to be more prone to produce extended grand minima. We further explore the possible existence of the dynamo in the subcritical regime in a Babcock-Leighton-type framework and in the presence of a small-scale dynamo. Keywords. Stars: magnetic field, stars: rotation, dynamo: sun-like stars", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Like our Sun, many other sun-like stars have magnetic fields and cycles as unveiled by various observations (Donati et al. 1992; Baliunas et al. 1995; Wright et al. 2011; Wright & Drake 2016; Metcalfe, Egeland, and van Saders 2016). According to these observations, a star's rotation rate plays an important role in determining its magnetic activity. Rapidly rotating (young) Sun-like stars exhibit a high activity level with no Maunder-like grand minimum (flat activity) and rarely display smooth regular activity cycles. On the other hand, slowly rotating old stars like the Sun and older have lower activity levels and smooth cycles with occasional grand minima (Skumanich 1972; Rengarajan 1984; Baliunas et al. 1995; Ol'ah et al. 2016; Boro Saikia et al. 2018; Garg et al. 2019). Recently, Shah et al. (2018) observed the decreasing magnetic activity of HD 4915, which might indicate it as the Maunder minimum candidate. Later Baum et al. (1995) confirmed that HD 166620 is entering into a grand minimum phase. Interestingly, these stars (including Sun) are slow rotators. Magnetic cycles in the Sun and other sun-like stars are maintained by dynamo action powered by helical convection and differential rotation in their convection zones (Karak et al. 2014a; Charbonneau 2020; Karak 2023). This is because the toroidal field is generated through the stretching of the poloidal field by the differential rotation, which is known as the Ω effect. There is strong evidence that the poloidal field is generated through a mechanism, so-called the Babcock-Leighton process (Dasi-Espuig et al. 2010; Kitchatinov & Olemskoy 2011; Mu˜noz-Jaramillo et al. 2013; Priyal et al. 2014; Mordvinov et al. 2020, 2022; Kumar et al. 2021). In this process, tilted sunspots (more generally bipolar magnetic regions; Sreedevi et al. 2023) decay and disperse to produce a poloidal field through turbulent diffusion, meridional flow, and differential rotation. While the systematic tilt in the BMR is crucial to generate the poloidal field, the scatter around Joy's law tilt (e.g, McClintock et al. 2014; Jha et al. 2020) produces a variation in the solar cycle (Lemerle & Charbonneau 2017; Karak & Miesch 2017, 2018; Karak et al. 2018; Karak 2020; Biswas et al. 2023a). Many observations suggest that as the stars rotate faster, the magnetic activity becomes stronger, but the relation between the activity cycle period and the rotation rate does not seem a straightforward trend. The cycle period tends to decrease with the rotation rate for the slowly rotating stars, whereas the trend is quite complicated for the fast rotators. Previous studies have explored the trend of magnetic field strength and the cycle period with the rotation rate of the stars (Karak et al. 2014; Hazra et al. 2019). Karak et al. (2020); Noraz et al. (2022) have also explored the possibility of magnetic cycle and reversals in slowly rotating stars possibly having anti-solar differential rotations (e.g., Karak et al. 2015, 2018). Here we aim to understand these observational trends of stellar magnetic activity using dynamo modeling. We shall extract the dependency of the rotation rate of the sun-like stars on its cycle variability and the occurrence of the grand minima using the dynamo models of Karak et al. (2014); Hazra et al. (2019) in which the regular behavior of the stellar cycle was simulated. As the stellar cycles are irregular, it is natural to explore the irregular features of the stellar cycles using these models. For this, we have included stochastic noise to capture the inherent fluctuations in the stellar convection, as seen in the form of variations in the flux emergence rates and the tilts of BMRs around Joy's law. To do so, we have included the stochastic fluctuations in the Babcock-Leighton source for the poloidal field in the dynamo.", "pages": [ 1, 2 ] }, { "title": "2. Model", "content": "In our work, we have developed three kinematic mean-field dynamo models, namely, Models I-III, by assuming the axisymmetry. Thus, the evolution equation of the poloidal ( ∇ × [ A ( r , θ ) e φ ] ) and toroidal ( B ( r , θ ) e φ ) fields are followings. where s = r sin θ , vvv p = v r ˆ r + v θ ˆ θ is the meridional flow and the Ω is the angular velocity whose profile is obtained from the mean-field hydrodynamic model of Kitchatinov & Olemskoy (2011), η is the turbulent magnetic diffusivity which is written as the function of r alone and take the following form, with r BCZ = 0 . 7 R s ( R s being the stellar radius), d t = 0 . 015 R s , d 2 = 0 . 05 R s , r surf = 0 . 95 R s , η RZ = 5 × 10 8 cm 2 s -1 , η SCZ = 5 × 10 10 cm 2 s -1 , and η surf = 2 × 10 12 cm 2 s -1 . S is the source for the poloidal field and its parameterized form is written as where B ( r t , θ ) is the toroidal field at latitude θ averaged over the whole tachocline from r = 0 . 685 R s to r = 0 . 715 R s , α 0 is the measure of the strength of the Babcock-Leighton process, which is expressed as the dependence on the rotation in the following way, where α 0 , s is the value of α 0 for the solar case, which is taken as 0 . 7 cm s -1 in Model I-II and T s and T are the rotation period of Sun and the star, respectively. And finally, α BL is the parameter for Babcock-Leighton process which is written in Model I-II as, where, r 4 = 0 . 95 R s , r 5 = R s , d 4 = 0 . 05 R s , d 5 = 0 . 01 R s , and for Model III, the α profile used is given by, where d = 0 . 01 R s . Thereafter, to study large-scale magnetic fields, we included fluctuations in the source of the poloidal field. The main reason for including the randomness was that since the star cycle's amplitude is not equal, it varies from time to time. This happens due to the fluctuating nature of the stellar convection. The dynamo parameters fluctuate around their mean. In BabcockLeighton, the fluctuations are due to the scatter in the bipolar active region tilts around the Joy's law. This randomness changes the poloidal field and makes irregular magnetic cycles as observed in Sun and Sun-like stars. In order to include randomness in our Babcock-Leighton α , we include fluctuations in the α appearing in Eq. (4) as, α 0 , s = α 0 , s r , where r is the Gaussian random number with mean unity and standard deviation ( σ ) as 2.67. We keep the value of σ the same for all the stars. In our models, the value of α 0 is updated randomly after a certain time, which we take to be one month. The α 0 in all three models have the same form (Eq. (4)), except in Model III, α 0 , s = 4 cm s -1 and fluctuations in this model are included separately in the two hemispheres. We note that above α in Eq. (6) has a sin 2 θ cos θ dependence instead of sin θ cos θ as used in Models I-II to make the α effect strong (weaker) in low (high) latitudes. Also, the radial extent of this α is a bit wider than that used in Models I-II. Finally, In Model III, we have added radial magnetic pumping. This inclusion of pumping is inspired by Hazra et al. (2019), who found some agreement of the cycle period vs rotation trend with observations. It was realized that a downward magnetic pumping helps to make the magnetic field radial near the surface and reduce the toroidal flux loss through the surface, making the dynamo model in accordance with the surface flux transport models and observations (Cameron et al. 2012; Karak and Cameron 2016). The pumping has the following form: where the amplitude of the radial magnetic pumping is given by γ 0 which is 24 m s -1 in all the stars.", "pages": [ 2, 3 ] }, { "title": "3. Results & Discussion", "content": "The simulations were done for M ⊙ mass stars having rotation periods of 1, 3, 7, 10, 15, 20, 25.30 (Sun), and 30 days, respectively. Here we discuss the various aspects of magnetic cycles obtained from all the stars and from all the three models. Firstly, the regular polarity reversals in the toroidal field were noted. Model II presents an irregular magnetic cycle with significant hemispheric asymmetry. In contrast, Model III shows regular polarity reversals for fast rotators and the Sun but irregularity for slower ones. Depending on the rotation period, stars exhibit varying magnetic field configurations: slow rotators mainly have dipolar fields, while those rotating in 7 days or less can have quadrupolar configurations. The pictorial representation and detailed analysis are in Vashishth et al. (2023). One obvious feature in these simulations, as seen in Fig. 1, is that the magnetic field becomes strong in fast-rotating stars. This is because the strength of α increases with the rotation rate of the star (the shear, however, remains more or less unchanged in different stars). If a star rotates faster, the tilt of the BMR associated with the Babcock-Leighton α is expected to increase. Therefore, with age, as the rotation rate decreases, the dynamo process becomes weaker, and the dynamo number also decreases. This implies that the rapidly rotating stars will likely have stronger magnetic cycles. This result agrees with the Karak et al. (2014) and the observations (Noyes et al. 1984; Wright et al. 2011). We also computed the cycle periods for all three models. To achieve this, we analyzed the Fourier power spectrum peaks of the toroidal field time series within the tachocline. This analysis was conducted separately for the northern and southern hemispheres, distinguishing between symmetric and anti-symmetric cycles. The variations of the cycle duration in each case with the rotation rate are shown in Fig. 2. Notably, Models I and II exhibit an increasing trend in the cycle period with higher stellar rotation rates. This behavior is due to the weakening of the meridional flow with the decrease in the rotation period which leads to intensifying the flow speed in the thin layers near the boundaries. Although these two models reproduced various stellar observations, they failed to reproduce the magnetic cycle period vs. rotation trend correctly for the slowly-rotating stars. One way to resolve this discrepancy was to include radial magnetic pumping in the stellar CZs as done by Hazra et al. (2019). As a result, in Model III, we got the cycle-rotation period trend closer to the observations. When strong downward magnetic pumping is included in this model, the diffusion of the magnetic field across the surface becomes negligible, and then the dynamo allows it to operate at a low α (Karak and Cameron 2016). The lower the α , the longer the cycle period. We can see from Fig. 2 that at 30 days rotation period, while Models I-II were producing a cycle period of 6 years, Model III produced a much longer period of 13 years. Then with the decrease of the rotation period, the α becomes stronger and thus the poloidal field generation process becomes more efficient. This makes the reversal of the field faster. This effect in the pumping-dominated regime overpowers the increase of the cycle period due to a decrease in meridional flow speed. The focal point of our study was to understand how the long-term variability of stellar cycles is influenced by their rotation rates. We found that fast rotators manifest irregular cycles with less pronounced long-term variability, while slower rotators exhibit longer modulations in their cycles, interspersed with periods of weaker magnetic fields. We then identified extended periods of low magnetic activity, known as grand minima, using a method adapted from solar studies by Usoskin et al. (2007). We examined the number of grand minima observed in each case with the help of a time-series plot of the toroidal field at the base of CZ and radial magnetic field from simulations for 11,000 years. We infer that the number of grand minima observed in all the models shows an increasing trend with the rotation period. We saw that the rapidly rotating stars hardly produce any grand minima, whereas the slowly rotating stars produce some grand minima, and also, as the rotation period increases, the number of grand minima is seen to increase (see Fig. 3). This is because, with the increase of rotation period, the supercriticality of the dynamo decreases, and the dynamo is more prone to produce extended grand minima in this regime. This result is as per Vashishth et al. (2021) where we observed that as the supercriticality increases (i.e., as the dynamo number increases), the frequency of occurrence of grand minima decreases. This is also supported by the observation that the detected grand/Maunder minima candidates are the slow rotators. Notably, the average duration of these grand minima also increases with rotation duration, with most lasting under 150 years and the majority even below 70 years.", "pages": [ 3, 4, 5 ] }, { "title": "4. Conclusion", "content": "Based on the kinematic dynamo simulations of one solar mass at different rotation rates with stochastically forced Babcock-Leighton source, we make the following conclusions. In slowly rotating stars, the cycles are smooth and show long-term variation with occasional grand minima. Whereas the magnetic field is strong for rapidly rotating stars, cycles are more irregular, and no grand minima are detected. The number of grand minima increases with the decrease in the star's rotation rate. Details of this work have been presented in Vashishth et al. (2023). We further explore the possible existence of the dynamo in the subcritical regime in a BabcockLeighton-type framework and in the presence of a small-scale dynamo, whose details have been presented in Vashishth et al. (2021) and Vashishth et al. 2023 (under preparation).", "pages": [ 5 ] } ]

[ { "title": "Piggybacking astronomical hazard investigations on scientific Big Data missions", "content": "Gijs A. Verdoes Kleijn 1 , 2 , Teymoor Saifollahi 1 , Rees Williams 3 Oscar Stolk 1 , Georg Feulner 4 , 1 Kapteyn Astronomical Institute, University of Groningen, The Netherlands 2 Netherlands Research School for Astronomy, The Netherlands, 3 Donald Smits Centre for Information Technology, University of Groningen, The Netherlands, 4 Potsdam Institute for Climate Impact Research, Germany, Abstract. Current and upcoming large optical and near-infrared astronomical surveys have fundamental science as their primary drivers. To cater to those, these missions scan large fractions of the entire sky at multiple wavelengths and epochs. These aspects make these data sets also valuable for investigations into astronomical hazards for life on Earth. The Netherlands Research School for Astronomy (NOVA) is a partner in several optical / near-infrared surveys. In this paper we focus on the astronomical hazard value for two sets of those: the surveys with the OmegaCAM wide-field imager at the VST and with the Euclid Mission. For each of them we provide a brief overview of the astronomical survey hardware, the data and the information systems. We present first results related to the astronomical hazard investigations. We evaluate to what extent the existing functionality of the information systems covers the needs for the astronomical hazard investigations. Keywords. minor planets, asteroids, comets, stellar proper motions, climate change, astrometry, surveys, Big Data, Data Science, information systems", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the last four decades there has been an exponential growth in the observational data gathered by optical and near-infrared astronomical imaging surveys (see e.g., Fig. 1, Tyson, 2019, Verdoes Kleijn, 2023). This growth is continueing unabated. The Netherlands Research School for Astronomy (NOVA † ) is partner in several optical / nearinfrared surveys. In this paper we focus on two sets of those: the surveys with the OmegaCAM wide-field imager at the VST, in particular the Kilo-Degree Survey and the ground-based and space-based surveys part of the Euclid Mission. These missions have fundamental astronomical science as their primary driver. For this they perform observations of large fractions of the entire sky at multiple wavelengths and at multiple epochs. These aspects make these data sets also valuable for investigations into astronomical hazards for life on Earth. These survey missions will reach the tens of Terabytes regime in terms of catalogs and metadata databases and up to tens of Petabyte regime in terms of bulk data volume. This volume is spread over up to hundreds of thousands of exposures with each of them producing rich sets of metadata, such as catalogs. These in turn lead to millions of bulk data files. Therefore, the scientific exploitation and mining of these 'Big Data' sets requires information systems which have an advanced databasing system at their core. Furthermore, to support the calibration of the raw data and its subsequent scientific analysis these systems also need to interface to high performance compute clusters and massive storage systems. For both survey missions NOVA has a † https://nova-astronomy.nl/ leading role in the development and operation of the associated information systems. This offers NOVA a good opportunity to (re-)use both the observational data and the associated information systems for investigations into astronomical hazards, such as posed by asteroids, comets and close stellar encounters. This way NOVA can make a contribution to protecting society against astronomical hazards, on short and long terms. In this paper we describe two recently initiated pilot projects within NOVA investigating astronomical hazards that piggyback on the survey data and associated information systems developed for fundamental science. One project pertains to Near-Earth Objects using OmegaCAM data with the AstroWISE information system. The other project pertains to climate change due to comet impacts due to close stellar encounters. This will use the surveys of the Euclid Mission and its Euclid Data Processing System. For both each we present first results related to the astronomical hazards. We also provide an overview of the astronomical survey hardware, the data and the information systems. In this we highlight to what extent the existing functionality of the information systems covers the needs for the astronomical hazard investigations.", "pages": [ 1, 2 ] }, { "title": "2. Near-Earth Object precovery and discovery with OmegaCAM and AstroWISE", "content": "Near-Earth Objects (NEOs) are asteroids or comets whose perihelion occurs at less than 1.3 astronomical units (au), meaning that close approaches with the Earth might occur at some point. The size of these objects ranges from meters to tens of kilometers. The impact hazard poses a significant threat to life on Earth (Boroviˇcka et al., 2013, Popova et al., 2013, Brown et al., 2013 ) and calls for planetary defence strategies. The Tunguska and Chelyabinsk impacts are recent reminders. Characterization of the orbits and physical composition of Near-Earth Objects is thus valuable for planetary defence. Serendipitous recovery of NEO appearances in archival scientific astronomical observations not dedicated to Near-Earth Objects can contribute to this characterization. In particular, NEO precovery -detection of a known NEO in an observing dataset prior to its discovery -provides kinematic information about the NEO at a location in its orbit possibly valuable in addition to the discovery and immediate follow-up observations. This is because the orbit uncertainty depends on the fraction of the orbital arc that is covered during discovery and follow-up. Precovering one or more points far away from the discovery and immediate follow-up observations can significantly improve the accuracy of the orbital parameters. In this way, science-driven missions not only have a scientific purpose but can also provide a societal spin-off. Therefore we initiated with ESA and within NOVA an exploratory pilot that evaluates both the re-use of astronomical imaging surveys, in this case those of the OmegaCAM wide-field imager, and the re-use of the information system which was developed to handle the production and scientific analysis of these surveys: AstroWISE . OmegaCAM is a wide-field camera on the VLT Survey Telescope at ESO's Cerro Paranal Observatory. OmegaCAM has 32 science CCDs with a field of view of approximately 1 square degree. Over the first decade of its operation, OmegaCAM has covered a significant portion of the southern hemisphere (Fig 2) in over 400 000 exposures. AstroWISE stands for Astronomical Wide-field Imaging System for Europe which is an information system for data management, image processing, and calibration for a range of astronomical telescopes and instruments in a single data flow environment (Begeman et al., 2013, McFarland et al., 2013a). It has been used to do survey production for example for OmegaCAM's Kilo-Degree Survey (Kuijken et al., 2019) and the Fornax Deep Survey (Peletier et al., 2020). For the pilot we developed an AstroWISE Precovery Pipeline which re-used the AstroWISE functionality for data calibration, processing and analysis and added automated interfaces to webservices for ephemerides prediction (SSOIS, Gwyn et al., 2012, JPL Horizons † ) and includes the deployment of dedicated software for the detection of streaks ( StreakDet , Virtanen et al., 2016, Pontinen et al., 2020). The pilot with the AstroWISE Precovery Pipeline resulted in the recovery of 196 appearances of NEOs from a set of 968 appearances predicted to be recoverable. The achieved astrometric and photometric accuracy is on average 0.12arcsec and 0.1 mag. It includes 49 appearances from a set of 68 NEOs predicted to be recoverable and which were on ESA's and NASA's risk list at that point. ESA's risklist ‡ is provided by the ESA near-Earth Objects Coordination Centre (ESA-NEOCC) and consists of known NEOs with a non-negligible chance of impact in the next hundred years. The appearances of three NEOs constituted precoveries, i.e., appearances well before their discovery. The subsequent risk assessment using the extracted astrometry removed these NEOs from the ESA and NASA risk list. For an in-depth discussion of the methods and results we refer to Saifollahi, 2023. Using the experience of the pilot we attempt here to answer questions on the value † https://ssd.jpl.nasa.gov/horizons/app.html ‡ https://neo.ssa.esa.int/risk-list and challenges of re-using the astronomical data and associated information systems for planetary defence against NEOs. What is the detectability of NEO appearances in astronomical archives such as the OmegaCAM archive? We define NEO detectability as the fraction of detectable appearances among the total of occurrences that a NEO is predicted to be located within the FoV of the images. For the OmegaCAM pilot, the detectability varies as a function of the chosen threshold of signal-to-noise (SNR). The detectability rate is estimated to be ∼ 0.005 at an SNR > 3 for NEOs on the risk-list and for the full list of NEOs. We expect no significant improvement can be made in detectability given the low 3 σ threshold in predicted SNR and the fact that detected NEOs tend to be often a few tenths of magnitude fainter than predicted for this pilot. What is the precovery rate for NEOs predicted to be detectable? The precovery rate for SNR > 3 is 40% for NEOs on the risk-list and 20% for the full list of NEOs. The precovery rate increases to about 50% for SNR > 10. So a factor of up to 5 more NEOs can be precovered from the OmegaCAM archive through improved detection techniques (see below for discussion on new techniques). It will be a significant result as well if, after improving the recovery processes, the failed recoveries turn out to be in fact non-appearances. It would suggest that the actual orbital accuracy for those objects (including those on the risk list) is significantly worse than predicted. What astrometric and photometric accuracy can be achieved? The astrometric and photometric accuracies are 0.12arcsec (15% of the average FWHM of OmegaCAM/VST images of about 0.8arcsec) and 0.1 mag. Improvements in astrometric accuracy are expected from propagating the proper motions in the Gaia astrometric reference catalog to the observation date of the science image. Improvements in photometric accuracy can come from a more sophisticated modelling of SED, observational configuration and NEO shape modelling. The Solar System Open Database Network might facilitate this (Berthier et al., 2022). What are the challenges in deploying the Precovery Pipeline, also on imaging archives from other instruments available in AstroWISE? The Pipeline works mostly automatically through all steps to produce candidate recoveries. These are then inspected by an expert for confirmation/rejection. Thanks to the common data model for calibrated observations in AstroWISE (McFarland et al., 2013b) it could be deployed straightforwardly on calibrated observations for the imaging archives of about a dozen other cameras available in AstroWISE . A challenge is that precise photometric calibration for a range of instruments is hard to fully automate. This is because the derivation of the solution requires reference stars sometimes inside the science images, sometimes in separate calibration observations. A potential solution would be to construct a photometric reference catalog that spans the entire sky observable by OmegaCAM with sufficient stellar density. This appears possible by aggregating information from the multiple large-scale surveys of the Southern Sky. Another main challenge is robust NEO detection and segmentation. This is also a main reason behind the the obtained recovery rates. StreakDet is a great tool for detecting high SNR streaks with sizes between 520arcsec. However, its performance drops for faint and long streaks. Deep learning might be a solution to improve streak detection and ultimately NEO precovery (e.g., Pontinen et al., 2020). For an in-depth discussion about the recovery results for OmegaCAM and the feasibility of deploying it to other instruments, we refer the reader to Saifollahi, 2023.", "pages": [ 2, 3, 4, 5 ] }, { "title": "3. Close stellar encounters with Euclid surveys and systems", "content": "Close encounters of stars to the Sun can affect climate and life on Earth. The ionizing radiation and cosmic rays from supernovae could have a significant impact on both for encounters within 10 pc (Thomas, these proceedings). Stellar encounters within 1 pc can cause significant gravitational perturbations in our Solar System's Oort Cloud. These can lead to increased influx of comets and hence planetary impacts in the inner Solar System (Bailer-Jones, these proceedings). Close stellar encounters can also bring an increase in the influx of exocomets. They can originate in either the Oort cloud of the passing star or in the cloud's tidal streams (Portegies Zwart, 2021). Impacts by comets and asteroids", "pages": [ 5 ] }, { "title": "Positions of stars in Euclid survey area", "content": "may have caused climate changes in the past (Brugger et al., 2017) and might do so again in the future. Identifying close stellar encounters requires six dimensional phase space coordinates (three positions, three velocities) for stars in the Milky Way. The Gaia Mission has brought an enormous information leap in stellar phase space measurements. Its third data release provides an all sky astrometric reference frame sampled with almost 1.5 billion point sources down to 22nd magnitude (Gaia Collaboration et al., 2022). It also provides six-dimensional phase space coordinates for over 33 million stars down to G=14 with positions accurate at the milli-arcsecond (mas) level, proper motions at the mas / year level and radial velocities at the km/s level. This stellar sample allowed identification of 42 stars with encounters within 1 pc with a perihelion time up to 6 Myr in the past and future (Bailer-Jones, 2022). From a similar analysis on Gaia's second data release it was estimated that about 15% of all close stellar encounters within 5 pc and within 6 Myr were detected at that point (Bailer-Jones et al., 2018). The associated inferred rate of encounters within 1 pc is about 20 per million year. The final Gaia release might roughly double the completeness and be able to detect encounters with perihelion times of order 10 Myr in past and future. To increase this completeness level and perihelion time span one has to identify close encounters from stars fainter than observable by Gaia using deeper surveys. Five dimensions of the six-dimensional phase space can be obtained by combining imaging surveys observing the same sky area at multiple wavelengths and at multiple epochs. The multiple epochs allow to derive proper motions (in addition to the positions). The multiple wavelengths allow to derive photometric distances (see e.g., Chapter 3 in Speagle, 2020). ESA's Euclid Mission brings together space-based and ground-based surveys at multiple epochs and multiple wavelengths over almost 15 000 square degrees of sky. Observations at 9 wavelengths are gathered via 8 instruments, located at 7 telescopes in space and on the ground. The first observations which are now being re-used as part of the Euclid Mission occurred from the ground in August 2013. The space-based observations for the Euclid Mission will be obtained with ESA's Euclid satellite. It will be launched in July 2023 and observe the almost 15 000 square degrees of extragalactic sky in about 6 years (Laureijs et al., 2011, Euclid Collaboration et al., 2022). The ground-based observations are planned to be completed well before July 2029. The Euclid satellite will survey the extragalactic sky using a 1.2m telescope with two imagers. The visible imager (VIS) and Near Infrared Spectrometer and Photometer (NISP), sharing a 0.53 square degree Field of View. VIS will detect point sources down to a limiting magnitude of 25 (AB, 10 σ for a point source measured using a 2 arcsec diameter aperture) using a very broad filter (550-900 nm). The Near-infrared Spectrograph and Photometer (NISP) will measure their photometry through Y, J, and H filters down to a magnitude limit of 23.5 (using same definition as VIS). All space observations of a sky area will be done at a single epoch. This data will be combined with data from ground-based telescopes in the optical filter u, g, r, i, z to matching depth. In the Northern hemisphere this will be with four surveys. The Canada-France Imaging Survey (CFIS, Ibata et al., 2017) observes in u and r. CFIS observations started in the first semester of 2015 and are done to full depth in a single epoch. The Waterloo Hawaii IfA G-band Survey (WHIGS † ) observes in g. WHIGS started approximately 2022 and observes to full depth in a single epoch. The Panoramic Survey Telescope And Rapid Response Systems 1 and 2 (Pan-STARRS 1 & 2, Kaiser et al., 2010) observes in i band. Pan-STARRS observations cover the Euclid survey area since 2010, building up depth by many revisits over years until a few years after 2023. † https://www.skysurvey.cc/aboutus/ The Wide Imaging with Subaru HSC of the Euclid Sky (WHISHES ‡ ) survey observes in z. It observes since the second semester of 2020 to full depth in single epochs. In the Southern hemisphere the Euclid survey area is covered by the Dark Energy survey in g, r, i and z (DES, Abbott et al., 2021) and the Large Survey of Space and Time (LSST, Ivezi'c et al., 2019) in u, g, r, i and z. DES observed from August 2013 until January 2019, building up depth in yearly revisits. The Vera Rubin Observatory plans to start the LSST survey late 2024 and has 10 years of planned operations. It will build up depth through many visits over many years. Combining such a heterogeneous set of epochs and filters into a homogeneously set of order billion stellar positions, proper motions and distances requires a careful calibration approach using a information system that also allows ample quality control. The astrometric calibration might best be done via calibration against with zero proper motion and well-defined and consistent centroids across the optical and near-IR. For this reason Tian et al., 2017 used compact galaxies as calibrators. They combined Gaia (Data Release 1) with data from the SDSS, 2MASS and PanSTARRS surveys to obtain proper motions for 350 million sources with a characteristic systematic error of less than 0.3 mas/year and a typical precision of 1.5-2.0 mas/year. The Euclid survey area will contain of order a billion stars. For Euclid such an astrometric calibration effort can be performed and released multiple times as observations on space and ground progress. To do such a massive operation repeatedly that accurately for so many objects is facilitated by an advanced databasing system providing a rich and detailed description of all data items (Mulder et al., 2020). The Euclid information system (called the Euclid Archive System) is such a 'data-centric' system. It consists of two main components: the Euclid Science Archive System and the Euclid Data Processing System (Nieto et al., 2019). All bulk and metadata of calibration and science observations required for the re-use to determine close stellar encounters reside in the Data Processing System. Only a subset resides in the Science Archive System. For example, all bulk data, metadata and data quality reports related to individual ground-based exposures resides only in the Data Processing System. Combining Euclid's five-dimensional phase space (positions, distances and proper motions) with stellar radial velocities, from e.g., a spectral survey, establishes then finally the six-dimensional data set from which one can infer which of these billion stars (mostly fainter than observable by Gaia) lead/led to close stellar encounters. Simulations are on going to determine what completeness and accuracy to expect in terms of nearby stellar encounters using Euclid Mission's observations gathered over almost two decades at 9 wavelengths when combined with such a radial velocity survey. Fig 3 shows a simulated Milky Way stellar distribution as observed by Euclid. The simulation is made using the Galaxia modeling code (Sharma et al., 2011) in its wrapper code (Rybizki et al., 2018). The code simulates magnitudes in all 9 Euclid filters and 6D phase coordinates. Photometric distance estimates can be derived using the magnitudes as input to a Bayesian statistical framework and modeling (e.g., Speagle, 2020). Proper motions could be obtained from the positional catalogs of all observations following the approach of Tian et al., 2017. Finally stellar radial velocities have to be supplied by another mission than Euclid. The perihelion distance and time of close stellar encounters can then be estimated using the Linear Motion Approximation (Bailer-Jones et al., 2018). Very preliminary estimates are shown in Fig 4 for four different assumptions on accuracy of estimated distances, proper motions and radial velocities. Completeness and false positive rate in a sample of candidate close encounters clearly depend critically on the measurement errors. ‡ https://www.skysurvey.cc/aboutus/", "pages": [ 7, 8 ] }, { "title": "4. Lessons learned", "content": "Above's two pilots have shown to us that it is worthwhile to explore further the piggybacking of astronomical hazard investigations onto the data and information systems developed for astronomical scientific Big Data missions. During the execution of above's two pilots we noticed two general characteristics of the astronomical scientific information systems that are key in making them more amenable for re-use in astronomical hazard investigations:", "pages": [ 9 ] }, { "title": "5. Acknowledgments", "content": "This work was executed as part of ESA contract no. 4000134667/21/D/MRP (CARMEN) with their Planetary Defence Office. The pilots made use of the Big Data Layer of the Target Field Lab project 'Mining Big Data'. The Target Field Lab is supported by the Northern Netherlands Alliance (SNN) and is financially supported by the European Regional Development Fund. The data science software system AstroWISE runs on powerful databases and computing clusters at the Donald Smits Center of the University of Groningen and is supported, among other parties, by NOVA (the Dutch Research School for Astronomy). This research has made use of Aladin sky atlas (Bonnarel et al., 2000, Boch and Fernique, 2014) developed at CDS, Strasbourg Observatory, France and SAOImageDS9 (Joye and Mandel, 2003). This work has been done using the following software, packages and python libraries: Astro-WISE (Begeman et al., 2013, McFarland et al., 2013a), Numpy (van der Walt et al., 2011), Scipy (Virtanen et al., 2020), Astropy (Astropy Collaboration et al., 2018).", "pages": [ 9 ] }, { "title": "References", "content": "Abbott, T. M. C., Adam'ow, M., Aguena, M., Allam, S., Amon, A., Annis, J., Avila, S., Bacon, D., Banerji, M., Bechtol, K., Becker, M. R., Bernstein, G. M., Bertin, E., Bhargava, S., Bridle, S. L., Brooks, D., Burke, D. L., Carnero Rosell, A., Carrasco Kind, M., Carretero, J., Castander, F. J., Cawthon, R., Chang, C., Choi, A., Conselice, C., Costanzi, M., Crocce, M., da Costa, L. N., Davis, T. M., De Vicente, J., DeRose, J., Desai, S., Diehl, H. T., Dietrich, J. P., Drlica-Wagner, A., Eckert, K., Elvin-Poole, J., Everett, S., Evrard, A. E., Ferrero, I., Fert'e, A., Flaugher, B., Fosalba, P., Friedel, D., Frieman, J., Garc'ıa-Bellido, J., Gaztanaga, E., Gelman, L., Gerdes, D. W., Giannantonio, T., Gill, M. S. S., Gruen, D., Gruendl, R. A., Gschwend, J., Gutierrez, G., Hartley, W. G., Hinton, S. R., Hollowood, D. L., Honscheid, K., Huterer, D., James, D. J., Jeltema, T., Johnson, M. D., Kent, S., Kron, R., Kuehn, K., Kuropatkin, N., Lahav, O., Li, T. S., Lidman, C., Lin, H., MacCrann, N., Maia, M. A. G., Manning, T. A., Maloney, J. D., March, M., Marshall, J. L., Martini, P., Melchior, P., Menanteau, F., Miquel, R., Morgan, R., Myles, J., Neilsen, E., Ogando, R. L. C., Palmese, A., Paz-Chinch'on, F., Petravick, D., Pieres, A., Plazas, A. A., Pond, C., Rodriguez-Monroy, M., Romer, A. K., Roodman, A., Rykoff, E. S., Sako, M., Sanchez, E., Santiago, B., Scarpine, V., Serrano, S., Sevilla-Noarbe, I., Smith, J. A., Smith, M., SoaresSantos, M., Suchyta, E., Swanson, M. E. C., Tarle, G., Thomas, D., To, C., Tremblay, P. E., Troxel, M. A., Tucker, D. L., Turner, D. J., Varga, T. N., Walker, A. R., Wechsler, R. H., Weller, J., Wester, W., Wilkinson, R. D., Yanny, B., Zhang, Y., Nikutta, R., Fitzpatrick, M., Jacques, A., Scott, A., Olsen, K., Huang, L., Herrera, D., Juneau, S., Nidever, D., Weaver, B. A., Adean, C., Correia, V., de Freitas, M., Freitas, F. N., Singulani, C., VilaVerde, G., & Linea Science Server 2021, The Dark Energy Survey Data Release 2. ApJS , 255(2), 20. Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., Gunther, H. M., Lim, P. L., Crawford, S. M., Conseil, S., Shupe, D. L., Craig, M. W., Dencheva, N., Ginsburg, A., Vand erPlas, J. T., Bradley, L. D., P'erez-Su'arez, D., de Val-Borro, M., Aldcroft, T. L., Cruz, K. L., Robitaille, T. P., Tollerud, E. J., Ardelean, C., Babej, T., Bach, Y. P., Bachetti, M., Bakanov, A. V., Bamford, S. P., Barentsen, G., Barmby, P., Baumbach, A., Berry, K. L., Biscani, F., Boquien, M., Bostroem, K. A., Bouma, L. G., Brammer, G. B., Bray, E. M., Breytenbach, H., Buddelmeijer, H., Burke, D. J., Calderone, G., Cano Rodr'ıguez, J. L., Cara, M., Cardoso, J. V. M., Cheedella, S., Copin, Y., Corrales, L., Crichton, D., D'Avella, D., Deil, C., Depagne, ' E., Dietrich, J. P., Donath, A., Droettboom, M., Earl, N., Erben, T., Fabbro, S., Ferreira, L. A., Finethy, T., Fox, R. T., Garrison, L. H., Gibbons, S. L. J., Goldstein, D. A., Gommers, R., Greco, J. P., Greenfield, P., Groener, A. M., Grollier, F., Hagen, A., Hirst, P., Homeier, D., Horton, A. J., Hosseinzadeh, G., Hu, L., Hunkeler, J. S., Ivezi'c, ˇ Z., Jain, A., Jenness, T., Kanarek, G., Kendrew, S., Kern, N. S., Kerzendorf, W. E., Khvalko, A., King, J., Kirkby, D., Kulkarni, A. M., Kumar, A., Lee, A., Lenz, D., Littlefair, S. P., Ma, Z., Macleod, D. M., Mastropietro, M., McCully, C., Montagnac, S., Morris, B. M., Mueller, M., Mumford, S. J., Muna, D., Murphy, N. A., Nelson, S., Nguyen, G. H., Ninan, J. P., Nothe, M., Ogaz, S., Oh, S., Parejko, J. K., Parley, N., Pascual, S., Patil, R., Patil, A. A., Plunkett, A. L., Prochaska, J. X., Rastogi, T., Reddy Janga, V., Sabater, J., Sakurikar, P., Seifert, M., Sherbert, L. E., Sherwood-Taylor, H., Shih, A. Y., Sick, J., Silbiger, M. T., Singanamalla, S., Singer, L. P., Sladen, P. H., Sooley, K. A., Sornarajah, S., Streicher, O., Teuben, P., Thomas, S. W., Tremblay, G. R., Turner, J. E. H., Terr'on, V., van Kerkwijk, M. H., de la Vega, A., Watkins, L. L., Weaver, B. A., Whitmore, J. B., Woillez, J., Zabalza, V., & Astropy Contributors 2018, The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package. AJ , 156(3), 123. Bailer-Jones, C. A. L. 2022, Stars That Approach within One Parsec of the Sun: New and More Accurate Encounters Identified in Gaia Data Release 3. ApJL , 935(1), L9. Bailer-Jones, C. A. L., Rybizki, J., Andrae, R., & Fouesneau, M. 2018, New stellar encounters discovered in the second Gaia data release. A&A , 616, A37. Begeman, K., Belikov, A. N., Boxhoorn, D. R., & Valentijn, E. A. 2013, The Astro-WISE datacentric information system. Experimental Astronomy , 35(1-2), 1-23. Berthier, J., Carry, B., Mahlke, M., & Normand, J. 2022, SsODNet: The Solar system Open Database Network. arXiv e-prints ,, arXiv:2209.10697. Boch, T. & Fernique, P. Aladin Lite: Embed your Sky in the Browser. In Manset, N. & Forshay, P., editors, Astronomical Data Analysis Software and Systems XXIII 2014,, volume 485 of Astronomical Society of the Pacific Conference Series , 277. Bonnarel, F., Fernique, P., Bienaym'e, O., Egret, D., Genova, F., Louys, M., Ochsenbein, F., Wenger, M., & Bartlett, J. G. 2000, The ALADIN interactive sky atlas. A reference tool for identification of astronomical sources. A&A Supplements , 143, 33-40. Boroviˇcka, J., Spurn'y, P., Brown, P., Wiegert, P., Kalenda, P., Clark, D., & Shrben'y, L. 2013, The trajectory, structure and origin of the Chelyabinsk asteroidal impactor. Nature , 503(7475), 235-237. Brown, P. G., Assink, J. D., Astiz, L., Blaauw, R., Boslough, M. B., Boroviˇcka, J., Brachet, N., Brown, D., Campbell-Brown, M., Ceranna, L., Cooke, W., de Groot-Hedlin, C., Drob, D. P., Edwards, W., Evers, L. G., Garces, M., Gill, J., Hedlin, M., Kingery, A., Laske, G., Le Pichon, A., Mialle, P., Moser, D. E., Saffer, A., Silber, E., Smets, P., Spalding, R. E., Spurn'y, P., Tagliaferri, E., Uren, D., Weryk, R. J., Whitaker, R., & Krzeminski, Z. 2013, A 500-kiloton airburst over Chelyabinsk and an enhanced hazard from small impactors. Nature , 503(7475), 238-241. Brugger, J., Feulner, G., & Petri, S. 2017, Baby, it's cold outside: Climate model simulations of the effects of the asteroid impact at the end of the Cretaceous. Geophysical Research Letters , 44(1), 419-427. Euclid Collaboration, Scaramella, R., Amiaux, J., Mellier, Y., Burigana, C., Carvalho, C. S., Cuillandre, J. C., Da Silva, A., Derosa, A., Dinis, J., Maiorano, E., Maris, M., Tereno, I., Laureijs, R., Boenke, T., Buenadicha, G., Dupac, X., Gaspar Venancio, L. M., G'omez-' Alvarez, P., Hoar, J., Lorenzo Alvarez, J., Racca, G. D., Saavedra-Criado, G., Schwartz, J., Vavrek, R., Schirmer, M., Aussel, H., Azzollini, R., Cardone, V. F., Cropper, M., Ealet, A., Garilli, B., Gillard, W., Granett, B. R., Guzzo, L., Hoekstra, H., Jahnke, K., Kitching, T., Maciaszek, T., Meneghetti, M., Miller, L., Nakajima, R., Niemi, S. M., Pasian, F., Percival, W. J., Pottinger, S., Sauvage, M., Scodeggio, M., Wachter, S., Zacchei, A., Aghanim, N., Amara, A., Auphan, T., Auricchio, N., Awan, S., Balestra, A., Bender, R., Bodendorf, C., Bonino, D., Branchini, E., Brau-Nogue, S., Brescia, M., Candini, G. P., Capobianco, V., Carbone, C., Carlberg, R. G., Carretero, J., Casas, R., Castander, F. J., Castellano, M., Cavuoti, S., Cimatti, A., Cledassou, R., Congedo, G., Conselice, C. J., Conversi, L., Copin, Y., Corcione, L., Costille, A., Courbin, F., Degaudenzi, H., Douspis, M., Dubath, F., Duncan, C. A. J., Dusini, S., Farrens, S., Ferriol, S., Fosalba, P., Fourmanoit, N., Frailis, M., Franceschi, E., Franzetti, P., Fumana, M., Gillis, B., Giocoli, C., Grazian, A., Grupp, F., Haugan, S. V. H., Holmes, W., Hormuth, F., Hudelot, P., Kermiche, S., Kiessling, A., Kilbinger, M., Kohley, R., Kubik, B., Kummel, M., Kunz, M., Kurki-Suonio, H., Lahav, O., Ligori, S., Lilje, P. B., Lloro, I., Mansutti, O., Marggraf, O., Markovic, K., Marulli, F., Massey, R., Maurogordato, S., Melchior, M., Merlin, E., Meylan, G., Mohr, J. J., Moresco, M., Morin, B., Moscardini, L., Munari, E., Nichol, R. C., Padilla, C., Paltani, S., Peacock, J., Pedersen, K., Pettorino, V., Pires, S., Poncet, M., Popa, L., Pozzetti, L., Raison, F., Rebolo, R., Rhodes, J., Rix, H. W., Roncarelli, M., Rossetti, E., Saglia, R., Schneider, P., Schrabback, T., Secroun, A., Seidel, G., Serrano, S., Sirignano, C., Sirri, G., Skottfelt, J., Stanco, L., Starck, J. L., Tallada-Cresp'ı, P., Tavagnacco, D., Taylor, A. N., Teplitz, H. I., Toledo-Moreo, R., Torradeflot, F., Trifoglio, M., Valentijn, E. A., Valenziano, L., Verdoes Kleijn, G. A., Wang, Y., Welikala, N., Weller, J., Wetzstein, M., Zamorani, G., Zoubian, J., Andreon, S., Baldi, M., Bardelli, S., Boucaud, A., Camera, S., Di Ferdinando, D., Fabbian, G., Farinelli, R., Galeotta, S., Graci'a-Carpio, J., Maino, D., Medinaceli, E., Mei, S., Neissner, C., Polenta, G., Renzi, A., Romelli, E., Rosset, C., Sureau, F., Tenti, M., Vassallo, T., Zucca, E., Baccigalupi, C., Balaguera-Antol'ınez, A., Battaglia, P., Biviano, A., Borgani, S., Bozzo, E., Cabanac, R., Cappi, A., Casas, S., Castignani, G., Colodro-Conde, C., Coupon, J., Courtois, H. M., Cuby, J., de la Torre, S., Desai, S., Dole, H., Fabricius, M., Farina, M., Ferreira, P. G., Finelli, F., Flose-Reimberg, P., Fotopoulou, S., Ganga, K., Gozaliasl, G., Hook, I. M., Keihanen, E., Kirkpatrick, C. C., Liebing, P., Lindholm, V., Mainetti, G., Martinelli, M., Martinet, N., Maturi, M., McCracken, H. J., Metcalf, R. B., Morgante, G., Nightingale, J., Nucita, A., Patrizii, L., Potter, D., Riccio, G., S'anchez, A. G., Sapone, D., Schewtschenko, J. A., Schultheis, M., Scottez, V., Teyssier, R., Tutusaus, I., Valiviita, J., Viel, M., Vriend, W., & Whittaker, L. 2022, Euclid preparation. I. The Euclid Wide Survey. A&A , 662, A112. Gaia Collaboration, Vallenari, A., Brown, A. G. A., Prusti, T., de Bruijne, J. H. J., Arenou, F., Babusiaux, C., Biermann, M., Creevey, O. L., Ducourant, C., Evans, D. W., Eyer, L., Guerra, R., Hutton, A., Jordi, C., Klioner, S. A., Lammers, U. L., Lindegren, L., Luri, X., Mignard, F., Panem, C., Pourbaix, D., Randich, S., Sartoretti, P., Soubiran, C., Tanga, P., Walton, N. A., Bailer-Jones, C. A. L., Bastian, U., Drimmel, R., Jansen, F., Katz, D., Lattanzi, M. G., van Leeuwen, F., Bakker, J., Cacciari, C., Casta˜neda, J., De Angeli, F., Fabricius, C., Fouesneau, M., Fr'emat, Y., Galluccio, L., Guerrier, A., Heiter, U., Masana, E., Messineo, R., Mowlavi, N., Nicolas, C., Nienartowicz, K., Pailler, F., Panuzzo, P., Riclet, F., Roux, W., Seabroke, G. M., Sordoørcit, R., Th'evenin, F., Gracia-Abril, G., Portell, J., Teyssier, D., Altmann, M., Andrae, R., Audard, M., Bellas-Velidis, I., Benson, K., Berthier, J., Blomme, R., Burgess, P. W., Busonero, D., Busso, G., C'anovas, H., Carry, B., Cellino, A., Cheek, N., Clementini, G., Damerdji, Y., Davidson, M., de Teodoro, P., Nu˜nez Campos, M., Delchambre, L., Dell'Oro, A., Esquej, P., Fern'andez-Hern'andez, J., Fraile, E., Garabato, D., Garc'ıa-Lario, P., Gosset, E., Haigron, R., Halbwachs, J. L., Hambly, N. C., Harrison, D. L., Hern'andez, J., Hestroffer, D., Hodgkin, S. T., Holl, B., Janßen, K., Jevardat de Fombelle, G., Jordan, S., Krone-Martins, A., Lanzafame, A. C., Loffler, W., Marchal, O., Marrese, P. M., Moitinho, A., Muinonen, K., Osborne, P., Pancino, E., Pauwels, T., RecioBlanco, A., Reyl'e, C., Riello, M., Rimoldini, L., Roegiers, T., Rybizki, J., Sarro, L. M., Siopis, C., Smith, M., Sozzetti, A., Utrilla, E., van Leeuwen, M., Abbas, U., ' Abrah'am, P., Abreu Aramburu, A., Aerts, C., Aguado, J. J., Ajaj, M., Aldea-Montero, F., Altavilla, G., ' Alvarez, M. A., Alves, J., Anders, F., Anderson, R. I., Anglada Varela, E., Antoja, T., Baines, D., Baker, S. G., Balaguer-N'u˜nez, L., Balbinot, E., Balog, Z., Barache, C., Barbato, D., Barros, M., Barstow, M. A., Bartolom'e, S., Bassilana, J. L., Bauchet, N., Becciani, U., Bellazzini, M., Berihuete, A., Bernet, M., Bertone, S., Bianchi, L., Binnenfeld, A., BlancoCuaresma, S., Blazere, A., Boch, T., Bombrun, A., Bossini, D., Bouquillon, S., Bragaglia, A., Bramante, L., Breedt, E., Bressan, A., Brouillet, N., Brugaletta, E., Bucciarelli, B., Burlacu, A., Butkevich, A. G., Buzzi, R., Caffau, E., Cancelliere, R., Cantat-Gaudin, T., Carballo, R., Carlucci, T., Carnerero, M. I., Carrasco, J. M., Casamiquela, L., Castellani, M., Castro-Ginard, A., Chaoul, L., Charlot, P., Chemin, L., Chiaramida, V., Chiavassa, A., Chornay, N., Comoretto, G., Contursi, G., Cooper, W. J., Cornez, T., Cowell, S., Crifo, F., Cropper, M., Crosta, M., Crowley, C., Dafonte, C., Dapergolas, A., David, M., David, P., de Laverny, P., De Luise, F., De March, R., De Ridder, J., de Souza, R., de Torres, A., del Peloso, E. F., del Pozo, E., Delbo, M., Delgado, A., Delisle, J. B., Demouchy, C., Dharmawardena, T. E., Di Matteo, P., Diakite, S., Diener, C., Distefano, E., Dolding, C., Edvardsson, B., Enke, H., Fabre, C., Fabrizio, M., Faigler, S., Fedorets, G., Fernique, P., Fienga, A., Figueras, F., Fournier, Y., Fouron, C., Fragkoudi, F., Gai, M., GarciaGutierrez, A., Garcia-Reinaldos, M., Garc'ıa-Torres, M., Garofalo, A., Gavel, A., Gavras, P., Gerlach, E., Geyer, R., Giacobbe, P., Gilmore, G., Girona, S., Giuffrida, G., Gomel, R., Gomez, A., Gonz'alez-N'u˜nez, J., Gonz'alez-Santamar'ıa, I., Gonz'alez-Vidal, J. J., Granvik, M., Guillout, P., Guiraud, J., Guti'errez-S'anchez, R., Guy, L. P., Hatzidimitriou, D., Hauser, M., Haywood, M., Helmer, A., Helmi, A., Sarmiento, M. H., Hidalgo, S. L., Hilger, T., Hglyph[suppress]ladczuk, N., Hobbs, D., Holland, G., Huckle, H. E., Jardine, K., Jasniewicz, G., JeanAntoine Piccolo, A., Jim'enez-Arranz, ' O., Jorissen, A., Juaristi Campillo, J., Julbe, F., Karbevska, L., Kervella, P., Khanna, S., Kontizas, M., Kordopatis, G., Korn, A. J., K'osp'al, ' A., Kostrzewa-Rutkowska, Z., Kruszy'nska, K., Kun, M., Laizeau, P., Lambert, S., Lanza, A. F., Lasne, Y., Le Campion, J. F., Lebreton, Y., Lebzelter, T., Leccia, S., Leclerc, N., Lecoeur-Taibi, I., Liao, S., Licata, E. L., Lindstrøm, H. E. P., Lister, T. A., Livanou, E., Lobel, A., Lorca, A., Loup, C., Madrero Pardo, P., Magdaleno Romeo, A., Managau, S., Mann, R. G., Manteiga, M., Marchant, J. M., Marconi, M., Marcos, J., Marcos Santos, M. M. S., Mar'ın Pina, D., Marinoni, S., Marocco, F., Marshall, D. J., Polo, L. M., Mart'ınFleitas, J. M., Marton, G., Mary, N., Masip, A., Massari, D., Mastrobuono-Battisti, A., Mazeh, T., McMillan, P. J., Messina, S., Michalik, D., Millar, N. R., Mints, A., Molina, D., Molinaro, R., Moln'ar, L., Monari, G., Mongui'o, M., Montegriffo, P., Montero, A., Mor, R., Mora, A., Morbidelli, R., Morel, T., Morris, D., Muraveva, T., Murphy, C. P., Musella, I., Nagy, Z., Noval, L., Oca˜na, F., Ogden, A., Ordenovic, C., Osinde, J. O., Pagani, C., Pagano, I., Palaversa, L., Palicio, P. A., Pallas-Quintela, L., Panahi, A., Payne-Wardenaar, S., Pe˜nalosa Esteller, X., Penttila, A., Pichon, B., Piersimoni, A. M., Pineau, F. X., Plachy, E., Plum, G., Poggio, E., Prˇsa, A., Pulone, L., Racero, E., Ragaini, S., Rainer, M., Raiteri, C. M., Rambaux, N., Ramos, P., Ramos-Lerate, M., Re Fiorentin, P., Regibo, S., Richards, P. J., Rios Diaz, C., Ripepi, V., Riva, A., Rix, H. W., Rixon, G., Robichon, N., Robin, A. C., Robin, C., Roelens, M., Rogues, H. R. O., Rohrbasser, L., Romero-G'omez, M., Rowell, N., Royer, F., Ruz Mieres, D., Rybicki, K. A., Sadowski, G., S'aez N'u˜nez, A., Sagrist'a Sell'es, A., Sahlmann, J., Salguero, E., Samaras, N., Sanchez Gimenez, V., Sanna, N., Santove˜na, R., Sarasso, M., Schultheis, M., Sciacca, E., Segol, M., Segovia, J. C., S'egransan, D., Semeux, D., Shahaf, S., Siddiqui, H. I., Siebert, A., Siltala, L., Silvelo, A., Slezak, E., Slezak, I., Smart, R. L., Snaith, O. N., Solano, E., Solitro, F., Souami, D., Souchay, J., Spagna, A., Spina, L., Spoto, F., Steele, I. A., Steidelmuller, H., Stephenson, C. A., Suveges, M., Surdej, J., Szabados, L., Szegedi-Elek, E., Taris, F., Taylo, M. B., Teixeira, R., Tolomei, L., Tonello, N., Torra, F., Torra, J., Torralba Elipe, G., Trabucchi, M., Tsounis, A. T., Turon, C., Ulla, A., Unger, N., Vaillant, M. V., van Dillen, E., van Reeven, W., Vanel, O., Vecchiato, A., Viala, Y., Vicente, D., Voutsinas, S., Weiler, M., Wevers, T., Wyrzykowski, L., Yoldas, A., Yvard, P., Zhao, H., Zorec, J., Zucker, S., & Zwitter, T. 2022, Gaia Data Release 3: Summary of the content and survey properties. arXiv e-prints ,, arXiv:2208.00211. Gwyn, S. D. J., Hill, N., & Kavelaars, J. J. 2012, SSOS: A Moving-Object Image Search Tool for Asteroid Precovery. PASP , 124(916), 579. Ibata, R. A., McConnachie, A., Cuillandre, J.-C., Fantin, N., Haywood, M., Martin, N. F., Bergeron, P., Beckmann, V., Bernard, E., Bonifacio, P., Caffau, E., Carlberg, R., Cˆot'e, P., Cabanac, R., Chapman, S., Duc, P.-A., Durret, F., Famaey, B., Fabbro, S., Gwyn, S., Hammer, F., Hill, V., Hudson, M. J., Lan¸con, A., Lewis, G., Malhan, K., di Matteo, P., McCracken, H., Mei, S., Mellier, Y., Navarro, J., Pires, S., Pritchet, C., Reyl'e, C., Richer, H., Robin, A. C., S'anchez-Janssen, R., Sawicki, M., Scott, D., Scottez, V., Spekkens, K., Starkenburg, E., Thomas, G., & Venn, K. 2017, The Canada-France Imaging Survey: First Results from the u-Band Component. ApJ , 848(2), 128. Ivezi'c, ˇ Z., Kahn, S. M., Tyson, J. A., Abel, B., Acosta, E., Allsman, R., Alonso, D., AlSayyad, Y., Anderson, S. F., Andrew, J., Angel, J. R. P., Angeli, G. Z., Ansari, R., Antilogus, P., Araujo, C., Armstrong, R., Arndt, K. T., Astier, P., Aubourg, ' E., Auza, N., Axelrod, T. S., Bard, D. J., Barr, J. D., Barrau, A., Bartlett, J. G., Bauer, A. E., Bauman, B. J., Baumont, S., Bechtol, E., Bechtol, K., Becker, A. C., Becla, J., Beldica, C., Bellavia, S., Bianco, F. B., Biswas, R., Blanc, G., Blazek, J., Blandford, R. D., Bloom, J. S., Bogart, J., Bond, T. W., Booth, M. T., Borgland, A. W., Borne, K., Bosch, J. F., Boutigny, D., Brackett, C. A., Bradshaw, A., Brandt, W. N., Brown, M. E., Bullock, J. S., Burchat, P., Burke, D. L., Cagnoli, G., Calabrese, D., Callahan, S., Callen, A. L., Carlin, J. L., Carlson, E. L., Chandrasekharan, S., Charles-Emerson, G., Chesley, S., Cheu, E. C., Chiang, H.-F., Chiang, J., Chirino, C., Chow, D., Ciardi, D. R., Claver, C. F., Cohen-Tanugi, J., Cockrum, J. J., Coles, R., Connolly, A. J., Cook, K. H., Cooray, A., Covey, K. R., Cribbs, C., Cui, W., Cutri, R., Daly, P. N., Daniel, S. F., Daruich, F., Daubard, G., Daues, G., Dawson, W., Delgado, F., Dellapenna, A., de Peyster, R., de Val-Borro, M., Digel, S. W., Doherty, P., Dubois, R., Dubois-Felsmann, G. P., Durech, J., Economou, F., Eifler, T., Eracleous, M., Emmons, B. L., Fausti Neto, A., Ferguson, H., Figueroa, E., Fisher-Levine, M., Focke, W., Foss, M. D., Frank, J., Freemon, M. D., Gangler, E., Gawiser, E., Geary, J. C., Gee, P., Geha, M., Gessner, C. J. B., Gibson, R. R., Gilmore, D. K., Glanzman, T., Glick, W., Goldina, T., Goldstein, D. A., Goodenow, I., Graham, M. L., Gressler, W. J., Gris, P., Guy, L. P., Guyonnet, A., Haller, G., Harris, R., Hascall, P. A., Haupt, J., Hernandez, F., Herrmann, S., Hileman, E., Hoblitt, J., Hodgson, J. A., Hogan, C., Howard, J. D., Huang, D., Huffer, M. E., Ingraham, P., Innes, W. R., Jacoby, S. H., Jain, B., Jammes, F., Jee, M. J., Jenness, T., Jernigan, G., Jevremovi'c, D., Johns, K., Johnson, A. S., Johnson, M. W. G., Jones, R. L., Juramy-Gilles, C., Juri'c, M., Kalirai, J. S., Kallivayalil, N. J., Kalmbach, B., Kantor, J. P., Karst, P., Kasliwal, M. M., Kelly, H., Kessler, R., Kinnison, V., Kirkby, D., Knox, L., Kotov, I. V., Krabbendam, V. L., Krughoff, K. S., Kub'anek, P., Kuczewski, J., Kulkarni, S., Ku, J., Kurita, N. R., Lage, C. S., Lambert, R., Lange, T., Langton, J. B., Le Guillou, L., Levine, D., Liang, M., Lim, K.-T., Lintott, C. J., Long, K. E., Lopez, M., Lotz, P. J., Lupton, R. H., Lust, N. B., MacArthur, L. A., Mahabal, A., Mandelbaum, R., Markiewicz, T. W., Marsh, D. S., Marshall, P. J., Marshall, S., May, M., McKercher, R., McQueen, M., Meyers, J., Migliore, M., Miller, M., Mills, D. J., Miraval, C., Moeyens, J., Moolekamp, F. E., Monet, D. G., Moniez, M., Monkewitz, S., Montgomery, C., Morrison, C. B., Mueller, F., Muller, G. P., Mu˜noz Arancibia, F., Neill, D. R., Newbry, S. P., Nief, J.-Y., Nomerotski, A., Nordby, M., O'Connor, P., Oliver, J., Olivier, S. S., Olsen, K., O'Mullane, W., Ortiz, S., Osier, S., Owen, R. E., Pain, R., Palecek, P. E., Parejko, J. K., Parsons, J. B., Pease, N. M., Peterson, J. M., Peterson, J. R., Petravick, D. L., Libby Petrick, M. E., Petry, C. E., Pierfederici, F., Pietrowicz, S., Pike, R., Pinto, P. A., Plante, R., Plate, S., Plutchak, J. P., Price, P. A., Prouza, M., Radeka, V., Rajagopal, J., Rasmussen, A. P., Regnault, N., Reil, K. A., Reiss, D. J., Reuter, M. A., Ridgway, S. T., Riot, V. J., Ritz, S., Robinson, S., Roby, W., Roodman, A., Rosing, W., Roucelle, C., Rumore, M. R., Russo, S., Saha, A., Sassolas, B., Schalk, T. L., Schellart, P., Schindler, R. H., Schmidt, S., Schneider, D. P., Schneider, M. D., Schoening, W., Schumacher, G., Schwamb, M. E., Sebag, J., Selvy, B., Sembroski, G. H., Seppala, L. G., Serio, A., Serrano, E., Shaw, R. A., Shipsey, I., Sick, J., Silvestri, N., Slater, C. T., Smith, J. A., Smith, R. C., Sobhani, S., Soldahl, C., Storrie-Lombardi, L., Stover, E., Strauss, M. A., Street, R. A., Stubbs, C. W., Sullivan, I. S., Sweeney, D., Swinbank, J. D., Szalay, A., Takacs, P., Tether, S. A., Thaler, J. J., Thayer, J. G., Thomas, S., Thornton, A. J., Thukral, V., Tice, J., Trilling, D. E., Turri, M., Van Berg, R., Vanden Berk, D., Vetter, K., Virieux, F., Vucina, T., Wahl, W., Walkowicz, L., Walsh, B., Walter, C. W., Wang, D. L., Wang, S.-Y., Warner, M., Wiecha, O., Willman, B., Winters, S. E., Wittman, D., Wolff, S. C., Wood-Vasey, W. M., Wu, X., Xin, B., Yoachim, P., & Zhan, H. 2019, LSST: From Science Drivers to Reference Design and Anticipated Data Products. ApJ , 873(2), 111. Joye, W. A. & Mandel, E. New Features of SAOImage DS9. In Payne, H. E., Jedrzejewski, R. I., & Hook, R. N., editors, Astronomical Data Analysis Software and Systems XII 2003,, volume 295 of Astronomical Society of the Pacific Conference Series , 489. Kaiser, N., Burgett, W., Chambers, K., Denneau, L., Heasley, J., Jedicke, R., Magnier, E., Morgan, J., Onaka, P., & Tonry, J. The Pan-STARRS wide-field optical/NIR imaging survey. In Stepp, L. M., Gilmozzi, R., & Hall, H. J., editors, Ground-based and Airborne Telescopes III 2010,, volume 7733 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , 77330E. Kuijken, K. et al. 2019, The fourth data release of the Kilo-Degree Survey: ugri imaging and nine-band optical-IR photometry over 1000 square degrees. A&A , 625, A2. Laureijs, R., Amiaux, J., Arduini, S., Augu'eres, J. L., Brinchmann, J., Cole, R., Cropper, M., Dabin, C., Duvet, L., Ealet, A., Garilli, B., Gondoin, P., Guzzo, L., Hoar, J., Hoekstra, H., Holmes, R., Kitching, T., Maciaszek, T., Mellier, Y., Pasian, F., Percival, W., Rhodes, J., Saavedra Criado, G., Sauvage, M., Scaramella, R., Valenziano, L., Warren, S., Bender, R., Castander, F., Cimatti, A., Le F'evre, O., Kurki-Suonio, H., Levi, M., Lilje, P., Meylan, G., Nichol, R., Pedersen, K., Popa, V., Rebolo Lopez, R., Rix, H. W., Rottgering, H., Zeilinger, W., Grupp, F., Hudelot, P., Massey, R., Meneghetti, M., Miller, L., Paltani, S., PaulinHenriksson, S., Pires, S., Saxton, C., Schrabback, T., Seidel, G., Walsh, J., Aghanim, N., Amendola, L., Bartlett, J., Baccigalupi, C., Beaulieu, J. P., Benabed, K., Cuby, J. G., Elbaz, D., Fosalba, P., Gavazzi, G., Helmi, A., Hook, I., Irwin, M., Kneib, J. P., Kunz, M., Mannucci, F., Moscardini, L., Tao, C., Teyssier, R., Weller, J., Zamorani, G., Zapatero Osorio, M. R., Boulade, O., Foumond, J. J., Di Giorgio, A., Guttridge, P., James, A., Kemp, M., Martignac, J., Spencer, A., Walton, D., Blumchen, T., Bonoli, C., Bortoletto, F., Cerna, C., Corcione, L., Fabron, C., Jahnke, K., Ligori, S., Madrid, F., Martin, L., Morgante, G., Pamplona, T., Prieto, E., Riva, M., Toledo, R., Trifoglio, M., Zerbi, F., Abdalla, F., Douspis, M., Grenet, C., Borgani, S., Bouwens, R., Courbin, F., Delouis, J. M., Dubath, P., Fontana, A., Frailis, M., Grazian, A., Koppenhofer, J., Mansutti, O., Melchior, M., Mignoli, M., Mohr, J., Neissner, C., Noddle, K., Poncet, M., Scodeggio, M., Serrano, S., Shane, N., Starck, J. L., Surace, C., Taylor, A., Verdoes-Kleijn, G., Vuerli, C., Williams, O. R., Zacchei, A., Altieri, B., Escudero Sanz, I., Kohley, R., Oosterbroek, T., Astier, P., Bacon, D., Bardelli, S., Baugh, C., Bellagamba, F., Benoist, C., Bianchi, D., Biviano, A., Branchini, E., Carbone, C., Cardone, V., Clements, D., Colombi, S., Conselice, C., Cresci, G., Deacon, N., Dunlop, J., Fedeli, C., Fontanot, F., Franzetti, P., Giocoli, C., Garcia-Bellido, J., Gow, J., Heavens, A., Hewett, P., Heymans, C., Holland, A., Huang, Z., Ilbert, O., Joachimi, B., Jennins, E., Kerins, E., Kiessling, A., Kirk, D., Kotak, R., Krause, O., Lahav, O., van Leeuwen, F., Lesgourgues, J., Lombardi, M., Magliocchetti, M., Maguire, K., Majerotto, E., Maoli, R., Marulli, F., Maurogordato, S., McCracken, H., McLure, R., Melchiorri, A., Merson, A., Moresco, M., Nonino, M., Norberg, P., Peacock, J., Pello, R., Penny, M., Pettorino, V., Di Porto, C., Pozzetti, L., Quercellini, C., Radovich, M., Rassat, A., Roche, N., Ronayette, S., Rossetti, E., Sartoris, B., Schneider, P., Semboloni, E., Serjeant, S., Simpson, F., Skordis, C., Smadja, G., Smartt, S., Spano, P., Spiro, S., Sullivan, M., Tilquin, A., Trotta, R., Verde, L., Wang, Y., Williger, G., Zhao, G., Zoubian, J., & Zucca, E. 2011, Euclid Definition Study Report. arXiv e-prints ,, arXiv:1110.3193. McFarland, J. P., Verdoes-Kleijn, G., Sikkema, G., Helmich, E. M., Boxhoorn, D. R., & Valentijn, E. A. 2013,a The Astro-WISE optical image pipeline. Development and implementation. Experimental Astronomy , 35a(1-2), 45-78. McFarland, J. P., Verdoes-Kleijn, G., Sikkema, G., Helmich, E. M., Boxhoorn, D. R., & Valentijn, E. A. 2013,b The Astro-WISE optical image pipeline. Development and implementation. Experimental Astronomy , 35b(1-2), 45-78. Mulder, W., de Jong, J. T. A., Verdoes Kleijn, G. A., Valentijn, E. A., Williams, O. R., Boxhoorn, D. R., Belikov, A. N., Fabricius, M., Helmich, E. M., Wetzstein, M., Vassallo, T., & George, K. The Astrometric Calibration Software System for Euclid's External Surveys. In Pizzo, R., Deul, E. R., Mol, J. D., de Plaa, J., & Verkouter, H., editors, Astronomical Data Analysis Software and Systems XXIX 2020,, volume 527 of Astronomical Society of the Pacific Conference Series , 615. Nieto, S., de Teodoro, P., Salgado, J., Altieri, B., Buenadicha, G., Belikov, A., Boxhoorn, D., McFarland, J., Valentijn, E. A., Williams, O. R., Droege, B., & Tsyganov, A. The Euclid Archive System: A Data-Centric Approach to Big Data. In Molinaro, M., Shortridge, K., & Pasian, F., editors, Astronomical Data Analysis Software and Systems XXVI 2019,, volume 521 of Astronomical Society of the Pacific Conference Series , 12. Peletier, R., Iodice, E., Venhola, A., Capaccioli, M., Cantiello, M., D'Abrusco, R., Falc'onBarroso, J., Grado, A., Hilker, M., Limatola, L., Mieske, S., Napolitano, N., Paolillo, M., Spavone, M., Valentijn, E., van de Ven, G., & Verdoes Kleijn, G. 2020, The Fornax Deep Survey data release 1. arXiv e-prints ,, arXiv:2008.12633. Pontinen, M., Granvik, M., Nucita, A., Conversi, L., Altieri, B., et al. 2020, Euclid: Identification of asteroid streaks in simulated images using streakdet software. Astronomy & Astrophysics , 644, A35. Pontinen, M., Granvik, M., Nucita, A. A., Conversi, L., Altieri, B., Auricchio, N., Bodendorf, C., Bonino, D., Brescia, M., Capobianco, V., Carretero, J., Carry, B., Castellano, M., Cledassou, R., Congedo, G., Corcione, L., Cropper, M., Dusini, S., Frailis, M., Franceschi, E., Fumana, M., Garilli, B., Grupp, F., Hormuth, F., Israel, H., Jahnke, K., Kermiche, S., Kitching, T., Kohley, R., Kubik, B., Kunz, M., Laureijs, R., Lilje, P. B., Lloro, I., Maiorano, E., Marggraf, O., Massey, R., Meneghetti, M., Meylan, G., Moscardini, L., Padilla, C., Paltani, S., Pasian, F., Pires, S., Polenta, G., Raison, F., Roncarelli, M., Rossetti, E., Saglia, R., Schneider, P., Secroun, A., Serrano, S., Sirri, G., Taylor, A. N., Tereno, I., Toledo-Moreo, R., Valenziano, L., Wang, Y., Wetzstein, M., & Zoubian, J. 2020, Euclid: Identification of asteroid streaks in simulated images using StreakDet software. A&A , 644, A35. Popova, O. P., Jenniskens, P., Emel'yanenko, V., Kartashova, A., Biryukov, E., Khaibrakhmanov, S., Shuvalov, V., Rybnov, Y., Dudorov, A., Grokhovsky, V. I., Badyukov, D. D., Yin, Q.-Z., Gural, P. S., Albers, J., Granvik, M., Evers, L. G., Kuiper, J., Kharlamov, V., Solovyov, A., Rusakov, Y. S., Korotkiy, S., Serdyuk, I., Korochantsev, A. V., Larionov, M. Y., Glazachev, D., Mayer, A. E., Gisler, G., Gladkovsky, S. V., Wimpenny, J., Sanborn, M. E., Yamakawa, A., Verosub, K. L., Rowland, D. J., Roeske, S., Botto, N. W., Friedrich, J. M., Zolensky, M. E., Le, L., Ross, D., Ziegler, K., Nakamura, T., Ahn, I., Lee, J. I., Zhou, Q., Li, X.-H., Li, Q.-L., Liu, Y., Tang, G.-Q., Hiroi, T., Sears, D., Weinstein, I. A., Vokhmintsev, A. S., Ishchenko, A. V., Schmitt-Kopplin, P., Hertkorn, N., Nagao, K., Haba, M. K., Komatsu, M., Mikouchi, T., & aff34 2013, Chelyabinsk Airburst, Damage Assessment, Meteorite Recovery, and Characterization. Science , 342(6162), 1069-1073. Portegies Zwart, S. 2021, Oort cloud Ecology. I. Extra-solar Oort clouds and the origin of asteroidal interlopers. A&A , 647, A136. Rybizki, J., Demleitner, M., Fouesneau, M., Bailer-Jones, C., Rix, H.-W., & Andrae, R. 2018, A Gaia DR2 Mock Stellar Catalog. PASP , 130(989), 074101. Saifollahi, T., V.-K. G. W. O. 2023, Mining archival data from wide-field astronomical surveys in search of near-Earth objects. A&A ,. Sharma, S., Bland-Hawthorn, J., Johnston, K. V., & Binney, J. 2011, Galaxia: A Code to Generate a Synthetic Survey of the Milky Way. ApJ , 730(1), 3. Speagle, J. S. 2020,. Mapping the Milky Way in the age of Gaia . PhD thesis, Harvard University, Massachusetts. Tian, H.-J., Gupta, P., Sesar, B., Rix, H.-W., Martin, N. F., Liu, C., Goldman, B., Platais, I., Kudritzki, R.-P., & Waters, C. Z. 2017, A Gaia-PS1-SDSS (GPS1) Proper Motion Catalog Covering 3/4 of the Sky. ApJS , 232(1), 4. Tyson, J. A. Cosmology data analysis challenges and opportunities in the LSST sky survey. In Journal of Physics Conference Series 2019,, volume 1290 of Journal of Physics Conference Series , 012001. van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, The NumPy Array: A Structure for Efficient Numerical Computation. Computing in Science and Engineering , 13(2), 22-30. Verdoes Kleijn, G. A. e. a. Object classification with Convolutional Neural Networks: from KiDS to Euclid. In Gwyn, S. e. a., editor, Astronomical Data Analysis Software and Systems XXXII 2023,, volume 530 of Astronomical Society of the Pacific Conference Series , in press. Virtanen, J., Poikonen, J., Santti, T., Komulainen, T., Torppa, J., Granvik, M., Muinonen, K., Pentikainen, H., Martikainen, J., Naranen, J., Lehti, J., & Flohrer, T. 2016, Streak detection and analysis pipeline for space-debris optical images. Advances in Space Research , 57(8), 1607-1623. Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, ˙ I., Feng, Y., Moore, E. W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A., Harris, C. R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P., & SciPy 1.0 Contributors 2020, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods , 17, 261-272.", "pages": [ 9, 10, 11, 12, 13, 14, 15, 16 ] } ]

[ { "title": "BURNETT'S CONJECTURE IN GENERALIZED WAVE COORDINATES", "content": "C ' ECILE HUNEAU AND JONATHAN LUK Abstract. We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose { g n } ∞ n =1 is a sequence of Lorentzian metrics (in arbitrary dimensions d ≥ 3) satisfying a generalized wave coordinate condition and such that g n → g in a suitably weak and 'high-frequency' manner, then the limit metric g satisfies the Einstein-massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "It is well-known that the 'high-frequency' limit g of a sequence of metrics { g n } ∞ n =1 to the Einstein vacuum equation Ric( g n ) = 0 may not be itself a solution to the Einstein vacuum equations. Nonetheless, Burnett conjectured that such limits must still be of a very specific type, where the 'effective matter' that arises in the limiting process takes the form of a massless Vlasov field: Conjecture 1.1 (Burnett [2]) . If g is a suitable 'high-frequency limit' of a sequence of vacuum metrics { g n } ∞ n =1 , then g solves the Einstein-massless Vlasov system (after defining a suitable Vlasov field). As described in [2], the conjecture, if true, shows in particular that in the limit, high-frequency gravitational waves do not interact directly, but they only interact via the Einstein equation through their influence on the geometry.", "pages": [ 1 ] }, { "title": "In this paper, we prove Conjecture 1.1 under the additional assumption that { g n } ∞ n =1 and g 0 satisfy suitable (generalized) wave coordinate conditions.", "content": "The Burnett conjecture was previously proven in an elliptic gauge when the metric admits a U (1) symmetry [12], and in the double null coordinates gauge when the metric is angularly regular [25]. This is the first proof of Conjecture 1.1 without symmetry assumptions or angular regularity assumptions. To discuss our result, we introduce the main assumptions of our theorem. Assumptions 1.2. Let d ≥ 3 . Consider a sequence of C ∞ Lorentzian metrics { g n } ∞ n =1 on a bounded open set U ⊂ R d +1 and suppose that the following holds: where the implicit constants are independent of n . where Γ α µν ( g n ) and Γ α µν ( g 0 ) denote the Christoffel symbols of g n and g 0 , respectively, there exists η ∈ (0 , 1] such that where the implicit constants are independent of n . Remark 1.3. We remark that instead of (3), the original conjecture in [2] only requires the bound sup n | ∂g n | < ∞ . Our stronger assumption (3) which concerns also the second derivative is imposed in the spirit of [16] , which fixes the frequency scale. However, it remains an open problem to prove (or disprove) Burnett's conjecture in generalized wave coordinates without any assumptions on the second derivatives of g n . We need a few definitions to introduce our main theorem. Definition 1.4. (1) Let T ∗ U ≡ U × R d +1 denote the cotangent bundle of U and let S ∗ U = U × (( R d +1 \\ { 0 } ) / ∼ ) denote the cosphere bundle, where ξ ∼ η if and only if ξ = αη for some α > 0 . The main result of the paper is the following theorem: Theorem 1.5. Burnett's conjecture (Conjecture 1.1) is true under Assumption 1.2. More precisely, under (1)-(4) in Assumption 1.2, define µ by where µ αβσρ is a Radon measure 1 on S ∗ U defined 2 so that for h = g n -g 0 and after passing to a subsequence (still labelled by n ), the following holds for all 0 -th order pseudo-differential operator A with principal symbol a ( x, ξ ) which is real and 0 -positively homogeneous in ξ : Then ( U, g 0 , µ ) satisfies the Einstein-massless Vlasov system, namely, the following all hold: (1) ( µ is a massless field, i.e., µ is supported on the light cone) where {· , ·} denotes the Poisson bracket { f, h } . = ∂ ξ µ f∂ x µ h -∂ x µ f∂ ξ µ h. The proof of the four parts of Theorem 1.5 can be found in Proposition 4.4, Proposition 4.6, Proposition 4.7 and Theorem 5.1, respectively. Remark 1.6. Theorem 1.5 includes as a special case when the classical wave coordinate condition is satisfied, i.e., when H n , H 0 ≡ 0 . We remark that one motivation to consider generalized wave coordinates instead of just wave coordinates is that the former seems to come up naturally in the construction of high-frequency limits; see [33, 35] . Remark 1.7. Notice that Theorem 1.5 is more specific than Conjecture 1.1 in the sense that we give an explicit construction of the massless Vlasov field as the microlocal defect measure. where P µν is a quadratic nonlinearity violating the classical null condition; see (3.14). To understand whether Ric( g 0 ) µν = 0, we need to understand the weak limit of the right-hand side of (1.7) as n →∞ . The linear terms obviously converge weakly to 0; the null condition terms also converge weakly to 0 due to compensation compactness. It turns out that the main quasilinear term g αα ' 0 g ββ ' 0 h α ' β ' ∂ 2 αβ h µν has a hidden null structure which makes it also converge weakly to 0 (see (4.5)). Thus the only term that does not converge weakly to 0 is P µν ( g 0 )( ∂h, ∂h ). The microlocal defect measure µ is defined so that we exactly have For the linear terms, we need an exact cancellation, which indeed holds true and can be verified with explicit computations. For the nonlinear (semilinear and quasilinear) terms, we need to control trilinear terms of the type. where A is a 0-th order pseudo-differential operator. In general, terms of the form (1.8) need not converge to 0. However, it turns out that if there is a null form, e.g., if Q is one of the classical null forms, then using also that h satisfies wave equations, it can be shown that 〈 ∂h, A ( Q ( ∂h, ∂h )) 〉 → 0. After reducing to a constant coefficient problem by freezing coefficients, we prove this with the help of a trilinear estimate by Ionescu-Pasauder [15]; see Proposition 2.6. In particular, this allows us to treat the null condition terms in (1.7). However, the P term in (1.7) violates the classical null condition. Nonetheless, there is a much more subtle trilinear structure, meaning that after suitable algebraic manipulations using the generalized wave coordinate condition, one can reveal a hidden null structure and can deal with this term using ideas in the paragraph above. We notice that this structure is present only when we consider the propagation equation for µ in (1.5), and does not hold for the individual µ αβρσ ! An additional interesting challenge comes from the quasilinear terms g µµ ' 0 g νν ' 0 h µν ∂ 2 µ ' ν ' h α ' β ' . Here, we need to decompose h µν into h µν = ∑ 3 i =1 h ( i ) µν according to their frequencies (see precise decomposition in Definition 5.9). The term h (1) µν has low frequency, so ∂ h (1) µν is o (1). This can then be exploited using the Calder'on commutator theorem in a similar manner as [12]. The term h (2) µν has high frequency, but the frequency lives away from the light cone of g 0 . In this frequency regime, the wave operator /square g 0 is elliptic, and thus the control of /square g 0 h that we get from the equation gives very strong compactness. Finally, the term h (3) µν has high frequency with the frequency possibly close to the light cone. Here is the most delicate case, but it can be dealt with after noting that there is a secret trilinear null structure if we use the generalized wave coordinate condition. The structure is a bit similar to the P terms discussed above. Under a codimensional 2 symmetry assumption, compactness questions as in Conjecture 1.1 was studied in [18, 19]. More generally, a complete characterization of high-frequency limits was obtained by LukRodnianski in [25] under angular regularity assumptions, where they consider a setting where the manifold is given by U 2 × S 2 , but instead of exact symmetries along S 2 they only required high regularity along directions of S 2 . Beyond the angularly regular settings, we proved Conjecture 1.1 in [12] for U (1) symmetric solutions under an elliptic gauge condition. In particular, we gave a formulation of Conjecture 1.1 in terms of microlocal defect measures of [6, 32]; it is also within the same framework that we discuss Conjecture 1.1 here. See also the subsequent work [9] by Guerra-Teixeira da Costa in the same setting with a slightly different treatment of the time-dominated frequency regime. We particularly draw attention to the works of Touati [33, 35] (see also [34]), where the construction is carried out in generalized wave coordinates, consistent with the general framework of the present paper. His works can be viewed as a justification and generalization of the considerations in the pioneering work [3]. We first establish some important analytic and algebraic facts which are central to our proof. In Section 2 , we prove some trilinear estimates for null forms. In Section 3 , we analyze the algebraic structure of the Einstein equations. We then turn to the proof of Theorem 1.5. In Section 4 , we prove the first three parts of the theorem. Part (4) of the theorem, which concerns the propagation equation of the Vlasov measure is the most difficult part and is proven in Section 5 . Some proofs of the results from Section 3 are given in Appendix A . Acknowledgements. We thank John Anderson and Arthur Touati for helpful discussions. J. Luk gratefully acknowledges the support by a Terman fellowship and the NSF grant DMS-2304445.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "2. Trilinear compensated compactness for null forms", "content": "The goal of this section is to establish estimates corresponding to the trilinear compensated compactness for null forms. As pointed out already in the introduction, we rely on the estimates proven by IonescuPasauder [15]. After introducing some notations in Section 2.1 , we will first deal with the simpler Q 0 case in Section 2.2 and then turn to the harder case for Q µν in Section 2.3 .", "pages": [ 5 ] }, { "title": "2.1. Notations.", "content": "Definition 2.1. For p ∈ [1 , ∞ ] and λ ≥ 2 , define the ( λ, g 0 ) -dependent norm X p λ ( g 0 ) by Remark 2.2. The reason that we use the X p λ ( g 0 ) norm is that by the assumptions of g n -g 0 in Assumption 1.2 and the equation that it satisfies (see Proposition 3.8 below), ‖ g n -g 0 ‖ X p λ ( g 0 ) /lessorsimilar 1 for every p ∈ [1 , ∞ ] , uniformly as λ → 0 . 2.2. The Q ( g ) 0 null forms. Definition 2.3. Given a Lorentzian metric g , define the null form Q ( g ) 0 by The following estimate can be proven easily with integration by parts (cf. [12, Proposition 12.2], [9, Lemma 3.6]). Proposition 2.4. Suppose g is a smooth Lorentzian metric on an open U ⊂ R d +1 and { φ ( i ) } i =1 , 2 , 3 ⊂ C ∞ c ( U ; C ) are supported in a fixed compact set K ⊂ U . ∞ Then for every f ∈ C c where the implicit constant may depend on f , g and K , but is independent of λ and ( φ (1) , φ (2) , φ (3) ) .", "pages": [ 5 ] }, { "title": "Proof. The key is to write", "content": "The integral of the last two terms is bounded by respectively, and are both acceptable by Definition 2.1. For the remaining term, we integrate by parts. The main contribution can be bounded by ‖ /square g 0 φ (3) ‖ L 2 ‖ ∂ α φ (1) ‖ L ∞ ‖ φ (2) ‖ L 2 or ‖ /square g 0 φ (3) ‖ L 2 ‖ φ (1) ‖ L ∞ ‖ ∂ α φ (2) ‖ L 2 and are therefore acceptable as before. The error terms arising from differentiating f in the process of the integration by parts are better. /square", "pages": [ 5 ] }, { "title": "Definition 2.5.", "content": "We have a result similar to Proposition 2.4. Notice that we need a global smallness condition (2.3) and that the power of λ that we gain is weaker. Proposition 2.6. Suppose g is a smooth Lorentzian metric on an open set U ⊂ R d +1 and { φ ( i ) } i =1 , 2 , 3 ⊂ C ∞ c ( U ; C ) are supported in a fixed compact set K ⊂ U . Assume, in addition, that g is C 0 close to the Minkowski metric in the following sense: where N , a i , b i , c i , B i j are smooth functions satisfying Then for every f ∈ C ∞ ( U ; C ) , where the implicit constant may depend on f , g and K , but is independent of λ and ( φ (1) , φ (2) , φ (3) ) . In the constant coefficient case, this proposition is a direct result of the work of Ionescu-Pasauder (which in fact proves a slightly stronger result). We will first recall the estimate of Ionescu-Pasauder, and then adapt it to our case after freezing coefficients to reduce to the constant coefficient case. We introduce some conventions for Lemma 2.7 and Propoosition 2.8, which concern some estimates on the Minkowski spacetime. In Lemma 2.7 and Propoosition 2.8, we take the metric and the volume form to be that of Minkowski. Denote a point in Minkowski by ( t, x ). We will use the spatial Fourier transform denoted by Let P k denote the standard Littlewood-Paley projection, but only in the spatial variables x . Define also as in [15] the following convention for angles The following lemma is from [15] up to extremely minor modifications. Lemma 2.7 (Ionescu-Pasauder, Lemma 2.9 in [15]) . Let ι, ι 1 , ι 2 ∈ { + , -} , b ≤ 2 , f, f 1 , f 2 ∈ L 2 ( R 3 ) , and k, k 1 , k 2 ∈ Z . Let m be one of the following symbols: Then where Φ ιι 1 ι 2 ( ξ, η ) is the phase given by Then Proof. This is almost explicitly as in [15] (and we have essentially follow their notation, except for rewriting Φ σµν as Φ ιι 1 ι 2 ). The only difference is that we used the specific symbols in (2.5), instead of a symbol satisfying ‖F -1 m ‖ L 1 ( R 3 × R 3 ) ≤ 1 in [15]. The estimates are therefore also modified accordingly. The point is that we can repeat the proof of [15, Lemma 2.9] and noting that the symbol m ( θ, η ) satisfies the pointwise bound | m ( θ, η ) | /lessorsimilar 2 b on the intersection of the supports of various relevant cutoffs. Moreover, for the operators that are used in the proof of [15, Lemma 2.9], the symbol m satisfies on the region where the integrand is non-vanishing. One can thus repeat the same proof to obtain the desired estimate. /square Proposition 2.8. The following estimate holds for all δ ≤ 1 and all β, µ, ν ∈ { 0 , 1 , · , d } : where /square M denotes the Minkowskian wave operator. Proof. Fix δ ≤ 1 and choose b ∈ Z such that δ ≤ 2 b ≤ 10 δ . We introduce some notations for the remainder of the proof. First, following [15], we denote Note that G , H are linear in the first two slots, but conjugate linear in the third. We also define R j and |∇| to be the following operators: We perform normal form as in [15]. Define ψ ( j, ± ) by In particular, the following two identities hold: Now taking spatial Fourier transform F , the term 〈 ∂ β P k 3 φ 3 , Q µν ( P k 1 φ 1 , P k 2 φ 2 ) 〉 can be written as linear combinations of terms of the form for ι 1 , ι 2 , ι 3 ∈ { + , -} and with m being one of the symbols in (2.5). Let us control the first type of terms in (2.13); the second type of terms are similar. We split the term into two parts: where χ 1 = χ 1 (2 -b Ξ ι 1 ,ι 2 ( ξ -η, η )) (recall (2.4)). For G χ 1 m [ ψ 1 ,ι 1 , ψ 2 ,ι 2 , ψ 3 ,ι 3 ], we use (2.6) and compact support in t to obtain For G (1 -χ 1 ) m [ ψ 1 ,ι 1 , ψ 2 ,ι 2 , ψ 3 ,ι 3 ], we note that where we used (2.11) in the last equality. We then integrate in t over R in t , and note that the last term, which is a total ∂ t derivative, drops. After using compact support, the time-integral of the remaining three terms can be bounded using (2.7). After summing over all dyadic scales 2 b ' ≥ 2 b , we obtain Combining (2.14) and (2.16), and recalling that δ ≤ 2 b ≤ 10 δ , we obtain the desired estimate. /square Corollary 2.9. Suppose g is a constant-coefficient metric where N , a i , b i , c i , B i j are constants such that B i j has vanishing diagonal entries and Then the following modification of the estimate in Proposition 2.8 holds: with constants independent of g , where the Littlewood-Paley projection P k is to be understood in the spatial coordinates (˜ x 1 , ˜ x 2 , ˜ x 3 ) for the coordinate system ( ˜ t, ˜ x 1 , ˜ x 2 , ˜ x 3 ) defined by Proof. The argument is exactly the same except for changing to the ( ˜ t, ˜ x 1 , ˜ x 2 , ˜ x 3 ) coordinates. The smallness condition (2.17) ensures that the implicit constants can be made independent of g . /square We can now prove Proposition 2.6. Proof of Proposition 2.6. By scaling, we can assume that Step 1: Spatial cutoffs. Let ε 0 = 1 2 . Cover the compact set K with O ( λ -4 ε 0 ) cubes of side-lengths λ ε 0 , labelled by C α . Let { ζ 3 α } be a smooth partition of unity corresponding to these cubes. The cut-off ζ α can be chosen so that ‖ ∂ k ζ α ‖ L 1 /lessorsimilar λ (4 -k ) ε 0 . Define φ ( i ) α . = ζ α φ ( i ) . Since the cutoff functions live at a larger scale than λ , it is easy to check using the X ∞ λ ( g 0 ) norm bound that Let g α be the constant coefficient Lorentzian metric given by g at the center of the ball. Then ‖ g - g α ‖ L (2 B ) /lessorsimilar λ ε 0 . ∞ α /square g α is now a constant coefficient wave operator. We will estimate the size of /square g α φ ( i ) α . First, we estimate Using this, we deduce For φ (2) and φ (3) , we only have L 2 -based bounds. For these, we argue similarly as above, but we use also orthogonality to obtain Step 2: A simple reduction. Let f α be the value of f at the center of C α . Since f is smooth, by the mean value theorem, | f -f α | /lessorsimilar λ ε 0 on the support of ζ α . Hence, where we used ε 0 = 1 2 in the last step. Step 3: Fourier cut-off. To proceed, we further introduce another spatial cut-off in Fourier space. Here, 'spatial' is to be understood with respect to the ( ˜ t, ˜ x 1 , ˜ x 2 , ˜ x 3 ) defined in (2.19) corresponding to the constant coefficient metric g α . Let a = 1 14 . For each i and α , define ˜ φ ( i ) α . = F -1 ( χ F ( ξ ) F φ ( i ) α ) , where χ F ( ξ ) is a smooth cutoff function such that supp( χ F ) ⊂ { 1 2 λ -1+ a ≤ | ξ | ≤ 2 λ -1 -a } and χ F ( ξ ) ≡ 1 when λ -1+ a ≤ | ξ | ≤ λ -1 -a . Using (2.20), (2.21) and (2.22), it is easy to see that Moreover, φ ( i ) α -˜ φ ( i ) α either has spatial frequency lower than 1 2 λ -1+ a or higher than 2 λ -1 -a . Denote σ ( i ) α = φ ( i ) α -˜ φ ( i ) α and decompose σ ( i ) α = σ ( i ) ,L α + σ ( i ) ,H α , where σ ( i ) ,L α and σ ( i ) ,H α have low- and high-frequency, respectively. Notice that 2 -k ˜ ∂P k and P k both have a kernel in L 1 and thus the operators are bounded both on L 2 and on L ∞ on a fixed dyadic frequency. Thus, using (2.24), it follows that the spatial derivatives of φ ( i ) α -˜ φ ( i ) α (denoted by ∂ ∈ { ∂ ˜ x 1 , ∂ ˜ x 2 , ∂ ˜ x 3 } ) obey improved estimates: and Combining and arguing similarly for φ ( i ) α -φ ( i ) α when i = 2 , 3, we obtain We now derive estimates similar to (2.25), but for the ˜ ∂ ˜ t derivative, using the estimates for /square g α ˜ φ ( i ) α and suitable elliptic estimates. We begin with ∂ ˜ t ( φ ( i ) α -φ ( i ) α ). For σ ( i ) ,L α , σ ( i ) ,H α defined above, note that and also ‖ σ (1) ,L α ‖ L ∞ ‖ ˜ ∂ 2 σ (1) ,L α ‖ L ∞ /lessorsimilar λ 2 a . Thus using the one-dimensional bound ‖ f ' ‖ 2 L ∞ ≤ 4 ‖ f ‖ L ∞ ‖ f '' ‖ L ∞ (see [28, Chapter 5, Problem 15]), we obtain ˜ where in the penultimate inequality we used (2.24) and (2.25) and in the final inequality we used 2 a = 1 7 < 1 2 . For ∂ ˜ t ( φ ( i ) α -φ ( i ) α ) when i = 2 , 3, we integrate by parts and use the wave operator bound as follows: where we used (2.24) and (2.25) similarly as in (2.26). Hence, using Holder's inequality and (2.24), (2.25), (2.26), (2.27), we obtain which is acceptable since a = 1 14 > 1 15 . By (2.23) and (2.28), it therefore suffices to bound the term Step 3: Applying the Ionescu-Pasauder normal form bounds. Now, each term in the summand in (2.29) can be treated with the Ionescu-Pasauder estimate (2.18). We decompose each ˜ φ ( i ) into Littlewood-Paley pieces. Notice that because of the Fourier cut-offs in the previous step, all factors of 2 k 1 -k 3 in (2.18) are at worst O ( λ -2 a ), and the 2 -min { k 1 ,k 2 } factor can be bounded above by O ( λ 1 -a ). Therefore, we have where in the last line we used (2.24). We then choose δ = λ 1 4 -a 2 (which is ≤ 1 as required for λ sufficiently small) to optimize the above bound so that Step 4: Concluding the argument. Finally, we sum over all α and ( k 1 , k 2 , k 3 ) in order to bound the term (2.29). First, note that the /lscript 2 sums in (2.24) allows us to sum over all α to obtain Due to the Fourier cut-offs, there are at most O ((log λ ) 3 ) terms in the sum in ( k 1 , k 2 , k 3 ) (one log λ from each k i ). We therefore bound the term in (2.29) as follows: which is acceptable after recalling a = 1 14 . This concludes the proof. /square", "pages": [ 5, 6, 7, 8, 9, 10 ] }, { "title": "3. Einstein equations in wave coordinates", "content": "The goal of this section is to derive the equation for h ; see Proposition 3.8. In order to achieve this, we carry out some computations for the Ricci curvature. 3.2. Localization of the problem. The following lemma is easy to check, and is an immediate consequence of the continuity of g 0 . Lemma 3.1. Given any x ∈ U , there exists a smaller open set U ' satisfying x ∈ U ' ⊂ U such that after a linear change of variables, where m is the Minkowski metric. Moreover, assumptions (1)-(4) in Assumption 1.2 continue to hold, except that the bounds in (1.1) , (3) and (1.4) may worsen by a fixed (i.e., λ -independent) multiplicative constant. From now on, we assume that (3.1) hold. We also expand out Assumption 1.2 in the following lemma, now in the conventions defined in Section 3.1 above. Lemma 3.2. Under the above conventions and Assumption 1.2, the following pointwise bounds for the metrics hold and the following pointwise bounds for H α = H α ( g λ )( ∂g λ ) and H α 0 = H α ( g 0 )( ∂g 0 ) hold Using (3.2) , the bound | H -H 0 | /lessorsimilar λ η also implies the following: 3.3. Computations for the Ricci tensor. Since these computations are straightforward but tedious, the proofs will be relegated to the appendix. Define the following notation for the Christoffel symbol: The Ricci curvature takes the following form. The proof can be found in Appendix A. Lemma 3.3 (Ricci curvature in generalized wave coordinates) . The following identity holds for any C 2 Lorentzian metric g : where 3 We now take the expression in Lemma 3.3 and derive an equation for /square g 0 ( g λ -g 0 ) by considering Ric( g λ ) -Ric( g 0 ). In the next few lemmas, we consider the contributions from the different terms in Lemma 3.3. The proofs of Lemma 3.3-Lemma 3.6 are relegated to Appendix A, while the proof of Lemma 3.7 is straightforward and omitted. Lemma 3.4 (Linear structure of B µν ) . Under the assumption (3.2) , the following holds (see (3.9) for definition of B µν ): where and )( ∂h ) = 4 g σρ 0 Γ ρ ( µ | ( g 0 ) ∂ σ h | ν ) α + D ασ ( µ | ( g 0 ) ∂ | ν ) h ασ (3.11) Lemma 3.5 (Nonlinear structure of B µν ) . Under the assumption (3.2) , B µν (see (3.9) ) can be decomposed as where Q µν and P µν are given by and In particular, Q µν is a linear combination of terms satisfying the classical null condition, i.e., it can be written as a linear combination of the null forms in Definition 2.3 and Definition 2.5. Lemma 3.6 (Quasilinear terms) . Under the assumption (3.2) , the following holds: ˜ /square g λ ( g λ ) µν -˜ /square g 0 ( g 0 ) µν = ˜ /square g 0 h µν -g αα ' 0 g ββ ' 0 h α ' β ' ∂ 2 αβ h µν + O ( λ ) . Lemma 3.7 (Terms related to the wave coordinate condition) . Under the assumptions (3.2) and (3.3) , the following holds: Putting together the above lemmas, we obtain the following main wave equation for h : . L µν ( g 0 α Proposition 3.8 (Structure of /square g 0 h ) . Under the assumptions (3.2) and (3.3) , the following holds: where /square g 0 is understood as the wave operator for scalar functions, i.e., /square g 0 = g αβ 0 ∂ 2 αβ -H α ( g 0 )( ∂g 0 ) ∂ α . In particular, we also have Proof. This is an immediate consequence of Lemmas 3.3, 3.4, 3.5, 3.6 and 3.7. Notice that we combine ˜ /square g 0 h µν and the term H ρ 0 ∂ ρ h µν in Lemma 3.7 to form /square g 0 h µν . /square", "pages": [ 11, 12, 13 ] }, { "title": "4. Microlocal defect measure and the limiting Ricci curvature", "content": "We now begin the proof of Theorem 1.5. We work under Assumption 1.2 and (3.1), and continue to use the conventions introduced in Section 3.1. Our goal in this section will be to prove parts (1)-(3) of Theorem 1.5.", "pages": [ 13 ] }, { "title": "4.1. Definition of the microlocal defect measure.", "content": "Definition 4.1. Define µ αβρσ so that the following holds for all A ∈ Ψ 0 with principal symbol a ( x, ξ ) which is real, positively 0 -homogeneous and has compact support in x : where we take µ αβρσ to be a measure on S ∗ U by acting on functions which are positively 2 -homogeneous in ξ (see Definition 1.4), and the lim λ → 0 limit is to be understood after passing to a (further) subsequence. Given µ αβρσ , we then define µ by (1.5) . Remark 4.2. (1) (Existence of µ αβρσ ) The existence of the measures µ αβρσ follows from the standard theory of microlocal defect measures. By our assumptions, ∂h is L 2 -bounded uniformly in λ . Thus, results in [6, 32] show that after passing to a subsequence, there is a measure ˜ µ γαβδρσ such that with ˜ µ γαβδρσ acting on positively 0 -homogeneous functions. Finally, using localization lemma for microlocal defect measures [6, Corollary 2.2] and the commutation of mixed partials, it can be shown that ˜ µ γαβδρσ = ξ γ ξ δ µ αβρσ . A similar and more detailed argument can be found for instance in [4] or [12, Sections 6.1, 6.2] . (2) (Symmetries of µ αβρσ ) It is easy to check that Indeed, the first equality follows from the fact that a is also the principal symbol of A ∗ and the second equality follows from the fact that h is a symmetric tensor. The following fact will be useful, and is a consequence of the generalized wave coordinate condition. Lemma 4.3. For any indices γ, σ, ν , Proof. This follows from (3.4) and Definition 4.1. /square 4.2. Proof of (1)-(3) in Theorem 1.5. With the definition of µ , we now prove parts (1)-(3) of Theorem 1.5. We start with showing that µ is supported on the null cone.", "pages": [ 13 ] }, { "title": "Proposition 4.4.", "content": "Proof. We prove the stronger statement that g αβ 0 ξ α ξ β d µ ρσγδ = 0 for any indices ρ , σ , γ , δ . This is easy and well-known; see for instance [4]. Since h → 0 (strongly in L 2 ) and /square g 0 h is uniformly bounded by (3.16) On the other hand, integrating by parts and using [ A, /square g 0 ] ∈ Ψ 1 , we obtain Since a is arbitrary, it follows that g αβ 0 ξ α ξ β d µ ρσγδ ≡ 0, as desired. /square Remark 4.5. Notice that in the proof of Proposition 4.4, we used the uniform boundedness of /square g 0 h , which in particular required the assumed bound on ∂ 2 h on ∂H . We mentioned in Remark 1.3 that the assumptions on ∂ 2 h and ∂H are not necessary. To see this, consider the unbounded term in (3.15) involving ∂ 2 h , i.e., g αα ' 0 g ββ ' 0 h α ' β ' ∂ 2 αβ h γδ . Its contribution to (4.3) is which also → 0 , which can be seen after integration by parts. A similar comment applies to the term involving ∂H . Next, we turn to part (2) of Theorem 1.5. Proposition 4.6. The limiting Ricci curvature is given by where µ = g αρ 0 g βσ 0 ( 1 4 µ ρβασ -1 8 µ ραβσ ) . Proof. We will use without comment the standard facts (1) p λ → p uniformly and q λ ⇀ q weakly in L 2 implies p λ q λ → pq in distribution and (2) p λ → p in distribution implies all its derivatives converge in distribution. We now consider Proposition 3.8, thought of as an expression of Ric( g 0 ), pair it with some ψ ∈ C ∞ c ( R d +1 ) and consider the lim λ → 0 limit. First note that the terms /square g 0 h and L ( g 0 )( ∂h ) are linear in (the derivatives of) h thus converge weakly to 0 as λ → 0. Similarly, the term 2( g 0 ) α ( µ ∂ ν ) ( H α -H α 0 ) is linear in the derivative of H α -H α 0 and thus converges weakly to 0 as λ → 0. For the quasilinear term g αα ' 0 g ββ ' 0 h α ' β ' ∂ 2 αβ h µν , we note that where in the second equality we used the wave coordinate condition (3.4). Notice now that term I is a total derivative of an O ( λ ) term and thus converges weakly to 0; term II also converges weakly to 0 because it can be written as 1 2 g αα ' 0 Q ( g ) 0 ( h α ' α , h µν ). For the null forms, we simply notice that Q ( g ) 0 ( φ, ψ ) = 1 2 /square g 0 ( φψ ) -1 2 φ /square g 0 ψ -1 2 ψ /square g 0 φ , and that Q αβ ( φ, ψ ) = ∂ α ( φ∂ β ψ ) -∂ β ( φ∂ α ψ ) so that in both cases Q ( φ λ , ψ λ ) ⇀ Q ( φ, ψ ). It thus follows that the only contribution from Proposition 3.8 that does not converge weakly to 0 comes from the term P µν ( g )( ∂h, ∂h ). However, this contribution is exactly ∫ S ∗ R d +1 ψξ µ ξ ν d µ by (3.14), (1.5) and Definition 4.1. /square Next, we turn to non-negativity of µ , i.e., statement (3) of the main theorem (Theorem 1.5). Let us remark that once we have obtained Proposition 4.6, the non-negativity of µ already follows from the fact, established in [7], that the weak energy condition holds for the limiting spacetime (irrespective of gauge conditions). Here, however, we give a direct proof of the non-negativity. Proposition 4.7. The measure µ is real-valued and non-negative. Proof. Step 1: Definition of null frame adapted to ξ . Given ( x, ξ α ) on the support of µ , denote n = g tα 0 ∂ α , ξ α = g αβ 0 ξ β and introduce the following ξ -dependent vector fields: /negationslash These vector fields are well-defined on the support of d µ since ξ t = 0 (by Proposition 4.4 and (3.1)). These vector fields are chosen so that the following relations hold: The relations (4.7) can be checked by direct computations, after noting that Step 2: Some computations. Before we proceed, we collect some computations. From now on, denote by g 0 the spatial part of g . In particular, g ij 0 denotes the ( ij )-component of the inverse of g 0 . It relates to the g ij 0 coming from the inverse of g 0 through which can be derived by standard formulas on inverses of block diagonal matrices. It is also convenient to note the following rewritings of g µν 0 ξ µ ξ ν = 0 (on supp( µ ), see Proposition 4.4): Next, we compute that L ( ξ ) t = ξ t , L ( ξ ) t = -g tt 0 ξ t , L ( ξ ) x /lscript = ξ /lscript and L ( ξ ) x /lscript = g tt 0 ξ /lscript ( ξ t ) 2 -2 g ti 0 ξ t . From this, and using (4.8), (4.9), it follows that / ∂ i t = 0 (so that / ∂ i is tangential to constantt hypersurfaces), and that g ij 0 ξ j / ∂ i t = g ij 0 ξ j / ∂ i x /lscript = 0. In particular, { / ∂ 1 , · · · , /∂ d } are linearly dependent, as they satisfy the relation Notice also that since { L ( ξ ) , L ( ξ ) , /∂ ( ξ ) 1 , · · · , /∂ ( ξ ) d } obviously span the whole tangent space, locally we can find d -1 linearly independent and spacelike elements in { / ∂ ( ξ ) 1 , · · · , /∂ ( ξ ) d } and perform Gram-Schmidt to obtain an orthonormal frame { /e ( ξ ) B } d -1 B =1 so that span { /e ( ξ ) B } d -1 B =1 = span { / ∂ ( ξ ) i } d i =1 and such that g 0 ( /e ( ξ ) B , /e ( ξ ) C ) = δ BC . From now on, fixed such a local orthonormal frame { /e ( ξ ) B } d -1 B =1 . We will use the convention that capital Latin indices run over B,C = 1 , · · · , d -1 and that repeated indices will be summed over this range. Using such an orthonormal frame and recalling the relations (4.7), we can write the inverse metric as follows: Step 3: Analyzing the microlocal defect measure using the orthonormal frame. We recall (4.2) and contract it with ( L ( ξ ) ) γ , ( /e ( ξ ) A ) γ and ( L ( ξ ) ) γ to get Since ( L ( ξ ) ) γ ξ γ = 0 and ( /e ( ξ ) A ) γ ξ γ = 0 (by (4.6), (4.7)), after recalling L ( ξ ) in (4.6), we use (4.12) and (4.13) to obtain Since ( L ( ξ ) γ ) ξ γ = - 2 (by (4.6), (4.7)), we use (4.14) to obtain We now expand the g 0 in µ = g αρ 0 g βσ 0 ( 1 4 µ ρβασ -1 8 µ ραβσ ) using (4.11) and (4.15) to get where the second line defines the notation µ / B/C / B ' / C ' . Note that ( L ( ξ ) ) ρ ( L ( ξ ) ) β ( L ( ξ ) ) α ( L ( ξ ) ) σ µ ρβασ cancels. Using also (4.16), we obtain /negationslash Step 4: Completing the proof. Since ξ t = 0 on supp( µ ), in order to prove the proposition, it suffices to show that for any χ ( x, ξ ) which is real, compactly supported in x and positively 0-homogeneous in ξ , By cutting off further if necessary, we assume the local orthonormal frame { /e ( ξ ) B } d -1 B =1 is well-defined on supp( χ ). For B = 1 , · · · , d -1, let A χ ( /e ( ξ ) B ) α be a 0-th order pseudo-differential operator with principal symbol χ ( x, ξ )( /e ( ξ ) B ) α (which is well-defined on supp( χ ) and is positively 0-homogeneous in ξ ). Then, using (4.18), which is manifestly non-negative. /square", "pages": [ 14, 15, 16 ] }, { "title": "5. Propagation of the microlocal defect measure", "content": "We continue to work under Assumption 1.2 and (3.1), and use the conventions introduced in Section 3.1. The goal of this section is to prove the following theorem. This proves part (4) of Theorem 1.5 and thus completes the proof of the main theorem. Theorem 5.1. The following identity holds for any smooth ˜ a : S ∗ U → R which is compactly supported in x and positively 1 -homogeneous in ξ : /negationslash For the remainder of the section, fix ˜ a that satisfies the assumption of Theorem 5.1 and define a ( x, ξ ) = ˜ a ( x,ξ ) ξ t . (Note that this is well-defined because ξ t = 0 on supp( µ ) by (3.1) and Proposition 4.4.) For K as above, fix another open set K ' ⊂ U such that K ⊂ ˚ K ' . Let χ be a smooth cutoff function such that supp( χ ) ⊂ K ' and χ ≡ 1 on K . From now on, we replace h by χh so that it is C ∞ c , which makes taking Fourier transforms easier. Moreover, we can now work globally in the whole space R d +1 , with g 0 extended outside ˚ K ' so that (3.1) holds globally and the C k norm is globally controlled for all k ∈ N . The choice of the extension of g 0 will not change the derivation of (5.1). Notice that after introducing the cutoff, all the estimates (3.2), (3.3), (3.4) and (3.16) still hold. However, the equation (3.15) no longer holds globally in U , but importantly it holds on supp( a ). This will already be sufficient in the proof of Proposition 5.3. We will work under these cutoff assumptions for the remainder of the paper. 5.2. Main identity for the propagation of µ . In this subsection, we derive the main propagation identity for µ ; see Proposition 5.4 below. We begin with a general lemma. Lemma 5.2. Let g 0 be as before. Let φ λ , ψ λ be smooth functions supported in a fixed compact set in R d +1 which (1) are uniformly bounded in H 1 , (2) satisfy φ λ , ψ λ → 0 in L 2 , and (3) are such that /square g 0 φ λ , /square g 0 ψ λ are uniformly bounded in L 2 . Define d µ φψ to be the cross microlocal defect measure, i.e., for any A ∈ Ψ 0 with positively 0 -homogeneous. principal symbol a , (up to a subsequence) Then for any A ∈ Ψ 0 whose principal symbol is a real, positively 0 -homogeneous Fourier multiplier m ( ξ ) , and any vector field X , we have Proof. In this proof, we write g = g 0 , φ = φ λ , ψ = ψ λ whenever it does not create confusion. Define Let ∇ be the Levi-Civita connection associated to g 0 . It is easy to check that It then easily follows that where ( X ) π µν . = ∇ ( µ | X | ν ) is the deformation tensor of X . Integrating (5.2) in the whole space using Stoke's theorem, and taking the λ → 0 limit, we obtain where we have used that g αβ ξ α ξ β = 0 on the support of µ φψ . We then compute where the last equation can be obtained using ∇ ρ g µν 0 = 0. Finally, we compute Putting all these together yields the conclusion. /square We now return to the setting of part (4) of Theorem 1.5, imposing, in addition, the assumptions in Section 5.1. Proposition 5.3. Let A ∈ Ψ 0 with principal symbol a ( x, ξ ) which is real. Define ˜ a ( x, ξ ) = ξ t a ( x, ξ ) . Then the following identity holds: Proof. By the Stone-Weierstrass theorem, it suffices to check the identity when a ( x, ξ ) = f ( x ) m ( ξ ), where f ∈ C ∞ c ( R d +1 ; R ) and m is real and positively 0-homogeneous. Define A m ∈ Ψ 0 with principal symbol m as above. By Lemma 5.2, given any vector field X , it holds that where we have used that [ A m , X ] , A m -( A m ) ∗ ∈ Ψ -1 Applying (5.5) with X = fg αα ' 0 g ββ ' 0 ∂ t , we obtain . We can compute ∫ S ∗ R d +1 g αβ 0 g α ' β ' 0 { g µν 0 ξ µ ξ ν , f ξ t m ( ξ ) } d µ αβα ' β ' in a similar manner. Thus, The desired conclusion hence follows from the definition µ . = 1 4 ( g αα ' 0 g ββ ' 0 -1 2 g αβ 0 g α ' β ' 0 ) µ αβα ' β ' . /square Combining the result above with Proposition 3.8, we obtain our main propagation identity: Proposition 5.4. The following identity holds: Proof. We start with (5.4) and use the equation for /square g 0 h from Proposition 3.8. Since g 0 , Ric( g 0 ) are smooth, it follows from integration by parts and | h | → 0 that The next five terms in Proposition 3.8 give the corresponding five terms in (5.8). Finally, the O ( λ η ) contribution in Proposition 3.8 vanishes in the λ → 0 limit and do not contribute to (5.8). /square The remainder of the paper thus involves handling the four terms on the right-hand side of (5.8). 5.3. The quasilinear term. The main goal of this subsection is the following proposition:", "pages": [ 16, 17, 18, 19 ] }, { "title": "Proposition 5.5.", "content": "5.3.1. A preliminary reduction. We start with a preliminary observation, namely that we can replace all instances of h in (5.9) by their frequency cutoff versions. The idea is related to that in Step 3 in the proof of Proposition 2.6, except now we use a spacetime Fourier cutoff. Lemma 5.6. Define χ : [0 , ∞ ) → [0 , 1] to be a cutoff function supported in {| ξ | ≤ 2 } and such that χ ≡ 1 when on [0 , 1] . Define h /LeftScissors αβ so that F h /LeftScissors αβ ( ξ ) . = χ ( λ 1 . 01 | ξ | ) F h αβ ( ξ ) . Then Proof. For notational convenience, we write ¯ h = h -h /LeftScissors in the remainder of this proof. The key estimate is that ‖ ∂ ¯ h ‖ L p /lessorsimilar λ 1 . 01 ‖ ∂ 2 h ‖ L p /lessorsimilar λ 1 . 01 λ - 1 /lessorsimilar λ 0 . 01 , ∀ p ∈ [1 , ∞ ] . (5.11) Importantly, this is better than the estimate for ∂h itself. We also have the following estimates for ¯ h and h /LeftScissors , which follow easily from the definitions of the cutoffs and (3.2): Writing h = h /LeftScissors + ¯ h in (5.10), we need to control the following three terms: For (5.13), we use that A : L 2 → L 2 is bounded and the Holder inequality to obtain where we used (5.11) together with (3.2) and (5.12). The term (5.14) can be treated similarly. For the term (5.15), we need to integrate by parts. First note that A -A ∗ , [ A,g 0 ] ∈ Ψ -1 and are bounded as maps L 2 → H -1 . Thus, using the bounds (3.2) and (5.12), we obtain We now integrate by parts the ∂ µ ' away so as to utilize (5.11). Notice that if ∂ µ ' hits on g 0 , this gives a much better term. Moreover, [ ∂, A ] ∈ Ψ 0 also gives better terms. We thus have where as before, we used (3.2), (5.11) and (5.12). /square The reason that it is useful to consider h /LeftScissors instead of h itself is the following lemma. Note that the bound for ‖ h /LeftScissors ‖ X p λ ( g 0 ) is no better than the bounds for h given in (3.2), but the frequency cut-off gives us access to third derivatives of h /LeftScissors and obtain a bound for ‖ ∂h /LeftScissors ‖ X ∞ λ ( g 0 ) . This improvement will be used in the proof of Proposition 5.15 below. Lemma 5.7. h /LeftScissors satisfies Moreover, ∂h /LeftScissors satisfies the estimates 5.3.2. Setting up the Fourier decomposition. In order to estimate the term in Proposition 5.5, we need to decompose h µν into three pieces using suitable frequency cutoff functions. The reader should think of h (1) as 'low frequency', h (2) as 'spatial frequency dominated,' and h (3) as 'temporal frequency dominated.' Moreover, the 'spatial frequency dominated' part is chosen so that the frequency is supported away from the light cone. We first fix a parameter δ Θ that we will use to define the decomposition. Lemma 5.8. Given ξ ∈ T ∗ R d +1 , denote its spatial part by ξ . Then there exists δ Θ > 0 such that Proof. This is a consequence of (3.1). /square From now on, fix δ Θ so that Lemma 5.8 holds. We now introduce the decomposition of h µν .", "pages": [ 19, 20 ] }, { "title": "Definition 5.9. Decompose", "content": "where we define for Θ : [0 , ∞ ] → [0 , 1] being a smooth cutoff function such that Θ ≡ 1 on [0 , 1] and Θ ≡ 0 on [2 , ∞ ] , and for being a fixed constant. It will be convenient to denote the corresponding projection operators by P ( i ) so that P ( i ) h = h ( i ) . In the next three subsections, we will consider the term (5.9) with h µν replaced by h (1) , h (2) and h (3) respectively. We will then combining the results to prove Proposition 5.5 in Section 5.3.6. 5.3.3. The h (1) term. We start with the h (1) term. The key property that we will use for h (1) is the following", "pages": [ 20 ] }, { "title": "Lemma 5.10.", "content": "We now control the term with h (1) . The key is that since ∂ h (1) is better by Lemma 5.10, we perform multiple integration by parts to ensure that a derivative falls on h (1) .", "pages": [ 20 ] }, { "title": "Proposition 5.11.", "content": "Proof. First, note that since g µµ ' 0 g νν ' 0 is smooth, the commutator [ A,g µµ ' 0 g νν ' 0 ] : L 2 → H -1 so that We then integrate by parts in ∂ µ ' . Note that if ∂ µ ' hits any factor of g 0 , we then have an integral of h ∂h∂h , which is o (1) by our assumptions. Hence, For term I , we use the estimate for ∂ h (1) in Lemma 5.10, which is sufficient to show that I = o (1). For term II , note that [ A,∂ µ ' ] is bounded on L 2 , and so using h /LeftScissors µν → 0 uniformly, we obtain II = o (1). Hence, Thus it remains to consider III in (5.23). For this term, we integrate by part in ∂ t . As before, if ∂ t hits any factor of g 0 , the resulting term in o (1). Hence, For III a in (5.25), we again use the Lemma 5.10 (as in the term I in (5.23)) to obtain smallness and get that As for III b in (5.25), we first relabel indices and then note that after commuting [ A, h (1) µν ], we obtain a term which is exactly -III . Combining (5.25)-(5.27), we thus obtain Combining (5.23)-(5.24), (5.28) then gives Finally, observe that [ A, h (1) µν ] ∂ 2 tµ ' h /LeftScissors αβ = [ A∂ t , h (1) µν ] ∂ µ ' h /LeftScissors αβ + o L 2 (1) using that ‖ ∂ h (1) µν ‖ L ∞ = o (1) by Lemma 5.10. On the other hand, the Calder'on commutator estimate (for the commutator of a pseudodifferential operator in Ψ 1 and a Lipschitz function, see, for instance, [29, Corollary, p.309]) imply Thus, using also Cauchy-Schwarz and Lemma 5.10, we obtain which gives the desired conclusion. /square 5.3.4. The h (2) term. The key property that we need for h (2) is the following improved estimate. This can be viewed as an elliptic estimate.", "pages": [ 20, 21 ] }, { "title": "Lemma 5.12.", "content": "Proof. Using Lemma 5.8, we can find S 2 ∈ Ψ -2 such that σ ( S 2 ) = ( g αβ 0 ξ α ξ β ) -1 when | ξ t | ≤ δ Θ | ξ | and | ξ | ≥ 1. In particular, there is an R 1 ∈ Ψ -1 such that h (2) = S 2 /square g 0 h (2) + R 1 h (2) (using the Fourier support of h (2) ). Therefore, where in the bound, we have used that (1 -Θ( λ -1 2 | ξ | ))Θ( 2 | ξ t | δ Θ | ξ | ) satisfies S 0 symbol bounds uniformly in λ and so ‖ /square g 0 h (2) ‖ L 2 /lessorsimilar ‖ /square g 0 h ‖ H 1 + ‖ h ‖ H 1 /lessorsimilar 1. Recalling that h (2) is defined to have frequency /greaterorsimilar λ -b , we thus have ‖ h (2) ‖ L 2 /lessorsimilar λ 2 b , ‖ h (2) ‖ H 1 /lessorsimilar λ b . /square With the bounds in Lemma 5.12, the desired estimate in this case is almost immediate.", "pages": [ 21 ] }, { "title": "Proposition 5.13.", "content": "Proof. By L 4 / 3 boundedness of A and Holder's inequality, we have 5.3.5. The h (3) term. The property that we will use for h (3) is captured in the following lemma. The lemma roughly says that h (3) has a nicely behaved anti-∂ t derivative. Lemma 5.14. For every λ ∈ (0 , λ 0 ) , there exists a ( λ -dependent) k µν such that the following holds: Recalling the Fourier support of h (3) , it follows that (1 -Θ(2 λ -b | ξ | ))(1 -Θ( 20 | ξ t | δ Θ | ξ | )) = 1 on the supp( F h (3) ). (3) Proof. Define S 1 ∈ Ψ -1 as a Fourier multiplier operator with Fourier multiplier m ( ξ ) . = 1 2 πi ξ -1 t ( 1 -Θ(2 λ -b | ξ | ) )( 1 -Θ( 20 | ξ t | δ Θ | ξ | ) ) , and for each component, let k µν . = S 1 h (3) µν = S 1 · P (3) h µν . From this, we obtain ∂ t k = h , which proves part (1). Before turning to parts (2) and (3), we derive some properties of S 1 . First, by Plancherel's theorem, we gain from the high ξ t frequency to obtain Second, notice that the multiplier m satisfies the symbol bounds | ∂ α ξ m ( ξ ) | /lessorsimilar | α | (1 + | ξ | ) -| α |-1 independently of λ . In particular, this implies by standard results on pseudo-differential operators that We now turn to the proof of (2). The first two estimates in (5.33) are simple consequence of (5.35) and (3.2). To prove the third estimate in (5.33), we note ‖ /square g 0 k ‖ L 2 /lessorsimilar ‖ [ S 1 · P (3) , /square g 0 ] ‖ L 2 → L 2 ‖ h ‖ L 2 + ‖ S 1 · P (3) ‖ L 2 → L 2 ‖ /square g 0 h ‖ L 2 /lessorsimilar min { λ, λ b } = λ b , (5.37) where we have used (5.35), (5.36), (3.2) and (3.16). Finally, turning to (3), we write = I + II + III. I , II are similar; we consider only I . Notice that the second estimate in (5.36) is by itself not sufficient to treat term I in (5.38) because we do not have improved bounds for ‖ h ‖ H -1 (compared to ‖ h ‖ L 2 ). Instead, we decompose h . = h high + h low , where h high and h low have frequency support /greaterorsimilar λ -b 2 and /lessorsimilar λ -b 2 , respectively. For h high , we use (5.36), ‖ h ‖ L 2 /lessorsimilar λ , the frequency support and Plancherel's theorem to obtain As for h low , we note that for λ small, the frequency support implies [ S 1 · P (3) , g νν ' 0 g µµ ' 0 ∂ µ ' ]( h low ) µν ' = S 1 · P (3) ( g νν ' 0 g µµ ' 0 ∂ µ ' ( h low ) µν ' ) and thus by (5.35) and then Plancherel's theorem, we obtain Combining (5.39), (5.40) for I in (5.38), and handling II similarly, we obtain Finally, for III in (5.38), we simply use (5.35) to obtain Combining (5.38), (5.41) and (5.42) yields (5.34). /square We now use Lemma 5.14 to handle the term (5.43) below. The key point is that after writing ∂ t k = h (3) (using Lemma 5.14), we can integrate by parts and use the wave coordinate type condition in (5.34) to reveal a null structure. The fact that the quasilinear terms have some hidden null structure is also used in [21], see also [15].", "pages": [ 22, 23 ] }, { "title": "Proposition 5.15.", "content": "Proof. Let k be as in Lemma 5.14. We now compute as follows: where in the first step we exchanged ∂ t and ∂ µ ' at the expense of a null form, in the second step we used the bound for H ( g 0 )( ∂ k ) in (5.34), and in the third step we noted that the commutation of A ∗ with g µµ ' 0 g αα ' 0 g ββ ' 0 ∂ γ and g µµ ' 0 g νν ' 0 g αα ' 0 g ββ ' 0 ∂ γ are in Ψ -1 . Finally, we claim that the remaining terms are o (1) after using the null form bounds in Proposition 2.4 and Proposition 2.6. For the first term, note that Proposition 2.4 allows us to put A ∗ h αβ , k µµ ' ∈ X 2 λ ( g 0 ) and ∂ ν ' h α ' β ' ∈ X ∞ λ ( g 0 ) so that since b ≥ 29 30 by (5.20). In the penultimate step above, we used that A ∗ : L 2 → L 2 is a bounded operator and applied the estimates in Lemma 5.7 and (5.33). The second term can be treated similarly except for using Proposition 2.6 instead so that we have since (by (5.20)) b -1 . 01 + 1 15 > 0. /square 5.3.6. Putting everything together. Proof of Proposition 5.5. This is an immediate consequence of the combination of Lemma 5.6, Proposition 5.11, Proposition 5.13 and Proposition 5.15. /square 5.4. The linear terms L µν ( g 0 )( ∂h ) . The goal of this subsection is the following bilinear estimate:", "pages": [ 23 ] }, { "title": "Proposition 5.16.", "content": "In order to prove Proposition 5.16, it is useful to first establish a lemma which relies on the wave coordinate condition.", "pages": [ 24 ] }, { "title": "Lemma 5.17.", "content": "Proof. Using the definition of the microlocal defect measure, we have which vanishes by Lemma 4.3. (We remark that the exact form of D does not play any role here.) /square We now return to the proof of Proposition 5.16: Proof of Proposition 5.16. To simplify the notations, we write Γ σ β α = Γ σ β α ( g 0 ). Recalling the definition of L in (3.11) and using Lemma 5.17 to handle the term involving D , we have LHS of (5.45)", "pages": [ 24 ] }, { "title": "We compute", "content": "We turn to term II . A completely analogous computation shows that Plugging (5.48) and (5.49) into (5.47) (and relabelling the indices) yields the desired conclusion. /square 5.5. The quadratic terms Q ( g 0 )( ∂h, ∂h ) and P ( g 0 )( ∂h, ∂h ) .", "pages": [ 24 ] }, { "title": "Proposition 5.18.", "content": "Proof. First using that A -A ∗ , [ A,g 0 ] ∈ Ψ -1 and [ A,∂ ] ∈ Ψ 0 and the usual bounds for h , we have Recall from (3.13) that Q ( g 0 )( ∂h, ∂h ) consists of a linear combination of null forms, thus after using Proposition 2.4 and Proposition 2.6, we obtain", "pages": [ 24, 25 ] }, { "title": "Proposition 5.19.", "content": "Proof. We treat the first term in (3.14) in detail; the other is very similar. The key point here is that after integrating by parts and using wave coordinate condition, we effectively have null condition terms: where in the first step, we used that the commutator [ A ∗ , g αα ' 0 g ββ ' 0 ∂ t ] gives rise to a lower order term; in the second step, we swapped the ∂ t and ∂ α derivative at the expense of a null form; in the third step, we used the wave coordinate condition in (3.3); finally, we use the trilinear estimate in Proposition 2.4 and Proposition 2.6 to handle the terms involving the null condition (in a similar manner as Proposition 5.18). /square", "pages": [ 25 ] }, { "title": "5.6. The wave gauge term.", "content": "Proposition 5.20. The following holds: Proof. We integrate by parts in ∂ t and ∂ β ' . The error terms from [ A,∂ t ], [ A,∂ β ' ] and the derivatives hitting on the smooth functions all give lower order contributions. Hence, where in the second step we used that the commutator [ A, ( g 0 ) α ' ρ ] or [ A, ( g 0 ) β ' ρ ] gives lower order contributions. Finally, by the wave coordinate condition (3.4) and the bound for ∂ t H in (3.3), we obtain which concludes the proof. /square", "pages": [ 25 ] }, { "title": "5.7. Putting everything together.", "content": "Proof of Theorem 5.1. We start with (5.8) in Proposition 5.4. By Proposition 5.16, the L µν ( g 0 )( ∂h ) term cancels the 1 4 ∫ S ∗ R d +1 g µν 0 ξ ν ˜ a ( x, ξ ) ∂ x µ (2 g αα ' 0 g ββ ' 0 -g αβ 0 g α ' β ' 0 ) d µ αβα ' β ' term in (5.8). By Propositions 5.5, 5.18, 5.19 and 5.20, all the remaining terms in (5.8) vanish. This concludes the proof of Theorem 5.1. /square Appendix A. Proofs of algebraic properties of the Einstein equations Proof of Lemma 3.3. To simplify notation, we will write H ρ = H ρ ( g )( ∂g ). We begin with the standard expression Note that Thus, and Plugging (A.3), (A.4), (A.5) into (A.1) yields the desired conclusion. /square Proof of Lemma 3.4. Using (3.2) and Holder's inequality, we deduce To simplify notations, we write g = g 0 and Γ ρ α µ = Γ ρ α µ ( g 0 ). We then compute using (3.9): which gives the conclusion. /square Proof of Lemma 3.5. This computation is similar to the computation in [20]. We need to manipulate the first term and the last term in (3.9). For the first term, using (3.3) twice, we obtain In a similar manner, we obtain Plugging (A.8) and (A.9) into (3.9) and using notations in Definition 2.3 and Definition 2.5, we obtain the desired decomposition. /square Proof of Lemma 3.6. Using (3.8), we compute ˜ /square g λ ( g λ ) µν -˜ /square g 0 ( g 0 ) µν = ( g αβ λ -g αβ 0 ) ∂ 2 αβ ( g λ ) µν + ˜ /square g 0 h µν = ( g αβ λ -g αβ 0 ) ∂ 2 αβ h µν + ˜ /square g 0 h µν + O ( λ ) , (A.10) where we have used (3.2) to control ( g αβ λ -g αβ 0 ) ∂ 2 αβ ( g 0 ) µν . Using (3.2) to compute the difference of the inverses, we obtain Plugging (A.11) into (A.10) and using the second derivative bounds in (3.2), we obtain the conclusion. /square", "pages": [ 25, 26, 27 ] }, { "title": "References", "content": "[17] R. A. Isaacson. Gravitational radiation in the limit of high frequency. II. nonlinear terms and the effective stress tensor. Phys. Rev. , 166:1272-1280, Feb 1968. CMLS, Ecole Polytechnique, 91120 Palaiseau, France Email address : [email protected] Department of Mathematics, Stanford University, CA 94304, USA Email address : [email protected]", "pages": [ 27, 28 ] } ]

[ { "title": "ABSTRACT", "content": "This paper argues why FLAMINGO (Fast Light Atmospheric Monitoring and Imaging Novel Gamma-ray Observatory) is the perfect name for an array of Cherenkov telescopes. Studies which indicate pink is the most suitable pigment for the structures of Cherenkov telescopes have passed with flying colors. Pink optimizes the absorption and reflectivity properties of the telescopes with respect to the characteristic blue color of the Cherenkov radiation emitted by high-energy particles in the atmosphere. In addition to giving the sensitivity a big leg up, a pink color scheme also adds a unique and visually appealing aspect to the project's branding and outreach efforts. FLAMINGO has a fun and memorable quality that can help to increase public engagement and interest in astrophysics and also help to promote diversity in the field with its colorful nature. In an era of increasingly unpronounceable scientific acronyms, we are putting our foot down. FLAMINGO is particularly fitting, as flamingos have eyesight optimized to detect small particles, aligning with the primary purpose of Cherenkov telescopes to detect faint signals from air showers. We should not wait in the wings just wishing for new name to come along: in FLAMINGO we have an acronym that both accurately reflects the science behind Cherenkov telescopes and provides a visually striking identity for the project. While such a sea change will be no easy feet, we are glad to stick our necks out and try: FLAMINGO captures the essence of what an array of Cherenkov telescopes represents and can help to promote the science to a wider audience. We aim to create an experiment and brand that people from all walks of life will flock to. Keywords: very-high-energy gamma rays; awesomeness; sparkling galaxies; non-thermal", "pages": [ 1 ] }, { "title": "Why FLAMINGO is the perfect name for an array of Cherenkov telescopes", "content": "P. Flock, 1 A. Laguna-Salina, 1 F. James, 1 G. Blossom, 1 B. Carotene, 1 C. Sparks, 1 D. Tarek, 1 A. Ahashia, 1 J. Donald, 1 (The FLAMINGO Collaboration) 1 Flamingo International College, Indonesia", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Cherenkov telescopes are a type of astronomical observatory that detect the Cherenkov radiation emitted by extensive air showers that are produced by high-energy cosmic rays or gamma rays interacting with the atmosphere. To achieve this, the telescopes use an array of mirrors that reflect the Cherenkov light onto a camera system that captures the images of these extensive air showers. Cherenkov telescopes play a critical role in the study of high-energy cosmic rays and gamma rays, which often are produced by the most energetic phenomena in the universe. These particles carry important information [email protected] about the sources and acceleration mechanisms of cosmic rays, as well as the properties of dark matter and other exotic astrophysical objects. With the development of next-generation Cherenkov telescopes, scientists will be able to study these particles with greater sensitivity and accuracy than ever before. These instruments are expected to make major contributions to many areas of astrophysics, including the study of active galactic nuclei, gamma-ray bursts, and the search for dark matter. They will also enable scientists to study the universe at extremely high energies, shedding light on the properties of cosmic rays and the behavior of particles in extreme environments. Finding a suitable name that encapsulates the essence of these missions is a crucial aspect of branding and public engagement efforts. In this paper, we present sev- eral reasons as to why FLAMINGO (Fast Light Atmospheric Monitoring and Imaging Novel Gamma-ray Observatory) is an excellent name for an array of Cherenkov telescopes. Section 2 outlines and discusses the merit of FLAMINGO as a name for an array of Cherenkov telescopes, while Section 3 summarizes these ideas and arguments.", "pages": [ 1, 2 ] }, { "title": "2.1. FLAMINGO's color", "content": "The color pink has been shown to be the most suitable color for Cherenkov telescopes. It optimizes the absorption and reflectivity properties of the telescope structure to enhance the detection efficiency of the predominately blue/UV Cherenkov light. Pink, being complementary to the blue Cherenkov light, provides the best properties to enhance visibility and improve the accuracy of the telescope's measurements. The connection between the color pink and flamingos comes from the bird's distinctive coloration. Flamingos are known for their pink or reddish-pink feathers, which are caused by pigments in their food sources. These pigments, called carotenoids, are found in algae, crustaceans, and other aquatic organisms that the flamingos consume. Over time, the carotenoids accumulate in the flamingo's feathers, giving them their characteristic pink coloration (Matthew 2017; KJ et al. 2015). Given this association with the color pink, the name FLAMINGO provides a natural connection to this color.", "pages": [ 2 ] }, { "title": "2.2. FLAMINGO's properties", "content": "The vision system of flamingos is optimized for their feeding behavior adopted to filter-feeding, which allows them to detect and feed on very small particles (Lisney et al. 2020; Mascitti & Kravetz 2002). Cherenkov telescopes are optimized for detecting faint flashes of light produced by high-energy cosmic rays and gamma rays as they interact with the Earth's atmosphere. These flashes are extremely brief and difficult to detect, requiring highly sensitive telescopes with advanced camera systems. In addition, the visual sensitivity of birds, compared to humans, reaches lower wavelengths into the near-ultraviolet band (Matthew 2017), which is the dominant band to detect Cherenkov light. Additionally, flamingos are known for their social behavior and their ability to communicate and coordinate with one another (Tindle et al. 2014). Similarly, arrays of Cherenkov telescopes often operate with multiple telescopes working in tandem, allowing them to coordinate their observations and improve the accuracy of their measurements. Building and operating large ar- rays of telescopes requires are large amount of moeny and workpower. This can only be realized by forming collaborations of multiple research groups across the world. The name FLAMINGO, therefore, not only captures the scientific goals of an array of Cherenkov telescopes but also embodies the qualities of teamwork and coordination that are essential for its success.", "pages": [ 2 ] }, { "title": "2.3. FLAMINGO's impact", "content": "The name FLAMINGO has a fun and memorable quality that can help to increase public engagement and interest in the field of astrophysics. Using a name that is striking and easy to remember can help to capture the public's attention and generate excitement about the project. In addition, the name FLAMINGO can be used to create a unique visual identity for an array of Cherenkov telescopes. By incorporating images of flamingos into the project's branding and outreach materials, such as logos (see, e.g., Fig. 1), social media graphics, and promotional videos, the project can create a recognizable and distinctive brand that sets it apart from other astronomical observatories. The use of a memorable name and distinctive branding (Samu & Krishnan 2010; Boersma 2018) can also help to increase public awareness and understanding of the science behind Cherenkov telescopes. By promoting the project through a variety of channels, including social media, press releases, and public outreach events, the project can engage with a broader audience and educate them about the fascinating science behind Cherenkov telescopes. Finally, the name FLAMINGO can help to make arrays of Cherenkov telescopes more accessible and approachable to people who may not have a background in astrophysics. By using a name that is fun and easy to remember, the project can help to break down barri- ers. This, in turn, can help to generate interest in the field and inspire the next generation of scientists and researchers (Otamendi & Sutil Mart'ın 2020).", "pages": [ 2, 3 ] }, { "title": "2.4. FLAMINGO's sensitivity", "content": "A structure painted in pink drastically enhances the sensitivity of imaging Cherenkov telescopes, allowing to reach an unprecedented performance in the very-highenergy gamma-ray range. In Fig. 2 the differential sensitivity of FLAMINGO, obtained by highly sophisticated Monte Carlo simulations of the array configuration, is shown in solid pink in comparison to the sensitivities of other current and planned gamma-ray instruments. The improvement in sensitivity reached by a FLAMINGO array translates in the possibility to observe and study sources beyond the Cosmic Gamma Ray Horizon, up to a pinkshift of 10. Coloring not only the supporting structure of the telescopes of the array, but also the outside of the cameras in pink could further enhance the performance of the FLAMINGO array. Recent studies conducted by researchers of the FLAMINGO collaboration point to an even stronger performance if glitter is added to the pink painting. Glitter can also be applied moderately on the electronics and on the operators of the telescopes. This will be the object of a dedicated paper by the FLAMINGO collaboration, currently under preparation. We aim to further flatten the sensitivity curve of FLAMINGO, and reach a more fair and inclusive distribution of the sensitivity for all energies. On this regard, it is interesting to note that the current name of such curves could be misleading and should be actually changed to 'insensitivity' curves. As will be published in a dedicated study, an optimal sensitivity curve should be completely flat.", "pages": [ 3 ] }, { "title": "2.5. FLAMINGO's diversity", "content": "Using the name FLAMINGO for an array of Cherenkov telescopes can also help to address diversity issues in the field of astrophysics. Astrophysics is one of the scientific fields that is still struggling with diversity and many demographic groups are underrepresented (Gaskin et al. 2023). The use of an inclusive name and branding strategy can help to create a more welcoming and diverse community of astronomers and astrophysicists. The use of a memorable name and distinctive branding can help to increase visibility for the project and attract a more diverse range of applicants for jobs and research positions. Moreover, by promoting the project through social media campaigns, public outreach events, and educational programs, the project can help to inspire and encourage a more diverse range of people to pursue careers in astrophysics. In particular, for the LGBTQ+ community FLAMINGO has a high potential. The color pink carries a large weight for many members of the LGBTQ+ community, and today it is often used in LGBTQ+ pride flags and other symbols. By using a name and brand that recall the color pink, such as FLAMINGO, new experiments can help to promote awareness and support for the LGBTQ+ community. In addition, the use of colorful imagery can further emphasize the project's commitment to diversity and inclusion. A working group within the collaboration, FLAMINGO Pride , is already tirelessly to help plan a number of events for Pride Month in June. Overall, using the name FLAMINGO for an array of Cherenkov telescopes can help to create a more inclusive and diverse community of astronomers and astrophysicists, promoting a more equitable and representative field.", "pages": [ 3 ] }, { "title": "3. SUMMARY", "content": "In this paper, we have argued that FLAMINGO is the perfect name for an array of Cherenkov telescopes for several reasons. Firstly, the color pink, which is associated with flamingos, has been shown to be the most suitable color for Cherenkov telescopes, providing the best characteristics to enhance sensitivity. Secondly, the keen visual ability of flamingos aligns well with the primary purpose of Cherenkov telescopes to detect faint signals of high-energy cosmic rays and gamma rays. In addition, the name FLAMINGO has a fun and memorable quality that can help to increase public engagement and interest in the field of astrophysics. Using a name like FLAMINGO can also help to address diversity issues in the field of astrophysics by creating a more inclusive and welcoming community of astronomers and astrophysicists. In conclusion, the name FLAMINGO represents a powerful branding strategy for any next-generation array of Cherenkov Telescopes, capturing the essence of the scientific goals of the project while creating a fun and approachable image. By using the name FLAMINGO and associated branding, the project can generate excitement and engagement among the public while also inspiring and supporting a more diverse flock of astronomers and astrophysicists. Visit https://flamingo. science. to stay up to date on the ongoing efforts of the FLAMINGO collaboration.", "pages": [ 3 ] }, { "title": "4. AUTHORS' CONTRIBUTIONS", "content": "Amazing Flamingo: paper editing, coordination of the project; Brilliant Flamingo: investigation (Pink Matter searches with FLAMINGO which strongly supported the sensitivity study); Colorful Flamingo: study of the sensitivity under different coloring of the structure; Enthusiastic Flamingo: endless positivity and enthusiasm and creation of an enjoyable atmosphere inside the collaboration; Glittery Flamingo: investigation (glitter theoretical studies) and paper review; Glowing Flamingo: study of the flattening of the sensitivity of the FLAMINGO array; Harmonious Flamingo: Technical support and outreach activities, such as the composition of the FLAMINGO collaboration song; Honorable Flamingo: guard of the FLAMINGO oath; Incredible Flamingo: Software Board approval and test of the MonteCarlo simulations; Radiant Flamingo: organization and supervision; Sparkling Flamingo: review of the paper, simulations and visualization (Fig. 2); Visionary Flamingo: paper editing, FLAMINGO acronym development. We would like to extend our sincere gratitude to everyone involved in the founding and development of the FLAMINGO project and all the currently active gamma-ray experiments. This includes the scientists, engineers, technicians, and administrative staff who have contributed their time, expertise, and resources to the project. Special gratitude goes to affordable beer and wine places, which laid the foundation for the creation of FLAMINGO. Finally, we would like to acknowledge the many members of the FLAMINGO collaboration who have contributed to the development of the project. Thank you all for your hard work, dedication, and support of the FLAMINGO project. FLAMINGO is here to FLAMINSTAY.", "pages": [ 3, 4 ] }, { "title": "REFERENCES", "content": "Boersma, R. 2018, PhD thesis, Wageningen University and Research Gaskin, J., O'Connell, W., Packham, C., et al. 2023, in American Astronomical Society Meeting Abstracts, Vol. 55, American Astronomical Society Meeting Abstracts, 167.02 KJ, Y., J, K., IT, C., et al. 2015, Sci Rep, 5, doi: https://doi.org/10.1038/srep16425 Lisney, T., Potier, S., Isard, P., et al. 2020, Journal of Comparative Neurology, 528, doi: 10.1002/cne.24902 Mascitti, V., & Kravetz, F. O. 2002, The Condor, 104, 73. http://www.jstor.org/stable/1370342 Matthew, A. 2017, Flamingos: Behavior Biology and Relationship with Humans, 1st edn. (Nova Publishers) Otamendi, F., & Sutil Mart´ın, D. L. 2020, Frontiers in Psychology, 11, doi: 10.3389/fpsyg.2020.02088 Samu, S., & Krishnan, H. 2010, The Journal of the Academy of Marketing Science, 38, 456, doi: 10.1007/s11747-009-0175-8 Tindle, R., Tupiza, A., Blomberg, S., & Tindle, E. 2014, Galapagos Research, 68", "pages": [ 4 ] } ]

[ { "title": "A generalized Λ CDMmodel with parameterized Hubble parameter in particle creation, viscous and f ( R ) model framework", "content": "G. P. Singh 1 * , Romanshu Garg 1† , Ashutosh Singh 2‡ 1 Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, Maharashtra, India. 2 Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura 281406, Uttar Pradesh, India", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this study, we construct a theoretical framework based on the generalized Hubble parameter form which may arise within the particle creation, viscous and f ( R ) gravity theory. The Hubble parameter is scrutinized for its compatibility with the observational data relevant to the late-time universe. By using Bayesian statistical techniques based on χ 2 minimization method, we determine model parameters's best fit values for the cosmic chronometer and supernovae Pantheon datasets. For the best fit values, the cosmographic and physical parameters are analyzed to understand the cosmic dynamics in model. We also analyze the model section criterion in comparison to the Λ cold dark matter model. Keywords: Flat FLRW metric, Particle creation, Bulk viscosity, Modified gravity.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The observations of astronomical origin indicates that the universe is expanding with an increasing rate [13]. At present, the precise nature of enigmatic fluid (or field) that is speeding up the expansion of cosmos is unclear to a great extent. This fluid (or field) is also referred as the 'dark energy' [4]. To elucidate the universe's accelerating expansion, the 'cosmological constant' ( Λ ) of General relativity model (visualized as the vacuum energy [5]) is widely accepted candidate for dark energy. Although, the cosmological constant appears to fit well with the observational evidences of the universe but this model faces two main problems namely, the coincidence problem and fine tuning problem [4]. There is a nearly 120-order of magnitude difference between its value from particle physics and the value needed to suit cosmic observations [4]. The universe's accelerated expansion may be explained by different cosmological mechanisms, see [6-13]. A modified gravity theory such as the f ( R ) gravity, a barotropic fluid model or a combination of both may be used to describe the early-time and late-time accelerated expansion epochs [6-8, 10-13]. An alternative explanation for the accelerated phase can be found in the particle production mechanism also [14, 15]. Schrodinger [16] proposed the particle generation mechanism which further studied by Parker [17, 18]. The particle creation mechanism by an external gravitational field in cosmological modeling has been studied by Parker [19]. There has been lots of discussion on the production of matter in an expanding universe [20-28]. The particle creation mechanism may yield cosmological scenarios with non-equilibrium thermodynamical descriptions ranging from inflation to late-time acceleration [29]. Viscous fluid presents a compelling and captivating concept within the realms of fluid mechanics [30] and cosmology [31, 32]. Murphy[33] has shown that initial singularity in homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology can be resolved by continuous bulk viscosity. The effect of viscosity introduction on the formation of singularity in the Friedmann cosmology framework with the idealized assumption of constant bulk viscosity coefficient has been demonstrated by Heller et al.[34]. Numerous authors have examined the impact of cosmic viscous fluids on universe evolution[27, 3549]. Λ CDM with viscous cosmology was studied in[50, 51]. The viscous fluids may also pave a way for graceful exit in the early universe during the inflationary epoch [47]. The resolution of the initial singularity problem in mainstream cosmology emphasized in the dynamics of dark matter with bulk viscosity effects [52]. In order to comprehend the different aspects of the observable cosmos, various pioneering concepts have been proposed with the modification of general relativity (GR), see for example [6-8, 10, 13] and references therein. A fundamental modification originates by replacing the Ricci scalar ( R ) with a general function of R in the Einstein-Hilbert action of GR, known as f ( R ) gravity [53, 54]. Λ CDM bounds for F ( R ) gravity were studied in[55, 56]. The f ( R ) gravity provides a good explanation for the cosmological expansion phenomenon and several studies have examined the limitations of feasible cosmological theories [6, 57]. A number of authors[58-60] have discussed observational deductions from the models of f ( R ) and, have talked about the solar system's constraints as well as the astrophysical phenomenon in f ( R ) gravity [61, 62]. In this paper, we show that a parameterized form of Hubble parameter may be a solution in particle creation model. Independently, the considered Hubble parameter may also solved in bulk viscous model and the f ( R ) gravity model. In cosmological modeling, the ansatzes of different cosmological quantities are studied to check whether they admit physically reasonable behaviors or not? The observational viability of ansatzes subjected to the observational data such as the Cosmic chronometer data and Pantheon supernovae data may aid in ruling out different kind of parametrizations. In this paper, we proceed along these lines and show that the cosmological solution based on parameterized Hubble parameter may be physically admitted. The paper has been arranged in 6 sections as follows: In Sec.(2), the characteristics of cosmological solutions are shown by using a particular Hubble function which may explain the transition from early era deceleration into late-time acceleration. In Section (3), the compatibility of cosmological solution is investigated by using Bayesian statistical techniques with two observational datasets, namely the cosmic chronometer (CC) and Pantheon datasets. In Sec.(4), we show that the considered Hubble parameter may be a solution in cosmological frameworks such as particle creation, viscous fluid model and f ( R ) gravity model. We construct the f ( R ) gravity form and examine the dynamical characteristics of universe in the model. In Sec.(5), we discuss the cosmographic parameters along with the age of current universe in model. Finally, the summary and conclusions are given in Sec.(6).", "pages": [ 1, 2 ] }, { "title": "2 The cosmological equations and background dynamics", "content": "The astronomical observations about the expanding universe points that the rate of universe's expansion is increasing although its spatial geometry seems be almost flat [3]. It is also well-established that the universe expansion history has a past of decelerating expansion which transits into the accelerating expansion phase during present times. For the observable universe, as par Aghanim et al. [3] the present day value of the expansion rate is H 0 = 67 . 4 ± 0 . 5 km / ( s · Mpc ) . Homogeneous and isotropic universe's expansion rate is described by H = ˙ a a , where a denotes the scale factor and overhead dot is time derivative. The spatially flat metric for homogeneous and isotropic universe may be written as In the general relativity context, the field equations can be expressed as where ρ and p represent the energy density and pressure respectively. In order to understand the transitional universe expansion era, we adopt a parametrization of the Hubble parameter H ( t ) . The parametrizations may be used in a model-independent manner for exploring the characteristics of dark energy dominated universe. Different kind of parameterization when tested against the observational data may single out the physically reasonable cases. In this approach, we proceed with the Hubble parameter having form Here, H 0 represents the Hubble parameter's current value and m , n are arbitrary constants. The above Hubble parameter may be seen as the generalization of Λ cold dark matter model defined by H ( a ) = H 0 √ Ω m a 3 + Ω Λ . Although, the Λ CDMmodel is supported by the latest observational datasets [1, 3, 63-65], the cosmological studies based on parametrizations of different parameters are widely explored [13, 21-29, 35-47, 66-72]. Above equation (3) may be solved using the relation a = a 0 1 + z with standard convention a 0 = 1 for present era to obtain Hubble parameter in terms of red-shift ( z ) as This kind of Hubble parameter may arise with affine equation of state p = n 1 ρ -n 2 , where n i , i = 1 , 2 are some constant [69] or in modified theory of gravity [66, 70]. When it comes to understanding the dynamics of the cosmic universe the cosmographical parameter[12] with the most influential is thought to be q . Based on the values of this parameter, the expansion dynamics of universe may be categorized into either accelerated or decelerated phases with q = 0 representing the transitional periods. The universe exhibits deceleration when q > 0 and power-law expansion when -1 < q < 0. The universe exhibits superexponential (de Sitter) expanding era for q < -1 ( q = -1) respectively. This cosmographic parameter may be defined as q = -1 + d dt 1 H and by using equation (4), we obtain Based on the model parameters' best fit values (see section 3), the evolution of deceleration parameter and effective equation of state (EoS) parameter ( defined as ω e f f = -1 -2 3 ˙ H H 2 ) of the reconstructed universe is shown in figure (1) and (2) respectively. For our model, the values of deceleration parameter are q 0 = -0 . 533 (for the CC data) and q 0 = -0 . 5441 (for the joint data) at present time having z = 0. This q 0 value is very close to q 0 = -0 . 55 for the Λ CDMmodel[12].", "pages": [ 2, 3 ] }, { "title": "3 Observational constraints on model", "content": "Here, we examine the compliance of parameterized from of Hubble parameter (3) with datasets of the cosmic chronometer (CC) and the joint data consisting of Pantheon and cosmic chronometer data nomenclated (in present paper) as CC+Pantheon sample. In order to perform statistical analysis, we use χ 2 minimization method with the Markov Chain Monte Carlo (MCMC) technique implemented with the emcee tool [73] and constrain the parameters to study of the cosmic behavior in the model.", "pages": [ 3 ] }, { "title": "3.1 The Cosmic chronometer data", "content": "The values of Hubble parameter at any instant describes the expansion rate of the universe at that particular instant and its observational values are important for exploration of dark energy and the evolution of universe. To constrain the model parameters, we use Cosmic Chronometer data composed of 31 data points [74, 75] which are determined by differential ages of galaxies technique within the red-shift region 0 . 07 ≤ z ≤ 1 . 965. In this case, we minimize the χ 2 function and thus the model parameters H 0 , m , and n are estimated with their median values. The corresponding χ 2 may be expressed as where the observed values are denoted by H obs ( z i ) and theoretical values of the Hubble parameter are denoted by H th ( θ , z i ) . The values σ 2 H ( zi ) are the standard deviation for each H obs ( z i ) observed value. illustrates the error bars of the CC points with the best fit Hubble parameter curve for the Hubble", "pages": [ 4 ] }, { "title": "3.2 The Pantheon data", "content": "We use the Pantheon sample, which includes 1048 supernovae Type Ia (SNIa) data points for the red-shift range 0 . 01 < z < 2 . 26 [76]. The CfA1-CfA4 [77, 78] surveys, Pan-STARRS1 Medium Deep Survey [76], SDSS [79], SNLS [80], Carnegie Supernova Project (CSP) [81] contributes to the SNIa sample. For the MCMCanalysis using Pantheon data, the theoretically expected apparent magnitude µ th ( z ) is given by where M is the absolute magnitude. Also, the luminosity distance d L ( z ) (having dimension of the Length ) may be defined as [82] where z represents SNIa's red-shift as determined in the cosmic microwave background (CMB) rest frame and and c is the speed of light. The luminosity distance ( d L ) is typically substituted with the dimensionless Hubble-free luminosity distance given by D L ( z ) ≡ H 0 d L ( z ) / c . The equation (7) could also be rewritten as The parameters M and H 0 can be combined to create a new parameter M , which may be identified as where H 0 = h × 100 Km / ( s · Mpc ) . We use this parameter with pertinent χ 2 for Pantheon data in the MCMC analysis as [83] where ∇ µ i = µ obs ( z i ) -µ th ( z i ) , C -1 i j is the covariance matrix's inverse and µ th will be provided by equation (9). The luminosity distance depends on the Hubble parameter. Therefore, we use the emcee package [73] and equation (4) to get the maximum likelihood estimate using the joint CC+Pantheon data set. The joint χ 2 for maximum likelihood estimate may be defined as χ 2 CC + χ 2 P . In Fig. ( 4 ) , we displays the posterior distribution with 1 σ and 2 σ from the Monte Chain Monte Carlo analysis using CC+Pantheon data. The Table (1) summarizes the best fit values for the model parameters in the MCMC analysis for the model.", "pages": [ 5 ] }, { "title": "4.1 The Particle Creation model", "content": "With the FLRW spacetime, the fundamental cosmological equations with the particle creation mechanism in a model may be written as ̸ where p and ρ represent to the thermodynamic pressure of the matter content and energy density respectively with p c is representing the creation pressure. The fluid particles are not conserved if thermodynamic system is regarded as open ( N i ; i = n Γ = 0, where particle number density is represented by n , while the fluid flow vector is denoted by N i = nu i ). The particle conservation equation can be expressed as follows, where the particle production rate Γ is the rate of change in particle number in the co-moving volume V . Depending on the particle production rate, one may identify the creation or annihilation of particles. Particle creation occurs when Γ > 0, particle annihilation occurs when Γ < 0, and no particle production occurs when Γ = 0. According to the second law of thermodynamics, Γ > 0 is required for entropy to never decrease. The gravitationally induced adiabatic particle formation rate and creation pressure p c are related by the relation[84, 85]. When particle production is present or absent, the creation pressure p c is negative or zero respectively. The energy conservation equation may be expressed as By considering the universe with matter having p m = 0, the energy density of matter may be written using Eqs. (4), (12) and (15) as In this scenario, the particle creation rate may be written as using Eqs. (4), (12), (14) as The above derived particle creation rate may interpolate between the early decelerating era with the latetime accelerating era in the model. The creation pressure being negative from the recent past may yield the mechanism for accelerating universe expansion in model. Matter's energy density behavior and creation pressure have been displayed in Fig. (5) and (6) respectively. The energy density remains positive during different cosmological evolution eras while the creation pressure has become negative from the recent past.", "pages": [ 5, 7 ] }, { "title": "4.2 Bulk viscous model", "content": "For the FRW metric (1), the field equations in the bulk viscous fluid model are where the bulk viscous pressure p ' may be written as with ξ being coefficient of bulk viscosity with ρ and p denoting the energy density and pressure of fluid respectively. In this paper, we consider the generic form of the inhomogeneous viscous fluid having form p ' = p -3 H ξ ( a , H , ˙ H , .... ) , where the bulk viscosity ξ ( a , H , ˙ H , .... ) is a general function of a , H and it's derivatives. The continuity equation may be expressed as where ξ is a natural candidate for actual fluid and contributes to the dissipative effects. By considering the matter having p m = 0, the ernergy density and viscous pressure may take the form The Fig. ( 7 ) depict the evolution of bulk viscous pressure ( p ' ) with red-shift. We can easily observe that bulk viscous pressure ( p ' ) has negative values which may be responsible for accelerating universe expansion in the model.", "pages": [ 8 ] }, { "title": "4.3 The f ( R ) gravity model reconstruction", "content": "The f ( R ) gravity theory action may be written as[11, 12] where f ( R ) denotes a general function of the Ricci scalar ( R ) and the action for the appropriate matter distribution is denoted by S m . The field equation may result by varying action (22) with respect to the metric tensor g i j as where f ' ( R ) = ∂ f ( R ) ∂ R and T µ v = -2 √ -g δ Sm δ g i j is stress-energy tensor. The field equations of f ( R ) gravity can be expressed in terms of the Einstein tensor, which includes an effective energy-momentum tensor T e f f µ v . We may write The Raychaudhuri equation in f ( R ) gravity for a time-like congruence with velocity vector u i may take the form [86, 87] The last term that is located on the right-hand side of equation (25) can be expressed as follows using the field equations (23) for f ( R ) theory: In this paper, we are dealing with the homogeneous and isotropic, spatially flat universe given by the metric (1). The Raychaudhuri equation (25) takes the following form for such a metric and a matter distribution of a perfect fluid T i j =( ρ + p ) u i u j + pg i j as [88, 89] It should be emphasized that for the fluid distribution, we have not assumed any equation of state until now, but by using field equations (24) in the Raychaudhuri equation (25), one may eliminates the fluid pressure p . In principle, the f ( R ) function of this modified gravity may be determined by using either the Hubble parameter or the scale factor. Reconstruction of f ( R ) for different Hubble parameter was initiated in Ref. [90]. The late- and early-time acceleration may be realized in f ( R ) models with reconstruction approach where dark energy may have the quintessence-like behavior [11]. A detailed summary of f ( R ) gravity reconstruction in metric formalism as well as Palatini formalism with the cosmological viability conditions has been presented in Ref. [12]. Here, we use Eq. (3) with Eq. (27) to determine f ( R ) gravity form. The Ricci scalar R for a spatially flat Friedmann-Robertson-Walker metric (1) is R = 6 ( ˙ a 2 a 2 + a a ) . Using the equation (3), we can write Here α =( 12 H 2 0 m -3 H 2 0 mn ) and β = 12 H 2 0 ( 1 -m ) . Using the equation (3), (27) and (28), we obtain For the matter having pressure p = 0, the energy density may be given by ρ = ρ m 0 a 3 and thus above Eq. (29) may take the form Solving Eq. (30), we obtained one solution for f ( R ) as where I =( m -1 ) H 2 0 -H 2 0 m β ( n 2 -1 ) α , α =( 12 H 2 0 m -3 H 2 0 mn ) and β = 12 H 2 0 ( 1 -m ) . We may also have the energy density of the matter given by For the median values given in Table (1), the behavior of energy density with red-shift have been displayed in Fig. (8). The energy density remains positive during the decelerating era and will preserve its positive nature as the universe evolves into the accelerating era in model. The energy density of matter (Eq. (32)) in the terms of red-shift decreases as the time evolves in the expanding model based on the expansion rate (4). The parameter involved in the expression of ρ m describes the evolution of matter density in accord to the observations, in which the energy density of matter is decreasing from the past. The present model describing early deceleration transiting into late-time acceleration have the f ( R ) form f ( R ) = a 1 R + a 2 , where a 1 , a 2 are recombined constants. The f ( R ) gravity may also have cosmologically viable models describing early era inflation as well as the late era accelerated expansion [91]. The powerlaw f ( R ) cosmology may describe the inflationary expansion and may relate with the Λ CDM dynamics in a natural way [92]. The non-singular exponential f ( R ) cosmology may also relate the early era and late era accelerated universe evolution [93]. In this sense, the present f ( R ) model may belong to the power-law f ( R ) model and it may relate the early deceleration to the late-time acceleration.", "pages": [ 8, 9, 10 ] }, { "title": "5 General issues", "content": "In this section, we study the cosmographic evolution of the universe governed by Hubble parameter (3). The cosmographic parameter helps to identify the sharp contrast between the dark energy model and the Λ CDM model. We also identify the universe's age in model for the best fit values in model.", "pages": [ 10 ] }, { "title": "5.1 Cosmographic parameters", "content": "The kinematic quantities like as jerk ( j ) and snap ( s ) parameter defines the cosmic scenario by using the geometric quantities like as the scale factor ( a ) and its derivatives. The cosmography study was initially discussed by Weinberg [94] by using a Taylor series to introduce the scale factor that increased around present time t 0 . The Hubble parameter H is regarded as a varying observable quantity prior to discovering evidence of the Universe's accelerating expansion. The deceleration parameter q illustrates its evolution, by using the second-order derivative of a [95]. The snap ( s ) and jerk ( j ) parameters provide insights about the cosmic evolution of the universe. Jerk can also sometimes be referred to as jolt, and jounce is another word for snap and may be defined as [12, 96]: The cosmographic analysis in absence of spatial curvature may be well understood with the expression (33). However, the presence of spatial curvature in a model leads to limitation in the standard cosmographic approach [12]. The standard cosmography may be improved with the method of Pad'e polynomials [98102], the method of Chebyshev polynomials [12, 103], for more details see, Ref. [12, 104-106]. For j and s in present analysis the equation (33) can be rewritten in red-shift's terms as [97]. For the median values, the jerk and snap parameter behaviors have been illustrated in figure ( 9 ) and ( 10 ) re- spectively. The snap and jerk parameters are provided by s 0 = -0 . 4467 and j 0 = 1 . 0186 respectively, for the median values derived from the CC data set. The values of the jerk parameter differ from the Λ CDMmodel with j 0 = 1. The obtained values of jerk and snap parameters respectively are j 0 = 1 . 0319, s 0 = -0 . 4482 for the median values derived from for joint data set. In the model independent reconstruction of f ( R ) gravity using Pad'e approximation [100], j 0 = 0 . 593 + 0 . 216 -0 . 210 . However, for third order Taylor's approximation j 0 = 1 . 223 + 0 . 644 -0 . 664 , s 0 = 0 . 394 + 1 . 335 -0 . 731 [12]. In case of Chebyshev polynomial method, j 0 = 1 . 585 + 0 . 497 -0 . 914 , s 0 = 1 . 041 + 1 . 183 -1 . 784 [103]. For the Λ CDM model with Ω m = 0 . 3, q 0 = -0 . 55 , j 0 = 1 , s 0 = -0 . 35 [12]. The jerk and snap parameter values of present model are consistent with these findings.", "pages": [ 10, 11 ] }, { "title": "5.2 Age of the Universe", "content": "In a cosmological model, the cosmic age t ( z ) of the universe can be calculated as We numerically calculate the aforementioned integral and get the present age of the universe for z = 0 and use the Hubble parameter H ( z ) (Eq. 3). For this model, the obtained value of universe's present age is t 0 = 13 . 52 + 1 . 73 -1 . 5 Gyr for CC data and t 0 = 13 . 4 + 3 . 30 -2 . 04 Gyr which are very close to the present age values obtained from the recent observations [3].", "pages": [ 11 ] }, { "title": "5.3 Information criteria", "content": "In order to test the statistical performance of model (4), we use the popular information criteria named as Akaike information criteria (AIC) and Bayesian information criteria (BIC). The expression for AIC is given by [107-109] where p is total number of free (fitted) parameters used in the present MCMC analysis. L max is the maximum likelihood of the considered model. The expression for BIC is given by [107, 109, 110] where p is total number of free (fitted) parameters used in the present MCMC analysis. Using the definitions of AIC and BIC, we calculate the △ AIC and △ BIC as compared to Λ CDMmodel. According to the criterion of calibrated Jeffrey's scale [111], the confronted models are consistent to each other for 0 < | △ AIC | < 2. If 2 ≤| △ AIC | < 4, there is certain disagreement between the confronted models. For | △ AIC |≥ 4, the model having large AIC value is disfavored by data. For 0 < | △ BIC | < 2, the model having large BIC value is weakly disfavored by the data. For 2 ≤| △ BIC | < 6, the model having large BIC value is strongly disfavored by the data. For | △ BIC |≥ 6, the model having large BIC value is very strongly disfavored by the data. The summary of AIC and BIC values have been given in Table 2. For the CC data, N = 31 and since Λ CDM (Generalized Λ CDM) model are having H 0 , Ω m ( H 0 , m , n ) parameters respectively, the △ AIC is highlighting that Generalized Λ CDM model has certain degree of disagreement with the Λ CDM model. For the joint analysis based on the Cosmic chronometer and Patheon data, one will have N = 1079 and 0 < | △ AIC | < 2, and thus these models may be said to be consistent with each other. On the basis of | △ BIC | value, we observe that BIC value of generalized Λ CDM model is greater than that of the Λ CDM model. In this sense, the generalized Λ CDM model is strongly (very strongly) disfavored by the Cosmic chronometer (joint) data respectively.", "pages": [ 11, 12 ] }, { "title": "6 Conclusions", "content": "In this paper, we investigated the a cosmological model having homogeneous and isotropic line element with flat spatial sections and a parameterized form of Hubble parameter. This kind of Hubble parameter may interpolate between the decelerating past to the accelerating present of the universe. We show that that this kind of Hubble parameter may be a solution in the particle creation, bulk viscous, and f ( R ) gravity framework of cosmological modeling for Γ ( H ) = α 1 H + α 2 H , ξ ( H ) = β 1 + β 2 H 2 and f ( R ) = a 1 f ( R ) + a 2 respectively where α i , β i , a i , i = 1 , 2 are some constants containing constrained model parameters. The Raychaudhuri equation has been used to get the form of f ( R ) function in the model. We scrutinize the observational viability of considered Hubble parameter form to the Cosmic chronometer and Pantheon data. By using Bayesian statistical technique with MCMC analysis, we obtain model parameters's best fit values. The obtained best fit are H 0 = 68 . 326 + 1 . 005 -1 . 045 Km / ( s · mpc ) , m = 0 . 307 + 0 . 059 -0 . 050 , n = 3 . 040 + 0 . 165 -0 . 164 subjected to the CC data and H 0 = 68 . 8 + 1 . 9 -1 . 9 Km / ( s · mpc ) , m = 0 . 297 + 0 . 046 -0 . 076 , n = 3 . 07 + 0 . 32 -0 . 32 subjected to the joint data of CC+Pantheon sample. At last, we find universe's present age is t 0 = 13 . 52 + 1 . 73 -1 . 5 Gyr for CC data and 13 . 4 + 3 . 30 -2 . 04 Gyr for joint CC+Pantheon data. Furthermore, the behavior of cosmographic parameters suggest that the universe in model will behave like Λ CDM model in the limiting limits z →-1. The early phase of the universe evolution is decelerating in nature which has been transitioned into the accelerating phase (see Figs. (1) and (2)). According to model parameters best fit values, the transition red-shift is z t = 0 . 63. Additionally for our model, the present values of deceleration parameter are q 0 = -0 . 533 (for the CC data) and q 0 = -0 . 5441 (for the CC+Pantheon data). The universe is dominated by the cold dark matter-like component at large red-shifts and subsequently the universe expands under the influence of quintessence kind of dark energy and will eventually approaches the cosmological constant limit having ω e f f = -1 as z →-1. It is evident in the model that the energy density will be decreasing with time while preserving positive nature during the complete cosmological history. From the trends obtained according to the observational data, the value of jerk parameter decreases from early to late times and, finally approaches to 1 which demonstrates that, in the early universe this model differs from the Λ CDM model and becomes similar to Λ CDM model in later times. Additionally, the jerk parameter's current values are j = 1 . 0186 for the CC data and j = 1 . 0319 for the CC+Pantheon data. In the early cosmos, the snap parameter ( s ) develops in the negative region. The values of snap parameter at the present times are s = -0 . 4467 for the CC data and j = -0 . 4482 for the CC+Pantheon data. These calculated values of cosmographic parameters are consistent with the findings in the literature[12]. In summary, we show that the parameterized Hubble parameter cosmology may be a observationally viable one and, it may be admitted as a solution in the particle creation, bulk viscous, and f ( R ) gravity framework also. The present generalized Λ CDM model may not be favored over the Λ cold dark matter model according to the Bayesian information criteria.", "pages": [ 12, 13 ] }, { "title": "Acknowledgments", "content": "G.P. Singh and A. Singh are thankful to the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India for support under the Visiting Associateship programme. Authors are thankful to the honorable reviewer for highlighting different issues with suggestions.", "pages": [ 13 ] }, { "title": "Data Availability Statement", "content": "This paper has no new associated data. All concepts as well as logical implications are stated in the paper with citations to the data sources.", "pages": [ 13 ] }, { "title": "Declaration of competing interest", "content": "The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper", "pages": [ 13 ] } ]

[ { "title": "ABSTRACT", "content": "a", "pages": [ 1 ] }, { "title": "August 13, 2024", "content": "Molly Burkmar a, 1 (corresponding author) and Marco Bruni a,b, 2 Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom b INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy 1 [email protected] 2 [email protected] Essay written for the Gravity Research Foundation 2024 Awards for Essays on Gravitation.", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the Emergent scenario, the Universe should evolve from a non-singular state replacing the typical singularity of General Relativity, for any initial condition. For the scalar field model in [1] we show that only a set of measure zero of trajectories leads to emergence, either from a static state (an Einstein model), or from a de Sitter state. Assuming a scenario based on CDM interacting with a Dark Energy fluid, we show that in general flat and open models expand from a non-singular unstable de Sitter state at high energies; for some closed models this state is a transition phase with a bounce, other closed models are cyclic. A subset of these models are qualitatively in agreement with the observable Universe, accelerating at high energies, going through a matter-dominated decelerated era, then accelerating toward a de Sitter phase.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Standard Model of Cosmology, a flat ΛCDM model with inflation, is based on General Relativity (GR) and provides a successful framework to describe the evolution of the Universe. However, the presence of singularities pose a problem and these are currently interpreted as points in space-time where GR breaks down [2, 3, 4, 5]. Observations are consistent with a flat universe [6, 7], however they do not rule out open or closed spatial curvature [8]. In particular, a closed universe is appealing as it avoids the problem of having an infinite universe [9]. The Emergent Universe has been widely studied as an alternative to an initial singularity, with models built with fine-tuned initial conditions on the assumption that the past repellor must be a static state represented by an Einstein model with positive curvature [1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Thus the problem with this scenario is that very special initial conditions are required, as the Einstein model is typically represented by an unstable saddle or a centre in the phase space of the relevant Ordinary Differential Equations (ODE), and so it is not a generic past repellor, as in the classical model by Eddington [20]. In this essay we contend that in the Emergent scenario, the Universe should evolve from a non-singular state replacing the typical singularity of GR for any initial condition. In this light, we first consider whether models in [1], based on a scalar field, are still non-singular for general initial conditions. Similarly to the analysis in [21] and [22], we complete a dynamical systems analysis of the Emergent scenario in [1]. We show that only a set of measure zero of trajectories emerge from the static Einstein model. We also find emergence from a non-singular de Sitter phase, however like with emergence from the Einstein state, only a set of measure zero of trajectories emerge from this phase. For trajectories not emerging from these states, we find that the past repellor is a singularity where the scalar field is kinetic dominated. 1 . We then consider a fluid-based model to show an example where emergence from a non-singular state is generic during expansion, regardless of initial conditions. Assuming a scenario based on Cold Dark Matter (CDM, represented by a dust fluid) interacting with a Dark Energy fluid with a nonlinear equation of state, we show that all but a set of measure zero of trajectories in phase space are either cyclic or expand from an unstable de Sitter state, a classical vacuum at high energies. This represents the asymptotic past of flat and open models, and a transition phase with a bounce for closed models. Independently from the initial conditions, all models are accelerating at high energies. A subset of models are qualitatively in agreement with the observed universe, emerging from a non-singular state, going through a matter-dominated decelerated era, then accelerating toward the future de Sitter phase.", "pages": [ 2 ] }, { "title": "2 Scalar-Field-Based Emergent Universe", "content": "We consider a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe, containing a scalar field ϕ with the potential where V 0 is the asymptotic value of the potential at ϕ → -∞ . Assuming 8 πG = c = 1, the Klein-Gordon and Raychaudhuri equations are where H = ˙ a/a is the Hubble expansion scalar ( a is the scale factor) and over-dots are derivatives with respect to time. The dynamics is complemented by the usual Friedmann first integral (the Hamiltonian constraint) Setting ψ = ˙ ϕ/ √ V 0 defines a 3-dimensional set of ODE. To analyse this dynamical system we define dimensionless variables ( ϕ is dimensionless by definition) and their compactified version, in order to study the behaviour of the system at infinity. Here, ϕ →±∞ corresponds to Φ = ± 1, ψ →±∞ to Ψ = ± 1 and y →±∞ to Y = ± 1. First, we consider the phase space of spatially flat models. The Friedmann equation (4) with k = 0 in our compactified variables becomes which we substitute into the equation for Ψ, and numerically solve alongside the equation for Φ. Here, Y > 0 corresponds to expansion and Y < 0 to contraction. The fixed points of the flat sub-manifold with their stability are summarised in Table 1. To understand the type of fixed points present at Φ = ± 1 and Ψ = ± 1, we need to understand what happens to the potential (1) and kinetic energy K = ψ 2 / 2 at these points. When Ψ = ± 1, K → ∞ . For the fixed points where Φ = -1, the potential is finite: V = V 0 . In this case the fixed points are kinetic dominated and so are stiff fluid singularities. At Φ = +1, both the potential energy and kinetic energy become infinite, and the fixed points cannot be linearised. We find that the kinetic energy grows more quickly than the potential, and so these fixed points represent singularities. Only the stability of the de Sitter fixed points dS 1 ± can be found from linearisation of the system; for the other fixed points there are infinities in the Jacobian. We can find their stability through plots of the phase space, however for repellor and attractor fixed points it is helpful to define Liapunov functions [23, 24] in order to obtain more rigorous results. For the Minkowski fixed point at ϕ = 0, we define the Liapunov function L as the energy density of the scalar field, where we approximate e ϕ ≃ 1 + ϕ such that V ≃ ϕ 2 around the origin: We then take the first derivative of L to be able to determine the stability of the fixed point. If L ' < 0 at least in a disc around the fixed point, then it is is an attractor, and if L ' > 0 then the fixed point is a repellor. Taking the first derivative of L , we find Taking Y > 0 ( Y < 0) for the expanding (contracting) case, we find L ' < 0 ( L ' > 0) for the whole phase space, therefore the Minkowski fixed point is a global attractor (repellor) in the expanding (contracting) phase space. For the past (future) singularity, we define the Liapunov function as a circle centered around the fixed point: We fix a radius of L ∗ = r 2 = 0 . 5 2 so that we only study the behaviour of trajectories around the singularities. The first derivative of L gives For the expanding (contracting) case, we find L ' > 0 ( L ' < 0) for L < L ∗ , therefore the past (future) Table 1: Fixed points for the expanding (+) and contracting (-) flat models in the 2-D sub-manifolds in Fig. 1, and the full 3-D phase space in Fig. 3. M denotes the Minkowski fixed point, dS the de Sitter fixed points and S the fixed points representing singularities. The phase space for the flat sub-manifold is shown in Fig. 1, where Fig. 1a shows the expanding case, and Fig. 1b the contracting case. In the expanding case, trajectories between the two separatrices expand from S 4+ , whereas trajectories outside the two separatrices expand from S 1+ . In both cases trajectories are kinetic dominated in the past. The past repellor S 1+ represents the scalar field starting on the flat part of the potential and rolling towards the minimum, and the fixed point S 4+ represents the field starting from the exponential part of the potential and rolling towards the minimum. There is also a special trajectory in the phase space, which emerges from the de Sitter fixed point dS 1+ and expands towards the Minkowski fixed point M . In this case, the field is initially on the asymptotic part of the potential, and rolls toward the minimum where it oscillates. Fig. 1b shows the contracting case. Here, trajectories contract from the Minkowski fixed point M in all cases, and in general contract towards a singularity at S 2 -or S 3 -. Trajectories between the two separatrices contract towards S 3 -, and trajectories outside the separatrices contract towards the singularity S 2 -. In both cases trajectories become kinetic dominated at the singularity. There is also the possibility that a trajectory contracts towards the de Sitter fixed point dS 1 -, although this requires a set of measure zero initial conditions. It is clear there are flat models for which trajectories are non-singular and can be past or future asymptotic to a de Sitter state dS 1 ± . However, these fixed points have saddle stability and are a set of measure zero in the phase space. For all other trajectories the past repellor is either S 1+ or S 4+ . Going back to the 3-D dynamics, we first consider the Φ = -1 sub-manifold, where the potential (1) reduces to V = V 0 . The fixed points of the Φ = -1 sub-manifold are shown in Table 2, and the phase space is shown in Fig. 2. The Einstein points E ± are both unstable saddles, with only two separatrix trajectories emerging from each. Therefore, these are not good candidates for emerging models as these are a set of measure zero for the whole phase space. In this sub-manifold, the past repellors for initially expanding models are the singularities S 1+ and S 2+ , and the de Sitter fixed point dS 1 -is the past repellor for models (a) Expansion ( Y > 0). (b) Contraction ( Y < 0). which initially contract. Trajectories emerging from the non-singular de Sitter state dS 1 -either collapse, bounce and expand towards another de Sitter state dS 1+ , or collapse to a singularity ( S 1 -or S 2 -). The full phase space for the 3-D system is shown in Fig. 3, where Fig. 3a shows examples of open (blue) and closed (purple) trajectories coming from and going to a singularity, and Fig. 3b shows examples of nonsingular trajectories. The fixed points that have not already been considered in the sub-manifolds are shown in Table 3. In Fig. 3a, there are two examples of closed models: one simply expands from a singularity, reaches a turn-around on the surface Y = 0 and then collapses to a singularity. The other expands and oscillates around the Minkowski fixed point (at the minimum of the potential), where it contracts, bounces and expands, before turning around on the Y = 0 surface and collapsing to the singularity. Whether there is a sufficient inflationary phase after the bounce remains to be seen. The expanding open model evolves from a singularity, and oscillates around the minimum of the potential, and the contracting open model oscillates around the Minkowski fixed point before collapsing to a singularity. Fig. 3b shows examples of non-singular models in the phase space, which emerge from the Einstein fixed points E ± (cyan) and the de Sitter fixed point dS 1+ (magenta). The trajectories that emerge from these fixed points require extremely fine-tuned initial conditions: they form a set of measure zero in phase space.Trajectories can also emerge from dS 2+ and Mi + , but these are also a set of measure zero in the phase space. For all other trajectories the past repellor is either S 1+ or S 4+ . The singularity S 4+ is a 3-D saddle, repelling the aforementioned subset and attracting in another sub-manifold, and the S 1+ point represents a stiff fluid singularity where the scalar field is kinetic dominated. Trajectories emerging from S 4+ require some fine-tuning, but regardless most trajectories expand from a singularity.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3 Emergence from Interacting Dark Energy and Dark Matter", "content": "In this essay, we contend that it is desirable for all trajectories to emerge from a non-singular state in the Emerging Universe scenario, regardless of initial conditions. In the following, we provide an example of a model with dark energy with a non-linear equation of state [25] interacting with a dust fluid [26], where the generic trajectory either represents cyclic models, or models that emerge from an unstable de Sitter phase during expansion. For flat and open models this is the asymptotic past repellor, whilst for a subset of closed models the de Sitter phase represents a transition through a bounce from contraction to expansion. In this scenario, the conservation equations for the dark matter and dark energy are where ρ m is the dark matter energy density and ρ x is the dark energy density. The dark energy has a non-linear equation of state, with w x defining its linear part, ρ ∗ the characteristic scale of the non-linear part, and ρ Λ representing its low energy attractor. ϵ is a free dimensionless parameter that fixes the sign and the strength of the quadratic term and the energy scale ρ i characterises the non-linear interaction. To close the system, we also require the Raychaudhuri equation to describe the evolution of the expansion scalar H , To analyse the dynamical system consisting of (11), (12) and (13), we first define dimensionless variables, as well as compactified variables, in order to see the behaviour of the system at infinity. For this system, Y = ± 1 corresponds to H →±∞ , and X = Z = 1 corresponds to ρ x , ρ m → + ∞ . An example of the full phase space for a subset of models that expand towards a future de Sitter state is shown in Fig. 4 with the fixed points shown in Table 4. We set the parameter values to ϵ = -0 . 25, q = 1, R = 0 . 05 2 and w x = -0 . 5, which defines a surface in the phase space. The reddish surface shows where the acceleration is zero, with models accelerating when trajectories are above this surface, and decelerating when they are below it. The Flat Friedmann Separatrix would be a surface in the phase space, however for the sake of clarity we just show the flat trajectories in green for the parameter values we set. There is a trajectory, not shown in the figure, which connects the flat fixed points dS 3+ and dS 3 -. This is a high energy de Sitter trajectory, representing a closed de Sitter model. In this example, all trajectories are non-singular. Generically they either emerge from a de Sitter phase or are cyclic models; a separatrix represents emergence from an Einstein state. Open (blue) and flat models emerge from a de Sitter fixed point (magenta) and expand (contract) to another de Sitter state. The Closed Friedmann Separatrix (CFS) passing through the saddle Einstein point is shown by the thick black curve. Closed models (purple) outside the CFS emerge from a contracting phase asymptotic in the past to the de Sitter fixed point dS 1 -, then bounce at high energy before expanding to dS 1+ . Trajectories inside the CFS either bounce once, contracting from dS 1 -and expanding to dS 1+ , or are cyclic around the Einstein fixed point E 2 , and repeatedly contract, bounce and expand. An emergent case which expands from the Einstein fixed point E 1 also exists along the CFS. In this example, two Einstein fixed points (cyan) exist in the phase space, which coincide with where the acceleration is zero. This means in this example, open, flat and closed trajectories outside of the CFS all evolve with an early- and late-time acceleration, connected by a decelerated period. These are the cases of interest as they qualitatively match the observed Universe. 1 0.5 0 0 99 Y (b) 5 1 0.5X 0", "pages": [ 8, 9, 10 ] }, { "title": "4 Conclusions", "content": "In this essay, we have carried out a dynamic systems analysis of the Emergent Universe scenario in [1], which consists of a scalar field with an asymmetric potential with a plateau for negative ϕ . Originally, this scenario was considered with specific initial conditions such that the universe emerged from a static Einstein state. The aim of this essay was to understand whether the past repellor was generically non-singular for any initial conditions. The de Sitter model is normally associated to a cosmological constant associated with an asymptotically stable future attractor [27, 28], but it can also represent a past repelling fixed point [29, 25]. We found trajectories which emerge from a de Sitter fixed point, however like with emergence from the Einstein fixed point, they form a set of measure zero in phase space. All other trajectories expand from a singularity. We remark that simply with the exchange ϕ →-ϕ the potential (1) for the model in [1] becomes the potential for the Einstein-frame version [30] of Starobinsky inflation [31], hence our analysis can easily be mapped into one for that model. Models of this type are currently the most favored by experimental results [32, 33]. With the aim of showing a scenario where emergence from a non-singular state is generic during expansion, we then considered a fluid model with interacting dark energy and dark matter. In this case, it is not necessary to have a quantum gravity era at high energy to avoid a singularity, e.g. see [34, 35]. Instead we have a classical de Sitter vacuum-dominated era at high energy from which our models emerge from, which is a past attractor for flat and open models, or a transition phase through a bounce for a subset of closed models. However, some models are cyclic and do not necessarily become close to this de Sitter phase. We also highlighted the models of interest that evolve with an early- and late-time acceleration connected by a decelerated period, which include open, flat and closed models. Therefore, at least qualitatively, these models are consistent with the observable Universe and emerge from a non-singular de Sitter state. For FLRW models that have a singularity, it is well known that this is very special in that it is matter dominated, while in more general GR models the shear anisotropy becomes dominant in approaching the singularity, which is said to be velocity dominated, e.g. see [36] and refs. therein. More in general, especially for bouncing models with a contracting phase, the question is if the singularity avoidance found in the FLRW case is stable against shear anisotropy [37]. In a long-wavelength approximation to inhomogeneities [38], this question can be investigated by generalising non-singular FLRW models with a bounce to anisotropic Bianchi IX models, see [39] and refs. therein. This will be the goal of our future investigation, generalising the nonsingular scenario presented here based on dark matter interacting with dark energy, to study the stability of the results [26].", "pages": [ 11 ] } ]

[ { "title": "Reinterpretation of the Fermi acceleration of cosmic rays in terms of the ballistic surfing acceleration in supernova shocks", "content": "Krzysztof Stasiewicz a,1 a Space Research Centre, Polish Academy of Sciences, Bartycka 18A, Warszawa, 00-716, Poland", "pages": [ 1 ] }, { "title": "Abstract", "content": "The applicability of first-order Fermi acceleration in explaining the cosmic ray spectrum has been reexamined using recent results on shock acceleration mechanisms from the Multiscale Magnetospheric mission in Earth's bow shock. It is demonstrated that the Fermi mechanism is a crude approximation of the ballistic surfing acceleration (BSA) mechanism. While both mechanisms yield similar expressions for the energy gain of a particle after encountering a shock once, leading to similar power-law distributions of the cosmic ray energy spectrum, the Fermi mechanism is found to be inconsistent with fundamental equations of electrodynamics. It is shown that the spectral index of cosmic rays is determined by the average magnetic field compression rather than the density compression, as in the Fermi model. It is shown that the knee observed in the spectrum at an energy of 5 × 10 15 eV could correspond to ions with a gyroradius comparable to the size of shocks in supernova remnants. The BSA mechanism can accurately reproduce the observed spectral index s = -2 . 5 below the knee energy, as well as a steeper spectrum, s = -3, above the knee. The acceleration time up to the knee, as implied by BSA, is on the order of 300 years. First-order Fermi acceleration does not represent a physically valid mechanism and should be replaced by ballistic surfing acceleration in applications or models related to quasi-perpendicular shocks in space. It is noted that BSA, which operates outside of shocks, was previously misattributed to shock drift acceleration (SDA), which operates within shocks. Keywords: Cosmic rays, Shock waves, Acceleration of particles, Supernova remnants", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Cosmic rays exhibit a power-law energy distribution with a spectral index of s ≈ -2 . 5 below the knee at an energy of K ≈ 5 × 10 15 eV, and a steeper slope, s ≈ -3, above the knee and below the ankle at 5 × 10 18 eV (Hillas, 1984; Helder et al., 2012). Understanding the spectrum and acceleration mechanisms operating at such high energies has been an important, yet not fully explained problem in astrophysics. Fermi (1949) proposed that the cosmic ray spectrum could correspond to ions accelerated by bouncing o ff magnetic clouds in the interstellar medium. It is now believed that these clouds represent magnetic turbulence responsible for second-order Fermi acceleration. On the other hand, first-order Fermi acceleration is generally regarded as the major mechanism responsible for the formation of cosmic rays via the di ff usive shock acceleration (DSA) process (Bell, 1978; Blandford and Eichler, 1987; Longair, 2011). The recent Magnetospheric Multiscale (MMS) mission (Burch et al., 2016), consisting of four satellites flying through the Earth's bow shock, has provided the best measurements of collisionless shocks to date. The MMS satellites fly occasionally with separation distances of 20 km, which is smaller than the thermal proton gyroradius of 100 km. In this mission the electric, E and magnetic, B fields are sampled at the rate of 8192 s -1 , while particles distribution functions are measured with time resolution of 30 ms for electrons and 150 ms for ions. Based on MMS data, it has been established that thermalization, heating, and acceleration of ions and electrons in collisionless shocks are related to four plasma processes: SWE (stochastic wave energization), TTT (transit time thermalization), BSA (ballistic surfing acceleration), and QAH (quasi-adiabatic heating), explained further in the text (see also list of acronyms). Understanding processes in the bow shock can help understand more powerful astrophysical shocks, such as those created during supernova explosions, which are most likely involved in the acceleration of cosmic rays. The processes SWE and TTT are stochastic in nature, related to deterministic chaos. They rely on strong gradients of the electric and magnetic fields that lead to randomization of particle orbits and e ffi cient stochastic heating on the timescale of one gyroperiod (Stasiewicz, 2023b). TTT thermalizes streaming ions on magnetic field gradients within a fraction of the gyroperiod and does not require any waves, instabilities, or anomalous collisions. SWE works on electric field gradients and can accelerate protons to a few hundred keV (Stasiewicz and Eliasson, 2021; Stasiewicz et al., 2021; Stasiewicz and Kłos, 2022). SWE of electrons is responsible both for heating and formation of flat-top electron distributions (Stasiewicz, 2023a). Quasiadiabatic heating (QAH) operates on particles with a conserved first adiabatic invariant ( v 2 ⊥ / B = constant), which requires the gyroradius rc to be much smaller than the width of the shock, D ≡ B | ∇ B | -1 . Ballistic surfing acceleration (BSA) operates on particles with a gyroradius rc ≫ D , which applies to superthermal ions and mildly relativistic electrons (Stasiewicz, 2023b). It bears resemblance to the shock drift acceleration (SDA) previously discussed by various authors (Jokipii, 1982; Jones and Ellison, 1991; Zank et al., 1996). Another similar process is shock surfing acceleration (SSA), where low energy particles drift along the shock front due to surface waves (Shapiro et al., 2001; Lever et al., 2001; Hoshino and Shimada, 2002). In all these processes, energization is due to particle motion along the convection electric field. However, the term SDA implies ∇ B drift acceleration, which occurs within the ramp. In contrast, BSA occurs outside the ramp, where particles with large gyroradii engage in ballistic surfing and do not experience ∇ B or wave e ff ects. As is well known, plasma can also be e ff ectively heated by waves at resonance frequencies: cyclotron, lower hybrid, upper hybrid, and plasma frequencies. Resonant heating is commonly used in laboratory plasmas, but does not appear to be important in shocks. A question arises, what is the relation of the Fermi mechanism and the DSA process with the above-described processes identified at the bow shock. In the next section we shall study heating of energetic ions in a model shock, searching for signatures of Fermi acceleration. We find that the first order Fermi acceleration represents a crude approximation of BSA. We show that the cosmic ray spectrum and the presence of the knee can be explained exclusively by BSA.", "pages": [ 1, 2 ] }, { "title": "2. Ballistic surfing acceleration", "content": "To understand acceleration of high energy particles we consider motion of test particles with velocity v much larger than the thermal velocity vTi of ions that maintain the shock. Trajectories of particles with rest mass m 0 and charge q are described by the momentum equation where p = γ m 0 v , γ = (1 -v 2 / c 2 ) -1 / 2 , and c is the speed of light. Here, we utilize the shock reference frame in a geometry where the ˆ x axis is in the negative direction to the shock normal, ˆ y is in the direction of the convection electric field, and the magnetic field is in the [ Bx 0 , 0 , Bz ( x )] plane. The kinetic energy increase of a particle implied by this equation is There are three types of electric fields that determine particle dynamics and energization processes in the shock frame: ˆ y Ey - the convection field, constant across one-dimensional structures, ˆ x ES ( x ) - the cross-shock electric field, maintained by the electron pressure gradient, and ˜ E ( r , t ) - the wave electric field. In the reference frame moving with the convection velocity, Ey vanishes, but acceleration is facilitated by the inductive electric field, ∇ × E = -∂ B /∂ t observed by a particle moving in a time varying magnetic field. SWE works best on thermal particles and is ine ff ective for high-speed particles, so we exclude electrostatic waves from consideration in this paper. The model shock is described in Stasiewicz (2023b) and represents a magnetic ramp with compression cB = Bd / Bu between downstream and upstream values. A normal component Bx 0 implies the upstream field angle cos η = Bx 0 / Bu from the normal direction. Magnetic turbulence is not included in the shock model, but it can be easily implemented similarly to electrostatic waves. It is expected to lead to the isotropisation of particle distributions and could also scatter particles back to the shock. Parameters that enter the model are as follows: the upstream sonic Mach number, M = Vu / vTi = 8, the ratio of the thermal ion gyroradius rci to the width of the shock ramp, rci / D = 1, and compression cB = 4. The transit time thermalization parameter is set to χ B = 8 and the stochastic parameter for the cross-shock electric field to χ S = 1 . 5; see Stasiewicz (2023b) for definitions. Two ions are injected into an oblique shock η = 85 · at x = -40 in units of the shock width and followed with di ff erential equations described in Stasiewicz (2023b). Fig. 1(a) shows ion trajectories in plane ( x , y ). The initial velocity consists of the E × B drift normalized by the upstream thermal speed, u E × B = V E × B / vTi , with additional velocities: ux = + 20 (blue), and uy = -20 (red). Panel (b) shows the total kinetic energies u 2 of ions along the respective trajectories, panel (c) shows gyration energies only ( u ⊥ -u E × B ) 2 , and panel (d) shows u 2 ∥ . Adiabatic projection of the initial perpendicular energy of the red ion is shown as the black curve b ( x ) u 2 0 ⊥ , where b ( x ) = B ( x ) / Bu . Both ions behave non-adiabatically and do not follow the adiabatic projection. We observe that the blue ion makes three crossings of the shock at x = 0, while the red ion makes seven crossings before being transmitted downstream. Each crossing is associated with an increase in gyration energy, as seen in panel (c). The kinetic energy in panel (b) correlates with particle movements in the ± y direction, along Ey , which has been defined as the BSA by Stasiewicz (2023b). This obvious acceleration process operating outside the shock was previously misattributed to SDA, which is related to ∇ B drift and operates within the shock. However, particles undergoing BSA in the gradient-free zone are unaware of the existence of the shock, and therefore cannot experience shock drift acceleration. Temperature variations in panel (c) and the transfer of energy into the parallel motion shown in panel (d) occur only during shock transitions at x ≈ 0 and are caused by TTT, which is related to the inductive electric field or to the changing magnetic field direction in the shock frame. By decreasing the shock angle to η = 70 · in Fig. 2 we observe the reflection of the red ion back into the upstream region. Particle reflections are more common for smaller angles η , and are not related to the magnetic mirror force, which does not apply for large gyroradius particles, rc ≫ D . For particles reflected upstream, the energy gain is directed into parallel motion rather than perpendicular heating. The transfer of energy into parallel motion, as shown in panels (d), is not related to the cross-shock electric field, which has a par- allel component E ∥ = ES ( x ) cos η , as the patterns remain the same when ES is set to zero. The cross-shock potential is e ∆Φ ≈ 2( cB -1) = 6 in normalized energy units of Teu ∼ Tiu , so it has negligible e ff ect even for zero temperature ions with energy u 2 = M 2 = 64; see Stasiewicz (2023b), not to mention the high-energy ions, u 2 ∼ 1000 in the simulations here. Furthermore, the cross-shock potential energy does not accumulate, but cancels out after each gyration around the shock. One should not confuse cyclotron turning points with shock reflections. Cyclotron turning points recur periodically, once during every gyroperiod, while shock reflections are singular events that occur only once. Turning points can manifest anywhere, contingent upon the initial conditions and the gyroradius. Sensitivity to initial conditions is a hallmark of deterministic chaos. Figures 1-2b illustrate that neither the process of reflection nor the transitions through the shock at x ≈ 0 are associated with significant changes in particle energy.", "pages": [ 2, 3 ] }, { "title": "3. BSA and formation of the cosmic ray spectrum", "content": "As observed in the previous section, energetic particles are accelerated outside shocks through the ballistic surfing acceleration process by the convection electric field Ey , rather than through reflections as suggested by Fermi (1949). A necessary condition for BSA is that the particle's gyroradius is larger than the shock width. The BSA process is elucidated in Fig. 3, where it can be observed that ions moving in the + y direction in the upstream part of the orbit increase kinetic energy ( qEyvy > 0), while those moving in the -y direction in the downstream orbit decrease energy ( qEyvy < 0). Because Bd > Bu , the downstream gyroradius rcd is smaller than the upstream one rcu , and particles always gain energy. Within the shock ramp denoted by vertical lines, the particles experience gyrocenter drifts induced by ∇ B and are subject to the combined convection and cross-shock electric fields, ˆ y Ey + ˆ x ES ( x ). The energy gain after a full rotation around the shock consists of three terms: where the first term corresponds to ballistic surfing acceleration outside the shock ramp, the second term to shock drift acceleration within the ramp, and the third term to acceleration by the cross-shock electric field. The first BSA term is the only significant one for high-energy particles because the integration domain is much larger than in the other two integrals (2 π rc ≫ 2 D ). It can be concluded that BSA is the primary mechanism for accelerating particles with large gyroradii ( rc ≫ D ), while the e ff ects of SDA are negligible due to the very short interaction time compared to the gyroperiod. Conversely, for small gyroradii particles ( rc ≪ D ), SDA serves as the primary mechanism for the acceleration and BSA does not apply. However, this process is equivalent to adiabatic heating, so SDA can be considered a subset of the more general QAH (Stasiewicz and Eliasson, 2023). After one gyration across the shock the energy gain implied by Eq. (3) is ∆ K ≈ 2 qEy ( rcu -rcd ), as illustrated in Fig. 3. For relativistic particles with kinetic energy K ≈ pc and gyroradius rc = p ⊥ / qB , the energy gain is where Vu = Ey / Bu is the upstream convection velocity, and the integral is the average over pitch angles, assuming isotropic distribution. Averaging over incident angles of particles entering the shock would introduce a numerical factor g ≲ 1 to this equation; see Fig. 3. Such a factor could be incorporated into a slightly modified value of cB , which is a free parameter of the model. The e ff ective compression will be then c ' B = cB / [ cB -g ( cB -1)]. Equation (4) is analogous to the expression derived through a wholly distinct approach by Bell (1978). This formulation is commonly referred to as the first-order Fermi acceleration: This equation was derived from the di ff erence in particle energy between two inertial frames, characterized by the velocity ratio Vu / Vd = nd / nu ≡ cN ; see Longair (2011) for derivation. Here, cN denotes the density compression, typically set to the standard value cN = 4. It's important to note that this equation was mistakenly linked to energization. However, the energy di ff erence between two inertial frames is simply a scalar value derived from the Lorentz transformation and doesn't relate to acceleration, which is defined by Eq. (2), and must be computed within the same reference frame. This yields the accurate expression (4), which depends on cB but remains independent of cN . Figs. 1-2b show that particles undergo energy gain over an extended period by ballistic surfing in the upstream region and lose energy in the downstream region, as described by d K / d t = qEyvy . Contrary to suppositions of the Fermi model, shock transitions, | x | < D , are not associated with significant energy changes, as can be seen in the aforementioned figures and in Eq. (3). Cumulative energy increases caused by multiple encounters of particles with shocks, combined with scattering by turbulence outside the ramp represent di ff usion in velocity space, described as DSA. Because the BSA energy increase in Eq. (4) is similar to the first-order Fermi acceleration (5), we can follow the standard approach to DSA, as described, for example, by Bell (1978) and Longair (2011). Equation (4) implies that the particle energy after one interaction with a shock is This acceleration is not related to the reflection or to the shock crossing process but to the full gyration across the shock ramp with cB > 1. Let P be the probability that particles remain in the shock region after one interaction or gyration. Then, after j interactions there are N = N 0 P j particles with energies K = K 0 h j . Eliminating j , one obtains N / N 0 = ( K / K 0) ln P / ln h , where N is the number of particles that reached energy K and can be accelerated further. Using Longair (2011) value for probability, P ≈ 1 -Vu / c , we find the spectrum of cosmic rays predicted by the BSA model where which can be compared with the spectral index predicted by the Fermi / DSA model (Longair, 2011) In deriving this equation, hF = 1 + 1 3 ( cN -1) Vu / c was used from incorrect Eq. (5). The Fermi / DSA formulation neglects the impact of the electric field, which is crucial for particle energization. Moreover, it introduces an erroneous dependence on density compression, inconsistent with the fundamental equations (1) and (2) which determine heating, but remain independent of plasma density. Eq. (5) is clearly non-physical, because it diverges for large values of cN , while the numerical factor in the correct Eq. (4) approaches unity for cB ≫ 1. While not being physically valid, Eq. (9) provides one correct value sF = -2 . 5 for cN = 3, coinciding with Eq. (8) for cB = 3, and both agree with the measured index s . This single, incidentally correct value of the Fermi / DSAmodel can possibly explain its success and popularity despite erroneous physical assumptions. The error has not been disclosed earlier, probably due to the observed relation n ∝ B in shocks, which yields cN ≈ cB . The physical dependence of particle energy gain on cB has been confused with a circumstantial dependence on cN . The BSA model would reproduce the observed index s ≈ -2 . 5 for the e ff ective shock compression cB ≈ 3. For the standard value cB = 4, implied from Rankine-Hugoniot relations (Kennel, 1988), we obtain s = -2 . 3. Smaller compressions cB = 2 would lead to s = -3 observed above the knee, while magnetic walls, cB ≫ 1 would lead to the asymptotic value s = -2. Although BSA would function e ff ectively at arbitrarily high energy, the acceleration in the upstream region will be counteracted by deceleration in the downstream region as the diameter of the orbit on the compressed side approaches the shock length L , see Fig. 3. The condition 2 rcd ∼ L would result in a knee in the spectrum located at energy Interestingly, this does not depend on shock velocity or particle mass. The observed distribution of supernova remnant sizes, L SNR, ranges from 1 pc ( = 3 × 10 16 m) to 200 pc (Badenes et al., 2010). The observed knee energy, KL ≈ 5 × 10 15 eV, is derived from Eq. (10) for ⟨ LBd ⟩ ∼ 1 nT pc, which could correspond to, for example, Bd ≈ 1 nT on the inner (compressed) side of a spherically expanding supernova shock and L ∼ 1 pc. Shocks with length L < 1 pc inside supernova remnants with an e ff ective compression cB = 3 would lead to s = -2 . 5, which could explain the energy spectrum below the knee. However, even particles with a gyroradius much larger than the shock length can undergo further acceleration. This scenario may occur when downstream flows become stagnant at some distance from the shock, with Vd ∼ 0, leading to the vanishing of the electric field in this region, Eyd ∼ 0. Particles gyrating in the downstream region would not experience deceleration, as depicted in Figs. 1-2b, but they would still be subject to acceleration in the upstream region, d K / d t = qEyvy > 0. This process could yield cosmic ray energies up to the ankle at 5 × 10 18 eV in shocks shorter than the gyroradius, L < rc . The spectral index s ≈ -3 in this energy range can be achieved with cB ≈ 2; however, other factors may also play a significant role in determining the observed index.", "pages": [ 3, 4, 5 ] }, { "title": "4. Discussion", "content": "BSA requires that the gyroradius of particles is larger than the shock width, rc / D > 1, which is fulfilled at the bow shock by protons with energy higher than 100 eV and by electrons with energy higher than 180 keV. Initial heating of cold ions in shocks is accomplished by TTT and SWE mechanisms, which can be seen in Fig. 4 of Stasiewicz (2023b), where streaming protons with temperature Ti ≈ 20 eV, are TTT thermalized and SWE energized to 400 Ti = 8 keV within a gyroperiod. In quasi-parallel shocks ions are accelerated by the SWE mechanism to ∼ 200 keV which corresponds to the E × B velocity in the wave electric field, VE × B = ˜ E ⊥ / B with ˜ E ⊥ ∼ 100 mVm -1 being much larger than the convection field Ey ∼ 5 mVm -1 (Stasiewicz and Eliasson, 2021; Stasiewicz et al., 2021). Further acceleration of these ions could continue by means of BSA in subsequent shock encounters. Let us assume that the injection energy for BSA is K 0 = 10 keV, and the shock parameters are cB = 4, and Vu = 10 , 000 kms -1 . Using the exact form of Eq. (4) with the kinetic energy defined as K ≡ ( m 2 0 c 4 + p 2 c 2 ) 1 / 2 -m 0 c 2 , we find that protons reach the knee at K = 5 × 10 15 eV after 657 BSA interactions without energy losses. In case of faster shock speeds, Vu = 20 , 000 km s -1 the final energy would be reached after 334 BSA, while Vu = 40 , 000 km s -1 requires only 172 BSA interactions. We have assumed here that transmitted particles could be scattered back to the shock by downstream turbulence, while particles reflected upstream can encounter another shock front, or can be turned back by upstream waves. The physical picture of acceleration in the shock reference frame adopted in this paper is easier to understand than the traditional (Fermi) analysis of ion acceleration in the plasma reference frame, where the convection electric field is removed. This approach ignores the fundamental laws of physics that particle trajectories are determined by the Lorentz force equation (1), as illustrated in Figs. 1-2, and the energization is determined by Eq. (2), and not by the energy transformation between two inertial systems as in the Fermi / DSA model. Furthermore, Fermi (1949) relies on the concept of reflections by the magnetic mirror force, which is valid for particles with rc < D , and clearly not applicable for high-energy, large gyroradius particles.", "pages": [ 5 ] }, { "title": "5. Conclusions", "content": "It is shown that ions and electrons with gyroradius rc ≫ D are accelerated by the convection electric field Ey in a process described as ballistic surfing acceleration. BSA operates outside of shocks on particles from approximately 100 eV for protons up to very high energies observed in the cosmic ray spectrum. BSA predicts a knee in the spectrum when the gyroradius becomes comparable to the size of the shock, which determines the knee energy given by Eq. (10). The spectral index in BSA is determined by the e ff ective shock compression cB , and can accurately reproduce the observed index s ≈ -2 . 5 below the knee energy as well as a steeper spectrum, s ≈ -3, above the knee. It is demonstrated that the Fermi / DSA model, which yields the spectral index (9), is inconsistent with the fundamental equations of electrodynamics (1) and (2), and is therefore not valid. The popularity of the Fermi model was unjustified, primarily due to the coincidental agreement of one spectral index value with the correct BSA model and with observations. It is found that to reach the knee energy of 5 × 10 15 eV, a proton starting from 10 keV in a collisionless environment needs only 334 BSA interactions in shocks moving with velocity Vu = 20 , 000 km s -1 . For protons moving in the average magnetic field of ⟨ B ⟩ = 1 nT, the net acceleration time of 334 gyroperiods would correspond to 227 years of sidereal time, accounting for time dilation during each gyroperiod.", "pages": [ 5, 6 ] }, { "title": "List of Acronyms", "content": "BSA - ballistic surfing acceleration DSA - di ff usive shock acceleration QAH - quasi adiabatic heating SDA - shock drift acceleration (a subset of QAH) SSA - shock surfing acceleration SWE - stochastic wave energization TTT - transit time thermalization", "pages": [ 6 ] }, { "title": "Data / software availability", "content": "The mathematical shock model used to make Figs. 1-3 was published in: https: // doi.org / 10.1093 / mnrasl / slad071.", "pages": [ 6 ] }, { "title": "Acknowledgements", "content": "This work has been supported by Narodowe Centrum Nauki (NCN), Poland, through grant No. 2021 / 41 / B / ST10 / 00823.", "pages": [ 6 ] }, { "title": "References", "content": "Blandford, R., Eichler, D., 1987. Particle acceleration at astrophysical shocks: A theory of cosmic ray origin. Physics Reports 154, 1 - 75. doi: 10.1016/ 0370-1573(87)90134-7 . Burch, J.L., Moore, R.E., Torbert, R.B., Giles, B., 2016. Magnetospheric multiscale overview and science objectives. Space Sci. Rev. 199, 1-17. doi: 10.1007/s11214-015-0164-9 . Fermi, E., 1949. On the origin of the cosmic radiation. Phys. Rev. 75, 11691174. doi: 10.1103/PhysRev.75.1169 . Hillas, A.M., 1984. The origin of ultra-high-energy cosmic rays. Annual Review of Astronomy and Astrophysics 22, 425-444. doi: 10.1146/ annurev.aa.22.090184.002233 .", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Traditional pulsar timing techniques involve averaging large numbers of single pulses to obtain a high signal-to-noise (S/N) profile, which is matched to a template to measure a time of arrival (TOA). However, the morphology of individual single pulses varies greatly due to pulse jitter. Pulses of different fluence contribute differently to the S/N of the pulse average. Our study proposes a method that accounts for these variations by identifying a range of 'states' and timing each separately. We selected two 1-hour observations of PSR J2145 -0750, each in a different frequency band with the Green Bank Telescope. We normalized the pulse amplitudes to account for scintillation effects and probed different excision algorithms to reduce radio-frequency interference. We then measured four pulse parameters (amplitude, position, width, and energy) to classify the single pulses using automated clustering algorithms. For each cluster, we calculated an average pulse profile and template and used both to obtain a TOA and TOA error. Finally, we computed the weighted average TOA and TOA error, the latter as a metric of the total timing precision for the epoch. The TOA is shifted relative to the one obtained without clustering, and we can estimate the shift with this weighting using the same data. For the 820-MHz and 1400-MHz bands, we obtained TOA uncertainties of 0 . 057 µ s and 0 . 46 µ s, compared to 0 . 066 µ s and 0 . 74 µ s when no clustering is applied. We conclude that tailoring this method could help improve the timing precision for certain bright pulsars in NANOGrav's dataset. Keywords: Pulsar timing - Automatized Classification - Compact Objects - Gravitational Waves", "pages": [ 1 ] }, { "title": "Improving Pulsar Timing Precision with Single Pulse Fluence Clustering", "content": "1 School of Physics and Astronomy, Rochester Institute of Technology, Rochester, NY 14623, USA Laboratory for Multiwavelength Astrophysics, Rochester Institute of Technology, Rochester, NY 14623, USA 2 3 SETI Institute, 339 N Bernardo Ave Suite 200, Mountain View, CA 94043, USA 4 Department of Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV 26506, USA 5 Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "1.1. TOA error due to single pulse variability Pulsar timing is a technique that involves using observations of radio pulsar pulse profiles to calculate pulse arrival times, which are then compared with mathematical models that incorporate a wide variety of astrophysical phenomena. The many merits of this technique include characterizing the orbits of binary systems (Fonseca et al. 2016), enabling tests of general relativity (Kramer et al. 2006), constraining nuclear equations of state (Fonseca et al. 2021; Cromartie et al. 2020; Demorest et al. 2013; Antoniadis et al. 2013), and detecting planetary-mass companions (Wolszczan & Frail 1992). Moreover, by monitoring variations in pulse times of arrival (TOAs) from an array of the most stable millisecond pulsars (MSPs), evidence has been detected for a signal consistent with nanohertz frequency gravitational radiation from a population of supermassive black hole binaries (Agazie et al. 2023; Hellings & Downs 1983; Detweiler 1979). As pulsars are generally weak radio sources, pulsar timing applications require the addition of several single pulses. This technique, known as folding, utilizes pulsars' remarkably stable rotation to time-average hundreds of thousands of single pulses. As a result, the background noise is decreased while the single pulses are added in phase, increasing the signal-to-noise (S/N) ratio. The average profile formed by averaging a large number of single pulses converges to a shape that appears to be epoch-independent (see Craft 1970; Backer 1975; Phillips & Wolszczan 1992; Hassall et al. 2012; Pilia et al. 2016). However, nearly every pulsar observed with high sensitivity shows intrinsic single-pulse stochastic variability in excess of that expected from radiometer noise (see Cordes & Downs 1985; Cordes et al. 1990; Liu et al. 2012; Shannon et al. 2014a; Dolch et al. 2014; Shannon & Cordes 2012). This includes variations in amplitude and phase that are correlated from pulse to pulse (such as the drifting sub-pulse phenomenon) and variations that are uncorrelated from pulse to pulse. When averaged to form a pulse profile, this variability causes the underlying pulse shape to differ from that of the template. This difference biases the measurements of arrival times, contributing to what is called 'jitter noise' in the TOAs (Shannon & Cordes 2010). The jitter TOA error is independent of S/N, so it cannot be mitigated with improved observing backends. Given the importance of precise timing for PTA experiments, several studies have attempted to quantify and mitigate the presence of pulse jitter in MSPs. Shannon & Cordes (2012) showed that intrinsic single-pulse variations in amplitude, shape, and pulse phase for the MSP J1713+0747 are largely responsible for the excess in timing errors for that pulsar. Moreover, they found that brighter single pulses tend to have earlier arrival times. Most importantly, they investigated two methods for correcting TOAs due to single-pulse variations: multi-component template fitting and principal component analysis. However, both algorithms were unsuccessful at improving the precision of arrival times using pulse-shape information. On the other hand, a growing body of evidence suggests that MSPs undergo short-scale mode changes: shapes. They classified single pulses into different categories based on their similarity (or dissimilarity) to the pulse template. Based on this classification, they devised timing algorithms that optimize the pulse template by creating a template basis that describes the single-pulse variations. In doing so, they reduced jitter error in an observation of J0835 -4510 (Vela) by 30-40%. Interestingly, they also find a strong correlation between the peak intensity of a pulse and the phase at which that peak falls. This body of work emphasizes the need for an in-depth study into single-pulse variability and how to account for it in the pulsar timing process.", "pages": [ 1, 2 ] }, { "title": "1.2. TOA Errors from Additive Noise", "content": "In the present work, we assess the merit of a method for improving TOA precision based on single-pulse analysis that, unlike Shannon & Cordes (2012)'s jitter-based approach, attempts to mitigate the error from additive noise. This error arises from single pulses having radiometer noise with a Gaussian probability density function that is additive when averaging them to create the integrated pulse profile. In turn, template matching yields TOA uncertainties that depend on the S/N of the average pulse (Cordes & Shannon 2010). Traditional pulsar timing techniques assume that the pulse intensity at a given observing frequency ν can be modeled as a function of time as: where U ( t ) is the pulse template shape, n ( t ) is additive noise with RMS amplitude given by σ n , t 0 is the TOA, and S is the S/N of the pulse profile (peak to RMS offpulse). In that case, the signal model for an integrated pulse profile is given by: where W eff is the effective pulse width (i.e., the width of a top-hat pulse with the same area), and N ϕ is the number of phase bins across the pulse (e.g., Dolch et al. 2014). The decrease in TOA uncertainty with S is the main reason why integrated pulse profiles are used for timing purposes. The number of single pulses that are needed to be able to achieve a stable profile that can be matched to a template is given by the profile stabilization timescale, and it is different for each pulsar (e.g.: Kloumann & Rankin 2011; Teixeira et al. 2012). In this work, we follow Osglyph[suppress]lowski et al. (2014)'s scheme of filtering single pulses based on their S/N. The authors sought to reduce the RMS timing residual by removing the lowest-S/N single pulses from the integrated pulse profile average. We found that such a scheme can potentially yield reduced TOA uncertainties for short observations ( ∼ 1000 single pulses). However, when a larger number of single pulses is available, we found that the improvement in S/N from single-pulse averaging outweighs the reduction in W eff when removing the lowestS/N pulses. Instead of using only the brightest single pulses in an observation, we resume Kerr (2015)'s idea that variations in the single pulse shape correspond to shifts between a range of pulse states, each with a characteristic fluence and integrated pulse profile. For the single pulses corresponding to the same state, the observed shape variability, while stochastic, is not without memory but rather correlated in time. From this perspective, states with different fluences weigh differently when contributing to the S/N of the epoch's pulse average. Therefore, assigning weights based on the fluence portrait when computing the epoch-TOA could potentially yield improved timing precision. In Shannon et al. (2014b), the authors found that the single pulses from MSPs with the highest intrinsic energy average to a different shape than when averaging all the pulses. Similarly, we will assume that single pulses belonging to different states average to different pulse shapes that can be modeled as: where ⃗s is a set of parameters (such as pulse energy, S/N, W eff , etc) that characterize each pulse state. Eq. 2 can then be rewritten as: From this perspective, conventional timing techniques involve averaging along the ⃗s -axis to increase S . However, we propose that for some pulse states ⃗s the decrease in W eff that was observed by Shannon et al. (2014b) can overcome the increase in S when averaging a larger number of single pulses and, therefore, provide a smaller σ TOA , as proved by Osglyph[suppress]lowski et al. (2014) for a smaller number of single pulses. For the sake of simplicity, we will assume discrete and disjoint pulse states ⃗s 1 , ⃗s 2 , ⃗s 3 , . . . . In the appendix, we consider a theoretical treatment of the timing implications of separating pulses belonging to one of two states: a high or low fluence state. However, in our main work, we do not restrict our analysis to a specific number of fluence-only states. Instead, we assume a range of pulse states, and that all single pulses belonging to the same state share similar morphological features such as energy, width, amplitude, S/N, etc. Under these assumptions, state assignment can be performed utilizing automated clustering techniques based on these single pulse features. These techniques are more sophisticated than manually binning pulses in the various dimensions and can potentially identify groupings in the pulse parameter phase space. We will show later in our work that the choice of algorithm does not change the overall impact of pulse clustering. Once the pulse states have been identified by a given algorithm, we calculate the average pulsar profile of each state which can then be compared to a pulse template to produce a TOA and its corresponding σ S / N . In Sec. 2 we provide information on the data collection and reduction methods for the observations we analyzed in this work, as well as the corresponding corrections for scintillation. In Sec. 3 we describe the quantification of the pulse features and the clustering algorithms we used to classify the single pulses. In Sec. 4, we summarize the main results, including the TOA error attained with each clustering method. In Sec. 5 we discuss the implications of this work. Processed data products presented here are publicly available 1 as of the date this work is published.", "pages": [ 2, 3 ] }, { "title": "2. OBSERVATIONS AND DATA PROCESSING", "content": "We will now describe the observations used in this work, how they were reduced, and how we accounted for the modulations in pulse intensity introduced by interstellar scattering/scintillation.", "pages": [ 3 ] }, { "title": "2.1. Dataset", "content": "In this work, we analyzed observations of PSR J2145 -0750. Discovered by Bailes et al. (1994), this low-mass binary pulsar is in a nearly circular orbit, with an orbital period of 6.8 days (Lohmer et al. 2004). It has a spin period of 16 . 05 ms, which is considerably longer than the average period of ∼ 4 ms of this type of binary pulsar, and a distinctive pulse profile with two components separated by 0.20 of the pulse period in phase (Bailes et al. 1994). Most importantly, this bright pulsar exhibits high flux densities, of 14 . 25 mJy at 800 MHz and 4 . 85 mJy at 1400 MHz (Alam et al. 2021). As a result, it is one of the few MSPs whose single pulses have been characterized. Its individual pulses were first detected by Edwards & Stappers (2003); however, only ∼ 100 pulses were detected and the statistics of the distribution of pulse energies were not explored. Observations of PSR J2145 -0750 were recorded in an approximately 2 hours-long session on February 1st, 2017, using the 100-m Green Bank Telescope (GBT) radio-telescope of the Green Bank Observatory in West Virginia, USA. The radio receivers used for this observation cover two frequency bands: the 820-MHz band and the 1400-MHz (L) band. The source was observed for approximately 1 hour in each frequency band. While the 820-band bandwidth is smaller (200 versus 800 MHz), the pulsar is observed to be brighter and the singlepulse S/N is higher at those frequencies (see Table 1). Moreover, the frequencies corresponding to the L-band are more heavily affected by radio-frequency interference (RFI). The data were collected in 'search' mode where intensity and polarization sampling occurred at a rate of 10.24 µ s in each frequency channel with coherent dedispersion applied. We used the DSPSR package (van Straten & Bailes 2011) to split the time series into individual time sub-integrations of one pulse each with 512 phase bins (31.3 µ s resolution) using the NANOGrav 12.5-yr timing model file to phase align the profiles (Arzoumanian et al. 2020). We then calibrated the profiles in polarization using PSRCHIVE (Hotan et al. 2004). Before each of the observations, a noise diode signal was observed so that differential gain and phase offset corrections were applied before we summed across the polarization channels. No absolute flux calibrations were performed. The end product of the observation was a set of total-intensity pulse profiles for a series of n chan = 128 (820-band) or 512 (L-band) frequency channels, resulting in a channel bandwidth of 1 . 5625 MHz. In the preprocessing stage, all the intensity single pulse profiles were averaged across the polarization channels, the average intensity of the off-pulse baseline was subtracted, and the single pulses were phase-shifted so that the main pulse window (see Sec. 3.1) is centered in the middle of the data array. To account for changes in pulse shape and intensity due to the interstellar medium, we will assume that all chromatic delays have been perfectly removed or are negligible over each narrowband channel. These include the dispersive delay from DM, scattering, and frequencydependent pulse profile evolution. We also assume that the signal polarization has been calibrated perfectly. Further details about the observations, their calibration, and data reduction can be found in NANOGrav's 12.5year Arzoumanian et al. (2020) and its earlier dataset papers (Arzoumanian et al. 2016; Lam et al. 2016). Under these assumptions, we model pulse shapes I ( t, ν, ϕ ) as a function of phase ϕ , centered on time t and in a sub-band centered on frequency ν .", "pages": [ 3, 4 ] }, { "title": "2.2. Corrections for Scintillation", "content": "In addition to jitter and additive noise, a third source of TOA variance in short scales is changes in the interstellar impulse response from multipath scattering, which depends strongly on radio frequency (Cordes et al. 1990). This results in a pulse broadening function (PBF) caused by diffractive interstellar scattering/scintillation (DISS). DISS will generally result in intensity modulation on timescales of minutes to hours, depending on observing frequency, DM, and direction. The shape perturbation is correlated over a time equal to the diffractive timescale t d and a frequency range equal to the scintillation bandwidth, ν d (Hemberger & Stinebring 2008). The DISS timescale and bandwidth vary strongly with observing frequency, approximately as ν -6 / 5 and ν -22 / 5 , respectively. As a result of DISS, the time-frequency plane (known as the dynamic spectrum ) will be made up of independent intensity fluctuations called scintles in a timefrequency resolution cell, with a characteristic time t d and frequency scales ν d . The scintillation structure is related to the temporal broadening of pulses, resulting in a time delay (Cordes & Shannon 2010). Because of the intensity fluctuations introduced by the scintles, some low-fluence single pulses will see their amplitudes artificially amplified. Therefore, in order to isolate the single pulses with intrinsic high amplitude, this effect must be mitigated before any single pulse classification can take place. In this work, we accounted for DISS effects with a three-step process: band) frequency channels and N ϕ = 512 phase bins per single pulse. olding of the wavelet coefficients (as implemented in the PSRCHIVE function psrsmooth ). The scale factor b = Sσ 0 (see Eq. 1) that results from correlating each pulse profile with the template was used as a measure of the pulse relative amplitude. We then used those values to construct a dynamic spectrum for each frequency band, presented in Fig. 1. Next, we averaged the observation in frequency. This process usually involves performing a weighted average where each frequency channel is weighted by its S/N for a given time sub-integration. However, in normalizing by the dynamic spectrum, we artificially modified the RMS noise and, therefore, the corresponding weights. Consequently, we computed new frequencychannel weights as w i = 1 /σ 2 i where σ i is the RMS of the off-pulse noise. We then used these weights to average in frequency and obtain I ( t, ϕ ).", "pages": [ 4, 5 ] }, { "title": "3. SINGLE PULSE CLUSTERING", "content": "In this section, we first describe the single-pulse clustering schemes we will use in our analysis. Then we outline an algorithm for calculating TOA measurements and uncertainties for an ensemble of clusters and then weight-averaging them. Finally, we discuss the impact of different RFI mitigation routines.", "pages": [ 5 ] }, { "title": "3.1. Pulse features", "content": "We classified the single pulses into fluence states based on their morphology. Given the extensive single-pulse variability, in a few single pulses dominated by background noise the intensity maximum in the pulse window would not coincide with the main pulse component. To account for such cases, we constructed a main component window . This was created by fitting the position of the template's peak and setting a window around it with a width equal to 12 . 5% of the template's phase bins. As a result, we obtained a window covering phase bins [224 , 288]. Similarly, we identified the three pulse components using Astropy (Astropy Collaboration et al. 2013, 2018, 2022) and created component windows around their peaks with a width equal to 2.5 times the component width (see Fig. 2). Since the pulse peak looks jagged, we smooth it by performing a least-squares Gaussian fit to the points inside the main component window (see panel A in Fig. 3). We then computed four pulse features: As a result, every single pulse will be represented by an (amplitude, width, position, energy) point in a 4-parameter space. We can visualize this space by marginalizing over one of the parameters and plotting the features in a 3-dimensional space, as presented in Fig. 4. In this figure, we can appreciate the diversity in single-pulse morphology, with a vast majority of single pulses conglomerating in a low-energy, low-amplitude cluster at the bottom of the diagram. As expected, this distribution shows that most single pulses have low S/N and are dominated by noise. On the other hand, in the middle section of Fig. 4 and above the low-amplitude cluster we find a smaller number of single pulses with higher amplitudes; these represent the high-fluence data we will attempt to isolate in this analysis.", "pages": [ 5, 6 ] }, { "title": "3.2. Clustering Algorithms", "content": "In order to assign the single pulse features (i.e., as shown marginalized in Fig. 4) into the corresponding pulse states, we surveyed different automatized clustering algorithms. We opted for unsupervised classification methods to eliminate the need for a training dataset. However, due to the large sample size ( N ∼ 219000 single pulses per frequency band), clustering algorithms with high computational costs are impractical for this analysis. Instead, we surveyed low-computation cost (memory usage ≤ O ( N 2 )) clustering algorithms from the SciKit Learn (Pedregosa et al. 2011) library. 820 Band", "pages": [ 6, 7 ] }, { "title": "3.2.1. K-Means", "content": "The K-Means algorithm (MacQueen et al. 1967) is a method of vector quantization that clusters data by trying to divide a set of N samples into k disjoint clusters, each described by the mean position µ j of its samples x i in the features space. The means are commonly called the cluster 'centroids'. This algorithm aims to find centroids that minimize the within-cluster sum of squares criterion (i.e., variance). The algorithm has three steps: Compute the difference between the old and the new centroids and repeat the last two steps until this difference is smaller than a threshold. This algorithm requires the number of clusters to be provided. Moreover, it assumes that clusters are convex and isotropic, and responds poorly to irregular clusters. However, for this work it is reasonable to assume that all high-fluence single pulses will be distributed in an approximately convex area near the high-amplitude, low-width top region of Fig. 4. While a higher number of clusters results in finer morphology resolution, it also reduces the number of samples per cluster and, therefore, the number of single pulses available to create an integrated pulse profile. Therefore, we surveyed different numbers of clusters, from k = 2 to 17. Moreover, given the high dimensionality of the problem, it is convenient to run the kmeans algorithm several times with different centroid seeds. Therefore, we ran the algorithm 3 times for each value of k . The final result is the best output of the consecutive runs in terms of intra-cluster inertia.", "pages": [ 7 ] }, { "title": "3.2.2. Mean Shift", "content": "Mean shift clustering (Comaniciu & Meer 2002) is a density-based clustering algorithm that can be summarized as follows: Mean shift is particularly useful for datasets where the clusters have arbitrary shapes and are not well separated by linear boundaries. Unlike the K-means algorithm, it is non-parametric and does not require specifying the number of clusters in advance, since this is determined by the algorithm with respect to the data. However, it is more computationally expensive ( ∼ O ( N 2 )), and it is not guaranteed that the resulting number of clusters is optimal for a given dataset.", "pages": [ 7, 8 ] }, { "title": "3.2.3. DBSCAN", "content": "The Density-Based Spatial Clustering of Applications with Noise (DBSCAN, Ester et al. 1996; Schubert et al. 2017) is an unsupervised clustering method based on a threshold for the number of neighbors, min samples , within the radius eps according to some metric (i.e., euclidean distance). A data point with more than min samples neighbors within this radius is considered a core point, and all those neighbors (called direct density reachable) are considered to be part of the same cluster as the core point. If any of these neighbors is again a core point, their neighborhoods are combined into the same cluster. Non-core points in this set are called border points, and points that are not density reachable from any core point are considered noise. Unlike K-Means, DBSCAN does not require assumptions about the shape and convexity of the clusters, and it can discover clusters of arbitrary shapes and sizes, including noise points (outliers) in the data. However, because eps is fixed for all points, the algorithm struggles when clusters have significantly different densities. Instead of specifying the number of clusters, DBSCAN requires values of min samples and eps to be provided. Besides some heuristic-based approaches (Sander et al. 1998), choosing appropriate values can pose a challenge. The choice depends on each dataset and experimentation with different values is usually needed to achieve the desired clustering results. To obtain results analogous to those obtained using K-Means, we varied eps = 0 . 51 , 0 . 52 , . . . , 1 . 1 using a Euclidean metric, and min samples to cover 1% , 1 . 5% , . . . , 5% of the total number of single pulses; we then retained the first combination of these values that resulted in k = 2 , 3 , . . . , 17 clusters. DBSCAN requires a function to calculate the distance between data points. However, for high-dimensional data, there is little difference in the distances between different pairs of points and the metric can be rendered almost useless due to the so-called 'curse of dimensionality'(e.g., Bellman 2003), making it difficult to find an appropriate value for eps . Finally, DBSCAN visits each point of the database, possibly multiple times. As a result, the worst-case memory complexity of DBSCAN is ∼ O ( N 2 ),", "pages": [ 8 ] }, { "title": "3.2.4. OPTICS", "content": "The Ordering points to Identify the Clustering Structure (OPTICS) algorithm (Ankerst et al. 1999) can be seen as a generalization of DBSCAN that relaxes the eps parameter from a single value to a range bounded by max eps , which is the maximum radius from each point to find other potential reachable points. Each point is then assigned two distances: Clusters can then be extracted by selecting a threshold on the reachability distance, or by different algorithms that try to detect steepness in a reachability plot , where the points are linearly ordered such that spatially closest points become neighbors in the ordering. By using a range of radius values, OPTICS handles clusters with varying densities more effectively than DBSCAN and can identify noise points at multiple levels. However, noise points are not as well-defined as in DBSCAN because the first samples of each area have a large reachability and will thus sometimes be marked as noise. The choice of its two hyperparameters can also pose a challenge, and we resorted to proving the same ranges for max eps and min samples as those used for DBSCAN (see Sec. 3.2.3). OPTICS clustering can be computationally expensive and slow for large datasets. It also requires more memory than DBSCAN, which can be a problem for datasets with limited memory (Schubert & Gertz 2018).", "pages": [ 8 ] }, { "title": "3.3. TOA Uncertainty Calculation", "content": "Osglyph[suppress]lowski et al. (2014) sought to reduce the RMS timing residual by removing the single pulses with the lowest S/N from the integrated pulse profile average. Such a scheme can potentially yield an improved TOA uncertainty for short observations. However, when larger numbers of single pulses are available, we found that the improvement in TOA uncertainty resulting from averaging a sufficiently large number of single pulses outweighs that from averaging only the brightest pulses. Therefore, instead of discarding the lower-S/N data, we aim to gather the information from all the pulse states while weighting the contribution of each pulse according to its fluence state. To this end, we devised the following algorithm to compute a weighted TOA measurement: Since the t i measurements are all independent, the variance of the weighted mean is given by (e.g.: Shahar 2017; Hartung et al. 2011): By using the weighted mean of the TOAs from all clusters, as opposed to removing the lowest S/N pulses like in (Osglyph[suppress]lowski et al. 2014), we ensure that all the available data is utilized in the TOA calculation. Moreover, by weighting each pulse state by the corresponding σ i , as opposed to weighting all the states equally like in traditional timing techniques, the high-fluence pulse states contribute more prominently to the TOA calculation than low-fluence states.", "pages": [ 8, 9 ] }, { "title": "3.4. RFI Mitigation", "content": "Radio frequency interference (RFI) can severely hinder the timing sensitivity of even the most sophisticated radio telescopes. Conventional pulsar timing techniques can reduce the effects of transitory RFI by folding large numbers of single pulses. However, since the present analysis requires clustering the data into smaller subsets of single pulses, the effects of unfiltered RFI will be more prominent in the resulting cluster average pulse profile, therefore greatly affecting the timing precision. As a result, our method is highly susceptible to RFIs. In particular, by visually inspecting pulse profile samples we found that the L-band data was more heavily affected by RFIs compared to the 820-band data, so employing RFI-excision techniques was paramount to this analysis. To such end, we evaluated the merit of five RFI removal algorithms: We find that the general conclusion of improvement via clustering does not depend on the choice of RFI excision algorithm, only the resulting TOA uncertainties overall. The results of this analysis are presented in Sec. 4.1.", "pages": [ 9, 10 ] }, { "title": "4. RESULTS", "content": "Here we present the TOA uncertainties obtained when clustering schemes are applied, we analyze the structure of the corresponding clusters in feature space, and we discuss how the results vary in the presence of different noise levels.", "pages": [ 10 ] }, { "title": "4.1. Impact of RFI Excision Algorithm", "content": "To assess the most efficient RFI-excision recipe and the impact on our clustering method, we implemented different combinations of the previously described RFIexcision algorithms (see Sec. 3.4) on the 820-band data and measured the resulting σ TOA as a function of the number of clusters when using the K-means clustering algorithm. The results are summarized in Fig. 5. We find that the resultant TOA uncertainty varies greatly depending on the RFI-excision algorithm, so we can expect that the efficiency of our timing method will be highly dependent on the RFI content of the observation. However, we see broadly that multiple clusters improve timing precision regardless of the algorithm used. In the next subsections, we opted for the RFI-excision algorithm that provided the lowest σ TOA median; for the 820-band dataset this is a combination of MeerGuard and paz -r , and for the more heavily RFI-affected Lband it is a combination of MeerGuard , clfd , and paz -r .", "pages": [ 10 ] }, { "title": "4.2. TOA Uncertainties", "content": "The TOA uncertainties obtained when clustering the 820-band data using each clustering algorithm are discussed in Fig. 6. In particular: k = 7 clusters resulting in σ TOA = 0 . 06 µ s, which represents a reduction of ∼ 6 ns (relative ratio of ∆ σ TOA /σ (0) TOA = 0 . 087). Sub-optimal results are obtained when the number of clusters is too small ( k ∼ 2). mber of single pulses) and eps =0.77. This is a reduction of ∼ 9 ns (∆ σ TOA /σ (0) TOA = 0 . 136) compared to uncertainty when no clustering is applied.", "pages": [ 10, 11 ] }, { "title": "4.3. Clustering Structure", "content": "Both K-Means and OPTICS provide the most optimal results; however, since K-Means only requires one hyperparameter to be specified, its implementation is significantly more straightforward. Therefore, we used it as a case study to examine its underlying clustering structure. In Fig. 7(a) we present the classified data in a 3-dimensional features space when using k = 7 clusters. In Fig. 7(b) we present some of the single pulses assigned to four of these clusters. We observe that each cluster corresponds to a distinctive single pulse morphology: is thereby comprised of bright, high-S/N single pulses. The last row of Fig. 7(b) shows the integrated pulse profile corresponding to each cluster. We readily observe that single pulses in different clusters average to different shapes, with varying peak heights, widths, and relative amplitudes between the pulse components. This result agrees with Shannon et al. (2014b), where the authors observed a similar behavior when using the single pulses with the highest intrinsic energy, and extends those findings to single pulses in other pulse states. In Fig. 8 we show the TOA and σ TOA resulting from each cluster, as well as the number of single pulses per cluster. We find that clusters 2, 3, 4 present reduced TOA uncertainties due to averaging a large number of mixed-fluence single pulses, which is the conventional procedure in pulsar timing. However, cluster 6 also attains a comparably low TOA uncertainty despite comprising the smallest number of single pulses. Indeed, this cluster represents a high-fluence state, and the reduction in W eff due to only averaging high-S/N single pulses is enough to outweigh the improvement in S that would result from averaging a large number of single pulses, resulting in a smaller σ TOA (see Eq. 4). Conversely, cluster 4 does not benefit from a large number of single pulses or being a high-fluence state, resulting in a large TOA uncertainty. In Fig. 9 we present a pairs plot showing the distribution of the clustered single pulses in single-pulse features space. By looking at the distribution in pulse position, we see that the clustering algorithm successfully identifies pulses that fall on the edges of the main pulse window, possibly because no significant pulse peak can be identified (clusters 4 and 1). We also note a clear separation in amplitude (with cluster 7 skewed to the highest amplitudes and energies) and width (cluster 3 skewed to the highest widths). In analyzing Fig. 9, we noticed a trend for peaks with larger amplitudes to appear thinner and have earlier positions than pulses with lower amplitudes. In particular, the distribution in peak location of the high-fluence cluster 6 has a mode towards earlier phase bins than lowfluence cluster 5; intermediate-fluence clusters 2 and 3 fall in the middle. To better study this behavior, we grouped the single pulses in intervals of increasing amplitude and, for each interval, we calculated the distribution across the position of the main pulse peak. The results are presented in Fig. 10. We find that, on average, a higher amplitude is correlated with a more leftward position of the main pulse and, therefore, an earlier time of arrival of the main pulse component. By calculating the median of the positions of the single pulses 820 Band in each interval (vertical red dotted lines in Fig. 10), we find that the pulses with the highest amplitudes arrive 3 phase bins earlier than the lowest-amplitude ones, which corresponds to a time offset of 94 . 04 µ s.", "pages": [ 11, 12 ] }, { "title": "4.4. L-band and Injected Noise", "content": "Finally, we performed a similar analysis for the single pulses in the L-band dataset. Due to the lower S/N for the observation in this band, the resulting distribution across the 4 single-pulse features is more narrow than for the 820-band data, with a predominance of low-amplitude, low-energy single pulses, which corresponds to a majority of single pulses dominated by noise. As a result, the clustering algorithms resulted in mixed results in determining meaningful clusters. Moreover, a wider exploration of the hyperparameter space was needed to initialize the algorithms correctly. We also noted that the high-fluence clusters are less populated in the L-band data and potentially do not meet the number of single pulses required to obtain a stable integrated pulse shape, resulting in a decreased TOA precision. The most robust results were found using a K-Means algorithm, presented in Fig. 11. When no clustering scheme is used and all single pulses are weighted equally, we obtained a TOA error σ (0) TOA = 0 . 74 µ s. When a K-Means classifier was applied to weigh the data, we obtained TOA errors as small as σ TOA = 0 . 46 µ s, which represents an improvement of 0 . 28 µ s ( δσ TOA = 0 . 37). The clustering algorithms we tested were more successful at correctly identifying meaningful fluence states for the 820-band data than for the L-band data. This behavior can be attributed to the difference in S/N between both frequency bands. Therefore, to quantify the robustness of this method when applied to datasets of varying S/N, we repeated the analysis on observations with injected artificial noise. For every single pulse in the 820-band dataset, we calculated the root-meansquare of the off-pulse intensities, σ OPI , and then added white noise with a standard deviation equal to 0 . 5 σ OPI , 1 . 0 σ OPI , 1 . 5 σ OPI , etc. For each level of injected noise, we classified the resulting data using K-means and calculated the weighted weight-averaged σ TOA using the algorithm described in Sec. 3; the results are presented in Fig. 12. We find that for injected noise levels up to 3 . 0 σ OPI , our method provides an improvement in σ TOA over conventional techniques with no clustering. However, for an injected noise amplitude of 3 . 5 σ OPI and higher, the conventional method outperforms our clustering approach.", "pages": [ 12 ] }, { "title": "5. CONCLUSIONS", "content": "In this work, we proposed that the stochastic variations in single-pulse morphology correspond to shifts between a range of pulse fluence states. Moreover, we created an algorithm for classifying single pulses according to their fluence to weigh their contribution to the TOA measurement for the epoch. We then tested the potential of this method to decrease the uncertainty in the TOA measurement by testing it on observations of PSR J2145 -0750. The algorithm performance depended on the observation, the choice of the clustering algorithm, and the corresponding hyperparameters. For an observation in the 820-MHz frequency band, we found that both K-Means and OPTICS provide similar improvements in the TOA uncertainty calculated ( σ TOA = 0 . 06 µ s and 0 . 057 µ s, respectively) compared to conventional timing techniques with no clustering ( σ TOA = 0 . 066 µ s). However, OPTICS requires a length hyperparameter tunning and is considerably slower and less robust when applied to high-dimensional data (see Sec 3.2.4). As a result, we find that OPTICS can be computationally expensive and difficult to initiate for single pulse datasets. Conversely, K-Means is simple to implement, for it only requires one hyperparameter, and works well on larger datasets. Therefore, OPTICS can 6 Cluster be optimized for more limited datasets, but K-Means can provide more robust and computationally efficient performances for larger datasets consisting of several observations and different pulsars. In this analysis, we used the weighted average TOA and TOA error as a metric to quantify the timing precision of our method. As a consequence of using different templates to calculate a TOA for each fluence cluster, the resulting TOA will be shifted relative to the one obtained from averaging all pulses with no intermediate clustering. Such shifts are only of consequence in terms of TOA accuracy whereas, for this work, we are only concerned about changes in TOA precision. However, since all single pulses belong to the same observations and are referenced to the same phase, we can potentially estimate the relative shift among the different clusters with this weighting. In analyzing the distributions of the single pulse features, we found a tendency for higher-amplitude single pulses to arrive earlier in the pulse window and have a reduced pulse width compared to pulses with lower amplitudes. A correlation between pulse latitude and intensity was first observed in observations of the Vela pulsar (PSR B0833 -45/J0835 -4510) by Krishnamohan & Downs (1983) and then again in PSR B0329+54 by McKinnon & Hankins (1993) using the VLA observatory. The latter interpreted this behavior in terms of Cheng & Ruderman (1980)'s ion outflow model as a change in the emission heights of the individual pulse components. A similar correlation was observed by Shannon & Cordes (2012) in PSR J1713+0747 using the Arecibo Telescope, and then again by Kerr (2015) in PSR J0835 -4510 (Vela) using the Parkes telescope. Lousto et al. (2022) and Zubieta et al. (2023) also found that pulses with larger amplitudes appear earlier than pulses with lower amplitudes in observations of the Vela pulsar at the Argentine Institute of Radioastronomy (Gancio et al. 2020); they used this behavior to support a pulsar model based on emission regions at different altitudes in the neutron star magnetosphere where the pulses of each cluster are emitted. The discovery of the same correlation in independent observations across various pulsars and using different backends suggests that this is not an instrumental artifact or an isolated pulsar behavior, but rather a robust trend. This relative shift in phase could be quantified and incorporated into TOA calculations for improved accuracy; however, each cluster was assigned a pulse template consistent with this shift, so it should be of no consequence to the TOA precision. A better understanding of this phenomenon could be obtained by performing single-pulses analyses in observations of other pulsars. Future work could also explore the evolution of the position of the minor pulse components as a function of the fluence and quantify the heights of the emitting regions. Such a study could provide new insights into sub-pulse structure. Among the limitations of this method, we recall that the dataset analyzed in this work benefited from the pulsar intrinsic high fluence, distinct pulse shape, and brightness magnification due to scintles in this particular observation (see Fig. 1). However, the L-band data presents noticeably higher levels of noise in the single pulses and a more narrow distribution across pulse features such as amplitude and width. Consequently, the clustering algorithms' ability to detect meaningful clusters was severely hindered when applied to this dataset. In that case, the σ TOA was reduced from 0 . 74 µ s when no intermediate clustering was applied, to 0 . 46 µ s. This behavior is also observed when different noise levels were artificially introduced to the 820-band dataset. In particular, we found that when the root-mean-square of the off-pulse noise in the 820-band dataset was amplified by a factor of 3.5 from its original value, clustering the data did not significantly improve the TOA uncertainty. We also found that the ability of the clustering methods to classify the single pulses into distinct fluence states is highly dependent on the RFI content of the observation. Indeed, unfiltered RFI can significantly change the single-pulse morphology, making low-S/N single pulses mistakenly identified as high-fluence due to the high amplitudes of the interference and, thereby, hindering classification efficiency. For this observation of J2145 -0750, we found that a combination MeerGuard and paz provides the most adequate RFI filtering for the 820-band. However, the choice of RFI cleaning recipe will depend on the observing backend, frequency band, and RFI environment. As a result, we expect that this method is better suited for analysis involving high-S/N observations and bright pulsars, and only after careful RFI excision and sensible clustering hyperparameter tuning. Another shortcoming is the large memory space needed to store single pulse datasets. A possible alternative to post-processing clustering is to develop a real-time clustering system. This would involve finding the optimal clustering and RFI-filtering scheme for a given pulsar ahead of the observing run. During the data acquisition process, subsets of the observed single pulses can be RFI-excised, classified in real-time into pre-established clusters, and then averaged into the corresponding cluster pulse profile. Once the observation is complete, the different cluster pulse averages can be timed separately and the TOAs from each cluster can be weighted average to obtain a single TOA. In doing so, we eliminate the need to store individual single pulse data while also preserving the underlying fluence structure. This study provides a proof-of-principle technique to be applied more extensively to observations of other pulsars. Potential applications include real-time clustering of single pulses in observations of new bright pulsars to improve the precision of the TOA estimations. With increasingly sensitive telescopes, this method could be applied to long-term monitoring of individual pulsars to improve the TOA precision at each epoch and gain sensitivity in various tests of fundamental physics requiring precision timing experiments. Acknowledgments and author contributions : S.V.S.F. undertook the analysis, developed the code pipeline, and prepared the figures, tables, and the majority of the text. M.T.L. developed the mathematical framework for this work, selected the analyzed data set, assisted with the preparation of the manuscript, wrote the appendix, provided advice interpreting the results, and supervised the project development. M.A.M. and M.T.L. designed the observational setup and undertook the data acquisition at GBO. 1 2 3 4 5 6 7 8 9 10 We thank David Nice, Matthew Kerr, Yogesh Maan, and NANOGrav's Noise Budget and Timing Working Groups for their valuable input and feedback on the manuscript. We also thank Capella and Rosie for their canine support. 11 12 13 14 15 SVSF is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 2139292. We acknowledge support from the NSF Physics Frontiers Center award number 2020265, which supports the NANOGrav project. The Green Bank Observatory and National Radio Astronomy Observatory are facilities of the NSF operated under a cooperative agreement by Associated Universities, Inc. MTL also acknowledges support from NSF AAG award number 2009468. S.V.S.F. acknowledges partial support from the NASA New York Space Grant, and from the Out to Innovate Career Development Fellowship for Trans and Non-binary People in STEM 2023. 16 17 18 19 20 21 22 23 24 25 26 27 28 Facilities: Green Bank Observatory (GBO). Software: PyPulse (Lam 2017), PINT (Luo et al. 2021), clfd (Morello et al. 2019), MeerGuard (Lazarus et al. 2020), LMFIT (Newville et al. 2014), Astropy (As- tropy Collaboration et al. 2013, 2018, 2022), dspsr (van", "pages": [ 13, 14, 15, 16 ] }, { "title": "A. TIMING PRECISION FORMALISM FOR TWO STATES", "content": "Here we consider the impact of two fluence states, one low and one high fluence state, in timing precision. We start with the RMS timing error, calculated from the template and the S/N as (Downs & Reichley 1983) where ρ ( τ = t -t ' ) ≡ ⟨ n ( t ) n ( t + τ ) ⟩ /σ 2 n is the autocovariance function of the noise. Following Eq. 1, again U ( t ) is the normalized (to unit height) pulse template shape, n ( t ) is additive noise with RMS amplitude σ n , and the S/N of the pulse profile is S . When the noise is uncorrelated and the pulse profile is sampled at intervals ∆ t much shorter than the pulse width, we have ρ ( t ) = δ ( τ )∆ t , which means the TOA error reduces to where we have converted to a form with N ϕ = P/ ∆ t phase bins sampled across the pulse period P . As in Lam et al. (2016), we define an effective width W eff such that σ S / N = W eff /SN 1 / 2 ϕ , which implies: We now consider the case where the pulsar emits in both high- and low-energy states (labeled h and l , respectively), which we will denote with template shapes U h ( t ) and U l ( t ). The template shape of all pulses, U a ( t ), will be weighted by the respective single pulse intensities of the two states, I h and I l , as well as the number of pulses contributing to the total shape, N h and N l . Quantitatively, The all-pulse average shape can then be calculated by substituting Eq. A4 into the left-hand side of Eq. A3 to obtain We can expand this expression, writing in W 2 eff ,a instead of W eff ,a for clarity, and replace ∫ dt [ U ' i ( t )] 2 = P/W 2 eff ,i using Eq. A3 to obtain Therefore, assuming that observationally N h ≪ N l (and there are not extreme differences between I h and I l so that N h I h /N l I l ≪ 1), we can approximate the above to find As a result, we obtain that the effective width of the all-pulse template is dominated by the low-fluence-state pulses. Since the S/N of the individual states are S i = I i N 1 / 2 i /σ 0 (by construction/proportionality), assuming that the off-pulse noise at the single pulse level is σ 0 and is the same for both states, then following Eqs. 2 and 4, the TOA errors for the individual states i = h, l are Assuming the peak intensities for both states are roughly aligned in phase, then the single-pulse intensity should be the weighted mean of the individual intensities, Then, we can use Eq. A9 but with the index i = a . In the case where N h ≪ N l , I a ≈ I l , and we can use Eq. A6 to replace the effective width W eff ,a of the all-pulse template with W eff ,l from the low-fluence-state template to find We see that as expected, using only the low-fluence-state pulses should yield slightly worse timing than using all pulses because, all other factors being equal, the only difference is the number of pulses averaged over given the approximations above. If the effective width of the high-fluence-state template is sufficiently narrow, or the single-pulse intensities sufficiently large, then if we shoud be able to improve upon template-fitting errors (i.e., if σ S / N ,h < σ S / N ,a or σ S / N ,l ).", "pages": [ 16, 17 ] }, { "title": "REFERENCES", "content": "Agazie G., et al., 2023, ApJL, 951, L8 Cordes J. M., Downs G. S., 1985, ApJS, 59, 343 Aggarwal K., et al., 2020, Journal of Open Source Software, 5, 2750 Alam M. F., et al., 2021, ApJS, 252, 4 Ankerst M., Breunig M. M., Kriegel H.-P., Sander J., 1999, ACM Sigmod record, 28, 49 Antoniadis J., et al., 2013, Science, 340, 448 Arzoumanian Z., et al., 2016, ApJ, 821, 13 Arzoumanian Z., et al., 2020, ApJL, 905, L34 Astropy Collaboration et al., 2013, A&A, 558, A33 Astropy Collaboration et al., 2018, AJ, 156, 123 Astropy Collaboration et al., 2022, ApJ, 935, 167 Backer D. C., 1975, ApJ, 182, 245 Bailes M., et al., 1994, ApJL, 425, L41 Bellman R., 2003, Dynamic Programming. Dover Books on Computer Science Series, Dover Publications, https://books.google.com/books?id=fyVtp3EMxasC Cheng A. F., Ruderman M. A., 1980, ApJ, 235, 576 Comaniciu D., Meer P., 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24, 603 Cordes J. M., Shannon R. M., 2010, arXiv e-prints, p. arXiv:1010.3785 Cordes J. M., Wolszczan A., Dewey R. J., Blaskiewicz M., Stinebring D. R., 1990, ApJ, 349, 245 Craft H. D., 1970, PhD thesis, Cornell University Cromartie H. T., et al., 2020, Nature Astronomy, 4, 72 Demorest P. B., et al., 2012, The Astrophysical Journal, 762, 94 Demorest P. B., et al., 2013, ApJ, 762, 94 Detweiler S., 1979, ApJ, 234, 1100 Dolch T., et al., 2014, The Astrophysical Journal, 794, 21 Downs G. S., Reichley P. E., 1983, ApJS, 53, 169 Edwards R. T., Stappers B. W., 2003, A&A, 407, 273 Ester M., Kriegel H.-P., Sander J., Xu X., 1996, in Knowledge Discovery and Data Mining. https://api.semanticscholar.org/CorpusID:355163 Fonseca E., et al., 2016, ApJ, 832, 167 Fonseca E., et al., 2021, ApJL, 915, L12 Gancio G., et al., 2020, A&A, 633, A84", "pages": [ 17 ] } ]

[ { "title": "C 0 -inextendibility of FLRW spacetimes within a subclass of axisymmetric spacetimes", "content": "Melanie Graf ∗ and Marco van den Beld-Serrano †", "pages": [ 1 ] }, { "title": "Abstract", "content": "Starting from the proof of the C 0 -inextendibility of Schwarzschild by Sbierski, the past decade has seen renewed interest in showing low-regularity inextendibility for known spacetime models. Specifically, a lot of attention has been paid to FLRW spacetimes and there is an ever growing array of results in the literature. Apart from hoping to provide a concise summary of the state of the art we present an extension of work by Galloway and Ling on C 0 -inextendibility of certain FLRW spacetimes within a subclass of spherically symmetric spacetimes, [2], to C 0 -inextendibility within a subclass of axisymmetric spacetimes. Notably our result works in the case of flat FLRW spacetimes with a ( t ) → 0 for t → 0 + , a setting where other known C 0 -inextendibility results for FLRW spacetimes due to Sbierski, [16], do not apply. MSC2020: 53C50, 53B30, 83C99", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "(In-)extendibility of spacetimes is a recurrent topic well known to play a central role in General Relativity both for specific models and for quite general classes. The usual procedure is to follow one (or both) of the following paths: either an explicit extension of the spacetime is found/constructed or it is proven that there is some general obstruction to extendibility within a certain class of extensions. For C 2 -inextendibility blow up of some curvature scalar (e.g., the scalar curvature or the Kretschmann scalar) gives such an extendibility obstruction because these scalars are invariant under diffeomorphisms. In other words, their blow up is a geometric property (i.e. coordinate independent) of the spacetime. Of course, the blow up of a curvature scalar only tells us that the manifold is C 2 -inextendible (as the curvature scalars contain second order derivatives of the metric). Hence, different strategies are required in order to explore the inextendibility of a spacetime in a lower regularity class (e.g. C 0 - or C 0 , 1 -regularity). Here a lot of new tools and techniques have been developed in the last six years. In particular, the question of C 0 -inextendibility was first tackled by Sbierski [13], who proved that the Minkowski and the maximally extended Schwarzschild spacetime are C 0 -inextendible. We now have a collection of low regularity inextendibility criteria the most stringent (though not always most useful in practice as one is often interested in inextendibility precisely in the cases where spacetimes do contain incomplete geodesics) of them being timelike geodesic completeness: in the first place, in [13] it was proven that if no timelike curve intersects the boundary of the extension ∂ι ( M ), then the spacetime is inextendible. This result already pointed to the idea that, under certain additional assumptions, timelike (geodesic) completeness would yield the inextendibility of a spacetime (in a low regularity class). Indeed, in [4] it was proven that a smooth globally hyperbolic and timelike geodesically complete spacetime is C 0 -inextendible. Moreover, in [5], it was shown that if the global hyperbolicity condition is dropped the spacetime still is C 0 , 1 -inextendible. Finally, in [10], using the Lorentz-Finsler framework, it was shown that a smooth timelike geodesically complete spacetime is C 0 -inextendible and in [6] an inextendibility result for timelike complete Lorentzian length spaces is shown. Beyond these general structural results special attention has been paid to FLRW spacetimes. FLRW spacetimes generally and especially flat or Euclidean FLRW spacetimes are of great interest in Cosmology in order to model the universe. For this reason one is particularly interested in (in-)extendibility results across a possible Big Bang. Part of the purpose of this note is to given an overview of the different concrete cases treated and the precise results available depending on the behaviour of the scale factor as one approaches the Big Bang or future infinity 1 and whether we are looking at spherical, Euclidean or hyperbolic FLRW spacetimes. This general review will be the focus of Section 2, see also Table 1 for a quick summary. The most open of the three cases is that of Euclidean FLRW spacetimes. While there is a C 0 , 1 loc -inextendibility result based on local holonomy, see [14], that works for flat FLRW spacetimes with a Big Bang at finite time and particle horizon the question of C 0 -(in-)extendibility is still very open. Galloway and Ling in [2] used an approach based on uniqueness of certain coordinate changes in flat and hyperbolic FLRW spacetimes to prove that such spacetimes are C 0 -inextendible within a certain subclass of spherically symmetric spacetimes. We present this result in Section 3. Apart from reviewing available literature the aim of this paper is to generalize their result to C 0 -inextendibility within a larger subclass of axisymmetric spacetimes which enjoy similar uniqueness properties for coordinates as the strongly spherically symmetric extensions considered in [2]. We show (cf. Corollary 27) Theorem 1. Let ( M,g ) be a (4-dimensional) flat future inextendible FLRW spacetime with scale factor a satisfying that a ' (0) ∈ (0 , ∞ ] . Then, there is no past natural strongly axisymmetric C 0 -extension of ( M,g ) compatible with the strongly spherically symmetric coordinates of [2]. referring to Section 4 for the definitions used here and to Corollary 27 for the precise statement (see also Theorem 14). We further mention that just as in [2] a similar result is available for hyperbolic FLRW spacetimes (cf. Corollary 28).", "pages": [ 1, 2, 3 ] }, { "title": "1.1 Notations and conventions", "content": "We conclude the introduction with fixing our notations and conventions. The term C k spacetime for us always denotes a connected time-oriented Lorentzian manifold ( M,g ), where M is a smooth manifold but the metric g is merely C k -regular. Furthermore, timelike curves are piecewise smooth curves whose right and left handed sided derivatives lie in the same connected component of the lightcone, which coincides with the convention in [2]. The following basic concepts play an important role in our study. Definition 2 ( C l -extension) . Fix k ≥ 0 and let 0 ≤ l ≤ k . Let ( M,g ) be a C k spacetime with dimension d . A C l -extension of ( M,g ) is a proper isometric embedding ι where ( M ext , g ext ) is C l spacetime of dimension d . If such an embedding exists, then ( M,g ) is said to be C l - extendible . The topological boundary of M within M ext is ∂ι ( M ) ⊂ M ext . By a slight abuse of notation we will sometimes also call ( M ext , g ext ) the extension of ( M,g ), dropping the embedding ι . Definition 3 (Future and past boundary) . Let ι : ( M,g ) ↪ → ( M ext , g ext ) be a C l -extension. We define the future boundary ∂ + ι ( M ) and past boundary ∂ -ι ( M ): /negationslash /negationslash where 'f.d.t.l. curve' stands for future directed timelike curve. If ∂ -ι ( M ) = ∅ (or ∂ + ι ( M ) = ∅ ), then ι is called a past C l -extension (resp. a future C l -extension )) and if no such extension exists, ( M,g ) is said to be past C l -inextendible (resp. future C l - inextendible ). Two of the earliest results about C 0 -extensions tell us that, while not every point point in ∂ι ( M ) has to belong to one of these boundary components (and neither do they have to be disjoint), it suffices to study the future and past boundary of a spacetime in order to prove its C 0 -inextendibility: Lemma 4 (Lemma 2.17 in [13], Proposition 2.3 in [2]) . Let ι : ( M,g ) → ( M ext , g ext ) be a C 0 -extension. Then ∂ + ι ( M ) ∪ ∂ -ι ( M ) = ∅ . /negationslash Further emptiness of one of these implies a rather nice structure for the other: Theorem 5 (Theorem 2.6 in [2]) . Let ι : ( M,g ) → ( M ext , g ext ) be a C 0 -extension. If ∂ + ι ( M ) = ∅ , then ∂ -ι ( M ) is an achronal topological hypersurface. As pointed out in the introduction we will be focusing on FLRW spacetimes, that is isotropic 2 cosmological spacetimes which are necessarily of the following form: Definition 6 (FLRW spacetimes) . A ( d +1)-dimensional FLRW spacetimes is a warped product of an open interval I ⊂ R with a complete and simply connected d -dimensional Riemannian manifold with constant scalar curvature K . Depending on the value of K , the FLRW spacetimes are divided in three subgroups: where the metrics g are written in FLRW coordinates ( t, r, ω ) ∈ I × (0 , ∞ ) × S d -1 and the scale factor a : I → (0 , ∞ ) is a smooth function. /negationslash If in addition it holds that lim t → t + inf a ( t ) = 0, where t inf := inf( I ), then we call ( M,g ) a FLRW spacetime with a Big Bang as t → t + inf and if t inf > -∞ we say the spacetime has a Big Bang at finite time. FLRW spacetimes with lim t → t + inf a ( t ) = 0 will be accordingly referred to as FLRW spacetimes without a Big Bang . If t inf = -∞ we say the spacetime is past eternal (with or without a Big Bang) and we note that the case of a Big Bang at finite time is qualitatively different from a past eternal FLRW spacetime having a Big Bang. Unless stated otherwise, we will always assume that I = (0 , ∞ ) and that there is a Big Bang as t → 0 + . More general intervals I ⊂ R will only be considered briefly in Section 2, when discussing some past eternal FLRW spacetimes. We can further classify FLRW spacetimes with a Big Bang into FLRW spacetimes with particle horizon and FLRW spacetimes without particle horizon depending on the integrability of 1 a ( t ) as t → 0 + . Definition 7 (Particle horizon) . Let ( M,g ) be an FLRW spacetime. It is said to have a particle horizon provided ∫ 1 0 1 a ( t ' ) dt ' < ∞ . Otherwise, it has no particle horizon . In the next section we will present some criteria which guarantee the ( C 0 - or C 0 , 1 loc -)inextendibility of FLRW spacetimes. These criteria mostly depend on the considered type of FLRW spacetime (flat, spherical or hyperbolic) and on properties related to the asymptotic behaviour of the scale factor. Furthermore, some examples of relevant C 0 -extendible FLRW spacetimes will be discussed. As having a particle horizon or not is related to the rate at which a ( t ) approaches zero it is perhaps not surprising that this will play a role in the inextendibility results we are considering. However, as the discussion and in particular Table 1 will show, this relationship does not appear to be straightforward in the sense that having or not having a particle horizon always leads to stronger inextendibility results. Acknowledgements This article originally started from work on MvdBS' Masters thesis written at the University of Tubingen. We would like to thank Carla Cederbaum for her support and bringing this collaboration together. We would further like to thank Eric Ling for bringing some of these problems to our attention and stimulating discussions. MG acknowledges the support of the German Research Foundation through the SPP2026 'Geometry at Infinity' and the Cluster of Excellence EXC 2121 'Quantum Universe', the University of Tubingen and the University of Potsdam. MvdBS thanks Carla Cederbaum for her financial support during the development of this research project, the Studienstiftung des deutschen Volkes for granting him a scholarship during his Master studies and for his PhD project, and Felix Finster for his support.", "pages": [ 3, 4 ] }, { "title": "2 Overview of the low-regularity inextendibility results for FLRW spacetimes", "content": "The aim of this section is to briefly review the main inextendibility results that have been proven for the different types of ( d +1)-dimensional FLRW spacetimes. In Table 1 the discussed low regularity inextendibility results are collected. We note that we will largely discuss future inextendibility and past inextendibility separately, focusing on past inextendibility. Of course time dual versions of all results hold as well, however future inextendibility results will always be for I = ( c, ∞ ), with c ∈ R , and mostly depend on the behaviour of a ( t ) as t →∞ so that their time duals will not be applicable to Big Bangs at finite time (and vice versa).", "pages": [ 5 ] }, { "title": "2.1 Future inextendibility", "content": "Before moving on to past inextendibility we start by stating a future C 0 -inextendibility result, the proof of which is essentially (though in the case of (i) somewhat anachronistically as the original proof preceded the general result about timelike geodesic completeness implying C 0 -inextendibility from [10]) based on showing that the given conditions on the scale factor already guarantee future timelike completeness. The importance for us will be that future inextendibility guarantees that the past boundary satisfies some nice dimensional and causal properties via Theorem 5. Theorem 8 (Theorem 3.2 in [2] and Theorem 3.5 in [14]) . Let ( M,g ) be a ( d + 1 )-dimensional FLRW spacetime (with d ≥ 1 ) which satisfies one of the following conditions: Then, ( M,g ) is future C 0 -inextendible and given any past C 0 -extension ι : M → M ext , the past boundary ∂ -ι ( M ) is an achronal topological hypersurface. Note that (as pointed out in [14]) the condition that ∫ ∞ 1 a ( t ) √ a ( t ) 2 +1 dt = ∞ is in particular satisfied if lim t →∞ a ( t ) = 0. /negationslash", "pages": [ 5 ] }, { "title": "2.2 Past inextendibility", "content": "While there exist quite general future C 0 -inextendibility results for the FLRW spacetimes, most past C 0 -inextendibility results depend on the particular type of FLRW spacetime we are considering (i.e. spherical, hyperbolic or flat), on the existence of particle horizons or on the symmetry properties of the considered extensions. Before presenting some of the strongest past C 0 -inextendibility results for each type of FLRW spacetimes, we state the following past C 0 , 1 loc -inextendibility result by Sbierski that applies to any FLRW spacetime with particle horizon. Theorem 9 (Theorem 3.7 in [14]) . Let ( M,g ) be a ( d +1 )-dimensional FLRW spacetime ( d ≥ 1 ) with particle horizon. Then ( M,g ) is past C 0 , 1 loc -inextendible. The proof is based on estimates involving blow-up of some local holonomy. More precisely Sbierksi shows that there exists a specific sequence of loops which approach the past boundary such that the parallel transport map along them is unbounded. This contradicts Lipschitz extendibility because, provided the metric components and its first order derivatives are bounded, the parallel transport map along curves in a bounded domain with a uniformly bounded tangent vector is uniformly bounded (which Sbierski proves using a Gronwall's inequality type of argument, cf. [14, Lemma 2.19]). With respect to the past C 0 -inextendibility criteria for general FLRW spacetimes, note that we will focus on the case that d ≥ 2. This is due to the fact that for d = 1 any FLRW spacetime is past C 0 -extendible (initially proven in Section 3.2 in [2], see also Section 6 in [15] or Section 1.2 in [16]). The main C 0 -inextendibility result available for spherical and hyperbolic FLRW spacetimes (without particle horizon) is the following. Theorem 10 (Theorem 1.5 and 1.6 in [16]) . Let ( M,g ) be a ( d +1 )-dimensional spherical FLRW spacetime (with d ≥ 2 ) without particle horizon. Assume one of the following conditions holds: Then, ( M,g ) is past C 0 -inextendible. The proof of (i) uses ideas from [13] and causality theoretic arguments while (ii) makes use of characterizing TIFs 3 in these spacetimes and finding a parametrization for the past boundary and relating TIFs to points on the past boundary. We remark that in condition (ii) of the previous theorem, that a ( t ) e ∫ 1 t 1 a ( t ' ) dt ' →∞ as t → 0 + plays a very important role: if a ( t ) e ∫ 1 t 1 a ( t ' ) dt ' → (0 , ∞ ) as t → 0 + , lim t → 0 + a ' ( t ) = 1 and a ( t ) has a sublinear growth (i.e. ( M,g ) is a so called Milnelike spacetime ), then the hyperbolic FLRW spacetimes do admit a C 0 -extension (cf. the next subsection). The case of flat FLRW spacetimes is more open: While there is a recent paper by Ling, [9], based on Sbierski's work in the spherical and hyperbolic setting that does consider the flat case, he looks at past eternal flat FLRW spacetimes with a Big Bang as t →-∞ as opposed to a Big Bang at finite time. Theorem 11 (Theorem 1.1 in [9]) . Let (M, g) be a flat simply connected FLRW spacetime M = R × R 3 with d ≥ 2 . Moreover, suppose the scale factor satisfies that Then ( M,g ) is past C 0 -inextendible. As already pointed out by Ling these two conditions on the scale factor imply that these spacetimes have no particle horizon. Similarly to the hyperbolic setting condition (ii) appears to be sharp: If the scale factor satisfies (i) and the spacetime has no particle horizon but lim t →-∞ a ( t ) ∫ 1 t 1 a ( t ' ) dt ' ∈ (0 , ∞ ) then it does admit a C 0 -extension (cf. [3] and the next subsection). Returning to our discussion of results on FLRW spacetimes with a Big Bang at finite time there unfortunately does at this point not exist a general C 0 -inextendibility result ('general' in the sense of only imposing conditions on the scale factor) for flat FLRW spacetimes. In the next section we will present a C 0 -inextendibility result by Ling and Galloway [2] that restricts the symmetry class of the metric of the extended spacetime. In particular, they prove the following result: Corollary 12 (Corollary 4.2 in [2]) . Let ( M,g ) be a flat FLRW spacetime with a ' (0) ∈ (0 , ∞ ] and which is future inextendible (for instance because it is future timelike geodesically complete or because it satisfies condition (i) in Theorem 8). Then there exists no natural strongly spherically symmetric C 0 -extension of ( M,g ) . Note that we have adapted the statement slightly from its original in [2], where they demand condition (i) in Theorem 8, to directly focus on the relevant consequence of this condition in the proof, namely a way to establish emptiness of the future boundary, which we now know to follow e.g. from future timelike geodesic completeness. We will introduce the relevant concepts and discuss this result more in depth in the next section. Note that in [2] an analogous result was also proven for hyperbolic FLRW spacetimes with the additional assumption that lim t → 0 + a ' ( t ) = 1: /negationslash /negationslash Corollary 13 (Corollary 4.4 in [2]) . Let ( M,g ) be a future inextendible hyperbolic FLRW spacetime with a ' (0) ∈ [0 , ∞ ] and a ' (0) = 1 . Then there exists no natural strongly spherically symmetric C 0 -extension of ( M,g ) . Building on the results by Galloway and Ling, in Section 4 we will prove that if flat and hyperbolic FLRW spacetimes satisfying some conditions on the scale factor a do admit a C 0 -extension, then the extended spacetime cannot belong to a certain subclass of axisymmetric spacetimes (for the proof see Corollaries 27 and 28): Theorem 14. Let ( M,g ) be a future inextendible (4-dimensional) FLRW spacetime. Assume one of the following conditions holds: /negationslash Then, there is no natural strongly axisymmetric C 0 -extension of ( M,g ) compatible with the strongly spherically symmetric coordinates appearing in [2].", "pages": [ 5, 6, 7 ] }, { "title": "2.3 Some important examples of C 0 -extendible FLRW spacetimes", "content": "Before closing this section, we present some of the main examples for C 0 -extendible hyperbolic and flat FLRW spacetimes: The common ground of these examples is that their scale factor behaves to some degree similarly to the behaviour of the scale factor when expressing certain proper open subsets of Minkowski or de Sitter spacetime in FLRW form. We want to reiterate that Minkowski and de Sitter (as well as anti de Sitter) spacetime themselves are all C 0 -inextendible: While the first proofs of these facts employed slightly different methods (and can be found in [13, Theorem 3.1] for Minkowski, [13, Remark/Outlook 1.3 i)] for de Sitter and [2, Theorem 2.12] for anti de Sitter), this now anachronistically follows immediately from them being timelike geodesically complete thanks to [10]. Our main examples for C 0 -extendible hyperbolic and flat FLRW spacetimes are:", "pages": [ 7 ] }, { "title": "· Milne (and Milne-like) spacetimes :", "content": "The Milne-like spacetimes (cf. [2, Definition 3.3]) are hyperbolic FLRW spacetimes M = (0 , ∞ ) × R d without particle horizon and with a scale factor that additionally satisfies that: Milne-like spacetimes always admit C 0 -extensions ([2, Theorem 3.4]) but depending on the exact form of the scale factor near t = 0 not necessarily C 2 or C ∞ extensions. Note that the 'classical' Milne spacetime (a hyperbolic FLRW spacetime M = (0 , ∞ ) × R d with a ( t ) = t which is well-known to be isometric to I + (0) ⊆ ( R d +1 , η ) and hence even smoothly extendible) is an example of a Milne-like spacetime (for the physical motivation and properties of the Milne-like spacetimes, see [8]). In particular the hyperbolic FLRW spacetime M = (0 , ∞ ) × H 3 with scale factor a ( t ) = sinh 2 ( t ) - and thus a Big Bang as t → 0 + - isometrically corresponds to a proper open patch of de Sitter spacetime (covering a quarter of full de Sitter) and as such is smoothly extendible. An explicit C 0 -extension where the extended manifold is conformal to the Minkowski spacetime was constructed in [8, Section 3.2].", "pages": [ 8, 9 ] }, { "title": "· Flatly sliced patches of de Sitter (and flat past asymptotically de Sitter) spacetimes :", "content": "The 4-dimensional flat FLRW spacetime M = R × R 3 with scale factor a ( t ) := e t is also isometric to a proper open patch of de Sitter spacetime (covering half of full de Sitter). The scale factor exhibits a Big Bang as t →-∞ and increases exponentially with time. Clearly this is smoothly extendible to full de Sitter as well. It is also C 0 -extendible (see [3, Section 3.2]) and similar to the Milne case the procedure for finding a C 0 -extension generalizes to a family of flat past asymptotically de-Sitter spacetimes (following the terminology of [3, Definition 2.5] these are 4-dimensional flat FLRW spacetimes with a Big Bang as t → -∞ where the Hubble parameter a ' a behaves similar enough to that of the flat slicing of de Sitter as t → -∞ ), cf. [3, Corollary 3.6]. They further give an explicit example of a scale factor satisfying a sufficient condition for C 0 -extendibility but where the spacetime is no longer C 2 extendible (cf. [3, Example (3.24)].", "pages": [ 9 ] }, { "title": "3 Summary of the proof of strongly spherically symmetric inextendibility in [2]", "content": "In the rest of this paper only FLRW spacetimes with a Big Bang as t → 0 + will be considered (and we will omit explicit reference to this initial singularity). Definition 15 (Strongly spherically symmetric spacetimes, cf. [2]) . Let ( M,g ) be a spacetime of dimension d +1. It is called a strongly spherically symmetric spacetime if for all p ∈ M there exists a neighbourhood V of p in M and (potentially) a curve L in V such that on U := V \\ L there exists a diffeomorphism ψ s = ( T, R, ω ) : U → ψ s ( U ) ⊂ R × (0 , ∞ ) × S 2 with T : U → R , R : U → (0 , ∞ ) and ω : U → S d -1 such that the pushforward of the metric takes the form: We call ( T, R, ω ) strongly spherically symmetric coordinates . Remark 16 . Note that we are allowing R to be the timelike coordinate. For why we call this strongly spherically symmetric and a discussion relating it back to coordinate invariant descriptions of spherical symmetry we refer to the brief discussion in the beginning of Section 4 in [2]. The main point is that we are excluding potential dTdR cross terms despite being in a regularity where the usual trick to do cannot be applied anymore. Our main reason for allowing the exclusion of a curve L is that we will want to be explicitly restrict the range of our R coordinate to (0 , ∞ ) for technical reasons. It also allows us to start with the FLRW metrics in the coordinates from Definition 6 where we technically already had to exclude an origin from our spherical coordinates on the spatial slice. A comparison of the metric of an FLRW spacetime in FLRW coordinates and (the pushforward of) the metric of a strongly spherically symmetric spacetime shows important similarities. Both locally correspond to a warped product metric on R × (0 , ∞ ) and a round d -1 dimensional unit sphere, either with warping factor a ( t ) r (if ( M,g ) is an FLRW spacetime and an origin for the spherical coordinates on R d has been chosen) or R (if ( M,g ) is a strongly spherically symmetric spacetime). Therefore, if a coordinate change from one to the other exists, the most natural one would be of the form T = T ( t, r ), R = R ( t, r ) and leave the angular part (i.e. the coordinates on the d -1 dimensional sphere) unchanged. It was observed in [2] that such a coordinate change exists on M \\ { r 2 a ' ( t ) 2 = 1 } and is unique (up to an initial condition): Theorem 17 (Theorem 4.1 in [2]) . Let ( M,g ) be a flat FLRW spacetime in FLRW coordinates ( t, r, ω ) where the scale factor a ( t ) satisfies a ' (0) := lim t → 0 + a ' ( t ) ∈ (0 , ∞ ] . Then: where F and G are regular (away from { r 2 a ' ( t ) 2 = 1 } where the Jacobian determinant J = ∂ ( T,R ) ∂ ( t,r ) vanishes). Remark 18 . For the full proof of the previous theorem, see [2]. However, let us briefly discuss some consequences of this Theorem and parts of its proof which will also play an important role in the next section. /negationslash that is g s degenerates as one approaches the past boundary: Note that as lim t → 0 + a ( t ) = 0 and lim t → 0 + R ( γ ( t )) = ∞ , it follows that /negationslash Hence, as the curve γ approaches the past boundary it holds that because of the assumption that lim t → 0 + a ' ( t ) = 0. /negationslash The importance of the previous theorem will become clearer by defining the following concepts and the follow-up corollary. Definition 19. Let ( M,g ) be a flat FLRW spacetime. Let ι : ( M,g ) → ( M ext , g ext ) be a C 0 -extension. We say that ι is a natural strongly spherical C 0 -extension provided that the following holds: where ψ s are the coordinates from Theorem 17 and ˜ M ⊆ M is the subset of M where ψ s is a local diffeomorphism (i.e. ˜ M is given by (6)). We may sometimes refer to ψ s as the natural strongly spherical change of coordinates . Note that the previous expression implies that on ψ s ( ι -1 ( U ) ∩ ˜ M ) for g s from (5). According to Theorem 17 the considered class of FLRW metrics in strongly spherically symmetric coordinates becomes degenerate (as G → 0) when approaching the past boundary. This property can be used to prove that, in fact, such a FLRW spacetime has no natural strongly spherical extension. Corollary 20 (Corollary 4.2 in [2]) . Let ( M,g ) be a future inextendible flat FLRW spacetime with a ' (0) ∈ (0 , ∞ ] . Then there exists no natural strongly spherically symmetric C 0 -extension of ( M,g ) . Proof. The proof is by contradiction. Let ( M,g ) be a flat FLRW spacetime satisfying that a ' (0) ∈ (0 , ∞ ]. Assume there exists a natural strongly spherical C 0 -extension ι : ( M,g ) → ( M ext , g ext ) and let p ∈ ∂ -ι ( M ). By definition there exists a neighbourhood V of p in M ext , a curve L in V and a diffeomorphism ψ ext : U := V \\ L ⊂ M ext → ψ ext ( U ) ⊂ R × (0 , ∞ ) × S d -1 such that, in U , g ext ,s := ( ψ ext ) ∗ g ext can be written in the strongly spherically symmetric form Equations (11) and (12) imply that ( T ext · ι, R ext · ι, ω ext · ι ) have to agree with ( T, R, ω ) in ι -1 ( U ) ∩ ˜ M and that Without loss of generality we may assume that p ∈ U : If p were not itself in U we could replace it by a point q ∈ U ∩ ∂ -ι ( M ). Note that such a q must exist for dimensional reasons since ∂ -ι ( M ) is a topological hypersurface, cf. Theorem 8, and U = V \\ L where V is an open neighbourhood of p ∈ ∂ -ι ( M ) and L is at most a curve. /negationslash Let γ : [0 , 1] → U ⊆ M ext be a future directed timelike curve with γ ((0 , 1]) ⊂ ι ( M ) and γ (0) = p (this curve exists by definition of ∂ -ι ( M )) parametrized by the time function t . In particular, lim s → 0 R ( γ ( s )) = 0 as R ext ∈ (0 , ∞ ) (by definition of the strongly spherical coordinates) and R and R ext agree in ι ( M ) ∩ U (so lim s → 0 R ( γ ( s )) = lim s → 0 R ext ( γ ( s )) = R ext ( p )). Thus, by the second part of Theorem 17 we have lim t → 0 + G ( ι -1 ( γ ( t ))) = 0. Further, γ ( t ) is contained in ι ( ˜ M ) for all t > 0 close enough to 0 by point 4 of Remark 18. Hence, also G ext ( p ) = lim t → 0 + G ext ( γ ( t )) = 0. Therefore, g ext ,s is degenerate at p , contradicting g ext ,s being the extended metric in strongly spherically symmetric coordinates near p . So no natural strongly spherical C 0 -extension can exist. Note that this corollary is not quite stating that, given a flat FLRW spacetime with a ' (0) ∈ (0 , ∞ ], there does not exist an extension ι : ( M,g ) → ( M ext , g ext ) such that ( M ext , g ext ) is strongly spherically symmetric. It shows that there is no strongly spherical extension that is compatible with the spherical symmetry of the original FLRW spacetime in the way described. This is certainly acknowledged in [2], but we decided to introduce the terminology of natural strongly spherical extension to make it even more explicit. /negationslash Remark 21 . For the sake of conciseness we do not present it in detail in this paper, but in [2] Galloway and Ling also proved a theorem and corollary analogous to Theorem 17 and Corollary 20 for a certain type of hyperbolic FLRW spacetimes (see also Corollary 13). Let ( M,g ) be a future inextendible hyperbolic FLRW spacetime, ι : M → M ext an arbitrary C 0 -extension and γ : (0 , 1] → M a future directed past inextendible timelike curve parametrized by the coordinate t with lim t → 0 + ( ι · γ )( t ) ∈ ∂ -ι ( M ). Ling and Galloway showed that if a ' (0) ∈ [0 , ∞ ] and a ' (0) = 1, then there exists a unique (subject to an initial condition and away from a subset of the spacetime where the determinant of the Jacobian of the strongly spherically symmetric change of coordinates vanishes) strongly spherically symmetric change of coordinates that leaves the spherical coordinate ω invariant, which furthermore satisfies that the metric coefficient G vanishes as t → 0 + provided that lim t → 0 + ( R ( γ ( t ))) ∈ (0 , ∞ ). Note that again R ( t, r ) = ra ( t ) and the metric coefficient G is given by (see [2, Theorem 4.3 and equation (4.24)]): /negationslash From the previous expression it becomes clear that it is important to impose that a ' (0) = 1 and lim t → 0 + ( R ( γ ( t ))) = 0 so that it holds that lim t → 0 + G ( γ ( t )) = 0. We will come back to this remark in Corollary 28. /negationslash", "pages": [ 9, 10, 11, 12 ] }, { "title": "4 Strongly axisymmetric inextendibility", "content": "In this section we will restrict ourselves to 4-dimensional spacetimes. The aim is to generalize the previously discussed results to a subclass of axisymmetric spacetimes. As in the previous section (when considering strongly spherically symmetric spacetimes), we will focus on a subclass of axisymmetric spacetimes for which the metric takes the following form with coordinates ( T, Z, ρ, ϕ ) and smooth functions A,B,C which are allowed to depend on T, z and ρ . We will call these spacetimes strongly axisymmetric spacetimes. Some specific examples of spacetimes whose metric takes such a form (but which do not necessarily fit into the FLRW framework) are: 1) the Godel Universe (for which D 2 = D 3 = 0, cf. [11, Section 2.10.1]), 2) cylindrically symmetric (in the sense of [17, Section 22.1]) spacetimes, where the metric coefficients are also independent of the z coordinate and D 1 = D 3 = 0, or 3) the Weyl class solutions to the Einstein equations (cf. [17, Section 20.2]), where the metric coefficients are time independent and D 2 = D 3 = 0. Definition 22 (Strongly axisymmetric spacetimes) . Let ( M,g ) be a 4-dimensional spacetime. It is called a strongly axisymmetric spacetime if for all p ∈ M there exists a neighbourhood V of p in M and (potentially) a two dimensional submanifold S ⊂ V such that for U := V \\ S there exists a diffeomorphism ψ a = ( T, z, ρ, ϕ ) : U → ψ a ( U ) ⊂ R 2 × (0 , ∞ ) × S 1 with T : U → R , z : U → R , ρ : U → (0 , ∞ ), and ϕ : U → S 1 such that in the coordinate neighborhood the pushforward of the metric takes the form: where dϕ 2 is the standard metric on S 1 and the metric coefficients are smooth functions of the coordinates T , z and ρ only. We call ( T, z, ρ, ϕ ) strongly axisymmetric coordinates . Remark 23 . Note that in the previous definition no assumption on the sign of the metric coefficients is being made, but we of course require the signature to remain Lorentzian. As one would expect strongly spherically symmetric spacetimes are also strongly axisymmetric. To motivate the changes of coordinates we are considering let us briefly consider the 3-dimensional Euclidean space ( R 3 , δ ) with spherical coordinates ( R,θ,ϕ ). The metric δ can then be expressed in cylindrical coordinates ( z, ρ, ϕ ) through the canonical change of coordinates ρ ( R,θ ) = R sin θ, z ( R,θ ) = R cos θ , which leaves ϕ fixed. In the following theorem we show that this type of 'natural' coordinate change generalizes to an arbitrary strongly spherically symmetric spacetime ( M,g ) and is essentially unique. Note that while we formulate the theorem for global spherically symmetric coordinates we can of course apply it to subsets U ⊆ M as well. Theorem 24. Let ( M,g ) be a four dimensional strongly spherically symmetric spacetime with strongly spherically symmetric coordinates ψ s : M → ψ s ( M ) ⊆ R × (0 , ∞ ) × S 2 and with g s = ( ψ s ) ∗ g the metric in strongly spherically symmetric coordinates ( T, R, ω ) . Then, subject to a suitable initial condition and the choice of coordinates θ and ϕ on S 2 \\ { pt ., -pt . } ∼ = (0 , π ) × S 1 (i.e., the choice of axis for the axial symmetry), there exists a unique local diffeomorphism of the form ψ a : ( R,θ ) ↦→ ( z, ρ ) such that g a := ( ψ a ) ∗ ( ψ s ) ∗ g is of the strongly axisymmetric form (15) . Moreover, the metric coefficients D 1 to D 3 vanish and A,B,C are regular (away from certain measure zero sets on which the change of coordinates is not well defined). Proof. In order to prove this theorem we closely follow the strategy used in [2] to prove Theorem 17: construct an explicit change of coordinates from a strongly spherically symmetric spacetime to a strongly axisymmetric spacetime. As an ansatz, we suppose there exists a smooth and invertible change of coordinates ψ a of the following form: such that g a := ( ψ a ) ∗ ( ψ s ) ∗ g can be written as in (15) and where ψ s is are the strongly spherically symmetric coordinates. Note that z R = ∂z ∂R and z θ = ∂z ∂θ (analogous for ρ R and ρ θ ). As the considered change of coordinates does not affect the T and ϕ coordinates it directly follows that D 1 = D 2 = D 3 = 0. Hence, under the change of coordinates (16), it holds that ( ψ a ) ∗ ( ψ s ) ∗ g = ( ψ s ) ∗ g : For now we do not specify if A , B and C are positive or negative (it will be discussed in the second half of the proof, once the explicit form of the change of coordinates is obtained). From the previous expression we can make two quick conclusions: Moreover, replacing (16) in the right hand side of (17), we get 3 new independent equations: /negationslash /negationslash The following calculations hold as long as G cos 2 θ +sin 2 θ = 0 and z θ = 0. The first corresponds to the three dimensional submanifold G ( T, R ) = -tan 2 ( θ ), so is in particular a measure zero set, and we will later discuss that the second cannot happen. If we square (18c), plug (18a) and (18b) in it and use that ρ θ = R cos θ and ρ R = sin θ we get an explicit expression for C: Note that from (18 b ) we have that B = R 2 (1 -C cos 2 θ ) z 2 θ . If we replace this equation and the expression for C in (18c) we obtain: This linear PDE corresponds to a transport equation with variable coefficients. In order to solve it, the method of characteristics is used so that the PDE becomes an ODE along a specific type of curves. In particular we choose a curve θ ( R ) in the ( R,θ )-plane that satisfies d dr z ( R,θ ( R )) = 0 and thus by the chain rule: /negationslash Combining this with (20) and restricting to θ = π 2 we get the characteristic equation of the considered curves: Integrating on both sides (using the substitution u = cos θ for the integral on the left hand side), we get the curve: These are the characteristic curves along which the solutions of the PDE (20) are constant. Therefore, a general solution to (20) is of the form: /negationslash  where f 1 , f 2 : R → R are arbitrary smooth functions, which correspond to the initial condition we can prescribe for our change of coordinates. Note that the reason that two arbitrary smooth functions f 1 and f 2 are obtained is due to the fact that, when integrating expression (21), the intervals (0 , π 2 ) and ( π 2 , π ) have to be considered separately (as θ = π/ 2). We use the freedom in choosing an initial condition for our change of coordinates to 'match' the functions f 1 and f 2 and make the following formulas nicer. Given an arbitrary smooth function f : R → R with f ' = 0 at each point (afterwards we will show that f ' = 0 actually leads to a contradiction with (18a)-(18c); note that by (24) f ' = 0 if and only if z θ = 0, where our derivation was not valid) and f (0) = 0, the functions f 1 and f 2 are fixed by demanding that f 1 = f · exp and f 2 = f · ( -exp). Then the expression for the coordinate z ( R,θ ) simplifies to /negationslash which is continuous across θ = π 2 . Note that which is always non-zero for our choice of f . Replacing this and expression (19) for C in (18b) we get that B is: Altogether, away from { G = -tan 2 ( θ ) } we have found the following change of coordinates ψ a : ( R,θ ) ↦→ ( z, ρ ) from a strongly spherically symmetric spacetime to a strongly axisymmetric spacetime, where: Under this change of coordinates, the metric coefficients of g c are: In particular, the coefficients A , B and C are positive or negative depending on the value of F and G . Recall that, by definition, for strongly spherically symmetric spacetimes the metric coefficients F and G are either both positive or both negative: In summary, we can write the strongly spherically symmetric metric in an explicit strongly axisymmetric form as follows: where ψ s is again the change of coordinates such that ( ψ s ) ∗ g can be written in strongly spherically symmetric coordinates. In order for ψ a : ( R,θ ) ↦→ ( z, ρ ) to be a local diffeomorphism, it has to be smooth and locally invertible. By the inverse function theorem we only have to check if the determinant of the Jacobian of this change of coordinates (26) vanishes or not. We obtain which is non-zero away from G = -tan 2 θ , where J vanishes. /negationslash /negationslash Finally, let us for completeness discuss why it isn't actually necessary to assume f ' = 0 (or equivalently z θ = 0): Plugging z θ = 0 into equations (18b) and (18c) with ρ = R sin θ we obtain /negationslash However these equations cannot hold simultaneously as C,R and sin( θ ) are nonzero (as we only consider θ ∈ (0 , π ) and demanded g a to be non-degenerate), so the first implies cos( θ ) = 0 while the second implies cos( θ ) = 0. Hence, we have shown that, subject to an initial condition and the choice of coordinates θ and ϕ on S 2 in the strongly spherically symmetric coordinates, there exists a unique transformation of coordinates ψ a of the form z = z ( R,θ ) , ρ = ρ ( R,θ ) which is well defined and a local diffeomorphism almost everywhere on M , more specifically away from the subsets { θ = 0 } ∪ { θ = π } , where the chosen coordinates on S 2 break down, and { G = -tan 2 θ } , where the change of coordinates breaks. Theorem 24 states that all strongly spherically symmetric spacetimes have a unique (subject to an initial condition and the choice of θ and ϕ ) natural strongly axisymmetric change of coordinates. Hence, it gives a uniqueness result within the class of changes of coordinates of the form ψ a : ( R,θ ) ↦→ ( z, ρ ). Similar to how Galloway and Ling obtained Corollary 20 from Theorem 17 we want to use Theorem 24 to show that certain strongly axisymmetric spacetimes have no natural strongly axisymmetric C 0 -extension. In order to do so, let us define the following concepts. Definition 25. Let ( M,g ) be a strongly spherically symmetric spacetime of dimension 4 together with strongly spherically symmetric coordinates ψ s defined on some ˜ M ⊆ M . Let ι : ( M,g ) → ( M ext , g ext ) be a C 0 -extension. We say that ι is a natural strongly axisymmetric C 0 -extension compatible with ψ s provided that the following holds: and such that there exists a diffeomorphism ψ ext,a : U ⊆ M ext → ψ ext,a ( U ) ⊆ R × R × (0 , ∞ ) × S 1 such that g ext ,a := ( ψ ext,a ) ∗ g ext takes the strongly axisymmetric form (15) with metric coefficients A ext , B ext , C ext , ( D 1 ) ext , ( D 2 ) ext and ( D 3 ) ext . In particular g ext ,a must be non-degenerate on ψ ext,a ( U ).", "pages": [ 12, 13, 14, 15, 16, 17 ] }, { "title": "(ii) We have", "content": "where ψ a is the change from spherically symmetric coordinates ( T, R, θ, ϕ ) to strongly axisymmetric coordinates from Theorem 24 and ˜ U ⊆ ι -1 ( U ) ⊆ ˜ M is the subset of ι -1 ( U ) where ψ a is a local diffeomorphism, that is We may sometimes refer to ψ a as a natural strongly axisymmetric change of coordinates . Recall that g ext | ι ( M ) = ι ∗ g . So the previous expression implies that on ψ s ( ˜ U ) This leads us directly to the next theorem which, essentially, gives conditions under which there is no natural strongly axisymmetric extension of ( M,g ). Theorem 26. Let ( M,g ) be a future inextendible four dimensional strongly spherically symmetric spacetime with with strongly spherically symmetric coordinates ψ s defined on some ˜ M ⊆ M . Let ι : ( M,g ) → ( M ext , g ext ) be a C 0 -extension. Let γ : (0 , 1] → M be a past inextendible curve with lim s → 0 ( ι · γ )( s ) = p ∈ ∂ -ι ( M ) . If γ ((0 , 1]) ⊂ ˜ M and lim s → 0 G ( γ ( s )) = 0 and lim s → 0 + R ( γ ( s )) ∈ (0 , ∞ ) , then either ι cannot be a natural strongly axisymmetric extension of ( M,g ) compatible with ψ s or p lies on the axis of symmetry, i.e., p ∈ S . Proof. Let ι : ( M,g ) → ( M ext , g ext ) be an arbitrary C 0 -extension and γ : (0 , 1] → ˜ M a past inextendible curve with lim s → 0 + ( ι · γ )( t ) = p ∈ ∂ -ι ( M ) and such that lim s → 0 + G ( γ ( s )) = 0 and lim s → 0 + R ( γ ( s )) ∈ (0 , ∞ ). Assume that ι is a a natural strongly axisymmetric extension of ( M,g ) compatible with the given realization of the strong spherical symmetry and p / ∈ S . Then there exists a neighbourhood U ⊆ M ext of p and a diffeomorphism ψ ext ,a : U → ψ ext ,a ( U ) ⊆ R 2 × (0 , ∞ ) × S 1 such that ι -1 ( U ) ⊆ ˜ M and (31) holds for ψ a from Theorem 24, i.e. We observe that the curve γ eventually lies in ˜ U as it does not intersect the region { G = -tan 2 θ } where the strongly axisymmetric change of coordinates is not well-defined: if γ were to intersect { G = -tan 2 θ } for arbitrarily small s , then as lim s → 0 + G ( γ ( s )) = 0 it should also hold that lim s → 0 + θ ( γ ( s )) ∈ { 0 , π } . But then ρ ( γ ( s )) = R ( γ ( s )) sin( θ ( γ ( s ))) → 0 (as by assumption lim s → 0 + R ( γ ( s )) ∈ (0 , ∞ )) which contradicts that ρ ( p ) ∈ (0 , ∞ ) since p / ∈ S . Note that this is an analogous argument to the one appearing in Remark 18 which shows that γ does not intersect the problematic region { r 2 a ' 2 ( t ) } . Now recall the explicit expression of the metric coefficient C in terms of the strongly spherically symmetric coordinates: on ˜ U . Then, it directly follows that i.e. the metric g a = ( ψ a ) ∗ g s = g ext ,a degenerates at p , a contradiction. Analogously to the discussion after Corollary 20, the previous theorem does not state that a strongly spherically symmetric spacetime cannot have a strongly axisymmetric extension. It states that there is no natural strongly axisymmetric extension of a strongly spherically symmetric spacetime compatible with some given strongly spherically symmetric coordinates. As before, the uniqueness argument used in the proof of the theorem only holds for the class of changes of coordinates of the class ψ a : ( R,θ ) ↦→ ( z, ρ ). Therefore, in principle, there could also exist a change of coordinates of the form ψ a : ( T, R, θ, ϕ ) ↦→ ( T,z, ρ, ϕ ) such that the pushforward of the extended metric g ext could be written in the strongly axisymmetric form (15). The previous theorem can be used to state an inextendibility result for FLRW spacetimes similar to Corollary 20. Recall that, if ( M,g ) is an flat FLRW spacetime with a ' (0) ∈ (0 , ∞ ], then there exists a unique (up to an initial condition) natural strongly spherical change of coordinates ψ s : ( t, r ) ↦→ ( T, R ). In order to find a change of coordinates from the flat FLRW spacetime to a strongly axisymmetric spacetime, one could either directly look for a change of coordinates ψ : ( t, r, θ, ϕ ) ↦→ ( T, z, ρ, ϕ ) or use the natural strongly spherical change of coordinates ψ s from Theorem 17 as part of a two-step transformation:   We note that, if ι : M → M ext is any C 0 -extension, then the coefficient G of ( ψ s ) ∗ g satisfies lim t → 0 + G ( γ ( t )) = 0 along any timelike curve γ : (0 , 1] → M with lim t → 0 + ( ι · γ )( t ) ∈ ∂ -ι ( M ) and lim t → 0 + R ( γ ( t )) ∈ (0 , ∞ ) by the second part of Theorem 17. Thanks to this we obtain the following Corollary from Theorem 26. Corollary 27. Let ( M,g ) be a future inextendible (4-dimensional) flat FLRW spacetime satisfying that a ' (0) ∈ (0 , ∞ ] . Let ψ s : ( t, r ) ↦→ ( T, R ) be the unique (subject to an initial condition and away from the subset { r 2 a ' ( t ) 2 = 1 } ) natural strongly spherical change of coordinates, so that g s = ( ψ s ) ∗ g can be written as in (4) . Then, there is no natural strongly axisymmetric C 0 -extension of ( M,g ) compatible with ψ s . Proof. Assume that ι : ( M,g ) → ( M ext , g ext ) be a natural strongly axisymmetric C 0 -extension compatible with the natural strongly spherical change of coordinates ψ s in M . Fix a point p ∈ ∂ -ι ( M ). Unwinding the definitions there on ψ ext,a ( U ) ⊆ R 2 × (0 , ∞ ) × S 1 and g ext ,a is non-degenerate. Without loss of generality we may assume that p ∈ U : If p were not itself in U we could replace it by a point q ∈ U ∩ ∂ -ι ( M ). Note that such a q must exist for dimensional reasons since ∂ -ι ( M ) is a topological hypersurface, cf. Theorem 8, and U is an open neighbourhood of p ∈ ∂ -ι ( M ) without (at most) a two dimensional submanifold. /negationslash Therefore, the conditions of Theorem 26 are satisfied and we get a contradiction. Let γ : (0 , 1] → M be a past inextendible timelike curve with lim t → 0 + ( ι · γ )( t ) = p ∈ ∂ -ι ( M ). In the first place, note that lim t → 0 + R ( γ ( t )) = 0 as otherwise ρ ext ( p ) = lim t → 0 + ρ ( γ ( t )) = lim t → 0 + ( R sin θ ) ( γ ( t )) = 0 contradicting that, by definition of axisymmetric coordinates, ρ ext ∈ (0 , ∞ ) on U . By the fourth point of Remark 18 (respectively the second part of Theorem 17) this implies that lim t → 0 + G ( γ ( t )) = 0 and γ ((0 , 1]) ⊆ ˜ M . Corollary 27 only relies on the existence of unique changes of coordinates ( ψ s from flat FLRW to a strongly spherically symmetric spacetime for which the metric coefficient G degenerates as one approaches ∂ -ι ( M ) at finite radii R . As discussed in Remark 21, also for certain hyperbolic FLRW spacetimes there exists a natural strongly spherically symmetric change of coordinates ˜ ψ s for which the metric coefficient G degenerates at ∂ -ι ( M ) as long as R takes a finite non-zero limit. Therefore, we can also state /negationslash Corollary 28. Let ( M,g ) be a future inextendible (4-dimensional) hyperbolic FLRW spacetime satisfying that a ' (0) ∈ [0 , ∞ ] and a ' (0) = 1 . Let ψ s : ( t, r ) ↦→ ( T, R ) be the unique (subject to an initial condition and to a subset of the spacetime where the determinant of the Jacobian vanishes) natural strongly spherical change of coordinates, so that g s = ( ψ s ) ∗ g can be written as in (4) . Then, there is no natural strongly axisymmetric C 0 -extension of ( M,g ) compatible with ψ s . /negationslash Proof. The proof is analogous to the one of Corollary 27 using that in [2] (recall the discussion in Remark 21) it was also proven that for hyperbolic FLRW spacetimes with a ' (0) ∈ [0 , ∞ ] and a ' (0) = 1 the metric coefficient G vanishes as t → 0 + provided that R has a finite positive limit as t → 0 + . And this condition holds again by the same argument than the one given in the proof of the previous corollary. To end let us briefly discuss the relevance of our results in the context of the other C 0 -inextendibility results in the literature which we reviewed in Section 2. Both our results are extensions of some of the earliest C 0 -inextendibility results through a Big Bang from [2] back in 2016, but just as their original counterparts are limited by our symmetry assumptions. One further immediately notices that the previous Corollary 28 is mostly interesting for hyperbolic FLRW spacetimes without particle horizons, as if they do have a particle horizon and satisfy that a ( t ) e ∫ 1 t 1 a ( t ' ) dt ' → ∞ as t → 0 + , then the recent stronger general C 0 -inextendibility result by Sbierski [16] already applies (cf. Theorem 10 and Table 1). Contrary to this, Corollary 27 remains more widely relevant as we are still lacking any general C 0 -inextendibility results for flat FLRW spacetimes with a Big Bang as t → 0 + . Of course the symmetry assumptions remain nevertheless limiting, so trying to establish a C 0 -inextendibility results in this case without requiring any symmetry (or at least only requiring weaker symmetry and not this strong compatibility with the symmetry structure of the original symmetry structure of the flat FLRW spacetime) in the extension continues be an interesting goal that was already formulated in [2].", "pages": [ 17, 18, 19 ] } ]

[ { "title": "ABSTRACT", "content": "The Gemini Planet Imager (GPI) is a high contrast imaging instrument that aims to detect and characterize extrasolar planets. GPI is being upgraded to GPI 2.0, with several subsystems receiving a re-design to improve its contrast. To enable observations on fainter targets and increase performance on brighter ones, one of the upgrades is to the adaptive optics system. The current Shack-Hartmann wavefront sensor (WFS) is being replaced by a pyramid WFS with an low-noise electron multiplying CCD (EMCCD). EMCCDs are detectors capable of counting single photon events at high speed and high sensitivity. In this work, we characterize the performance of the HNu 240 EMCCD from Nuvu Cameras, which was custom-built for GPI 2.0. Through our performance evaluation we found that the operating mode of the camera had to be changed from inverted-mode (IMO) to non-inverted mode (NIMO) in order to improve charge diffusion features found in the detector's images. Here, we characterize the EMCCD's noise contributors (readout noise, clock-induced charges, dark current) and linearity tests (EM gain, exposure time) before and after the switch to NIMO. ' Further author information: (Send correspondence to Clarissa R. Do O) Clarissa Do ' O: E-mail: [email protected]", "pages": [ 1 ] }, { "title": "GPI 2.0: Exploring The Impact of Different Readout Modes on the Wavefront Sensor's EMCCD", "content": "Clarissa R. Do ' O a , Saavidra Perera b , J'erˆome Maire b , Jayke S. Nguyen b , Vincent Chambouleyron k , Quinn M. Konopacky b , Jeffrey Chilcote c , Joeleff Fitzsimmons d , Randall Hamper c , Dan Kerley d , Bruce Macintosh k , Christian Marois d , Fredrik Rantakyro f , Dmitry Savranksy g , Jean-Pierre Veran d , Guido Agapito h , S. Mark Ammons i , Marco Bonaglia h , Marc-Andre Boucher j , Jennifer Dunn d , Simone Esposito h , Guillaume Filion j , Jean Thomas Landry j , Olivier Lardiere d , Duan Li g , Alex Madurowicz e , Dillon Peng c , Lisa Poyneer i , and Eckhart Spalding c a Department of Physics, University of California, San Diego, La Jolla, CA 92093 b Department of Astronomy and Astrophysics, University of California, San Diego, La Jolla, CA 92093 d National Research Council of Canada Herzberg, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada f Gemini Observatory, 670 N. A'ohoku Place, Hilo, HI 96720, USA g Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA h Arcetri, Largo Enrico Fermi 5, I - 50125 Florence, Italy i Lawrence Livermore National Laboratory, Livermore, CA 94551, USA j Opto-M'ecanique de Pr'ecision, 146 Bigaouette St. Quebec City, QC, Canada, G1K 4L2 k Center for Adaptive Optics, University of California Santa Cruz, Santa Cruz, CA 95064, USA", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Finding new exoplanets requires the development of state-of-the-art high contrast imaging instruments. Directly imaged planets are much fainter than their host stars, which is why most high contrast imaging instruments require a coronagraph, a device that blocks the light from the star such that an off-axis signal, like the planet's light, can be detected. 1 However, proper imaging of an off-axis source requires the coronagraph to be wellaligned with the starlight. Low-order wavefront aberrations, such as tip or tilt, can severely suppress the contrast capabilities of an instrument by allowing the starlight to 'leak' into the rest of the image. This necessitates efficient wavefront measurement and correction using a highly sensitive wavefront sensor and deformable mirrors. The Gemini Planet Imager (GPI) is a high contrast imaging instrument that operated at Gemini South for 6 years with the main goal of studying gas giant planet occurrence at wide orbits. Its goal was also to constrain whether these planets were in agreement with the formation via core-accretion or gravitational instability. Planets that form via core-accretion have lower entropy in their formation due to the formation of an accretion disk around the solid core of the forming planet. 2 This lower entropy causes these planets to be fainter than their gravitational instability counterparts, where the unstable region in the disk collapses directly to form a planet that retains much of its initial entropy. 3 Therefore the contrast requirements for finding both types of planets are quite demanding. The improvement in high contrast imaging technologies now allows for better contrasts at smaller separations from the host star, enabling not only the detection of closer-in planets but also for planets more consistent with the core accretion model, 4 . 5 GPI is going through an upgrade to become GPI 2.0, which is designed to achieve higher contrast than GPI 1.0. In order to achieve this goal, several subsystems are receiving upgrades, including the calibration unit, the coronagraphic system, the integral field spectrograph (IFS) and the adaptive optics system (AO). The predicted contrast after the upgrade to GPI 2.0 is shown in Figure 1. The figure shows that GPI 2.0 will unlock the detection of planets consistent with the cold-start models, as GPI 1.0 was already quite sensitive to hot-start planets, 4 enabling detections at the peak of the expected giant planet population distribution. 5", "pages": [ 2 ] }, { "title": "1.1 The Gemini Planet Imager 2.0: Wavefront Sensor Upgrade", "content": "AO subsystem upgrades occurred at the University of California, San Diego (UCSD). 6 GPI 2.0's pyramid wavefront sensor has higher sensitivity to low-order aberrations compared to its previous iteration (Shack-Hartmann) that will allow for the detection of fainter targets. GPI 2.0's wavefront sensor design is shown in Figure 2. The modulation around the tip of the pyramid occurs on the first fast steering mirror (FSM 1) in order to increase the dynamic range of the WFS. It also has a fold mirror which allows for focusing (focus stage) of the beam and a second fast steering mirror for tip and tilt adjustments. The light goes through the double four-sided pyramid and a triplet lens before hitting the detector, the Nuvu EMCCD.", "pages": [ 3 ] }, { "title": "1.2 The EMCCD", "content": "The GPI 2.0 wavefront sensor's detector is a NuVu Cameras electron-multiplying CCD (EMCCD). 7 EMCCDs are detectors capable of counting single photon events at high speed and high sensitivity. The EMCCD chip configuration is presented in Figure 3. The detector has 8 different outputs. Much like a traditional CCD, the EMCCD turns photons into electrons via the photoelectric effect. However, unlike the traditional CCD, the EMCCD has an extra register called the multiplication register. Once the photons hit the silicon body of the chip, the electrons in the imaging area travel row by row to the storage area. This mechanism allows for the next frame to be taken while the previous one is being processed. 7 Once in the storage section, the electrons travel to the multiplication register where hundreds of electrodes accelerate them, causing a phenomenon called impact ionization. Using high voltages, the captured electrons collide with the multiplication registers' silicon atoms, ripping an electron from the atom. This new electron then becomes part of the measured signal. 7 The EM Gain sets how much an electron signal will be multiplied by, which is achieved by changing the voltage in the multiplication register. This specific EMCCD chip, CCD220, has 8 outputs for a faster readout of these electrons, with 2 outputs sharing one multiplication register. EMCCDs are particularly useful for AO . wavefront sensor systems because of their high sensitivity at a high operating speed. This EMCCD's operating speed of 3,000 frames per second (FPS) and high sensitivity combined with low noise allows it to keep up with the changing atmosphere and therefore for better corrections of the wavefront to high order aberrations, 8 . 9 Our camera has a nominal temperature operation of -45 · C.", "pages": [ 3, 4 ] }, { "title": "1.3 Inverted Mode, Non-Inverted Mode and Charge Diffusion", "content": "After extensive testing at UCSD the EMCCD was found to have charge diffusion due to its 'inverted mode' set-up. EMCCDs have two readout modes for operation: inverted vs. non-inverted mode (IMO and NIMO). The difference in the modes can be set by the voltages across the sensor. The IMO is at a lower voltage, generating 'holes' in the sensor's substrate, which recombine with dark current electrons before reaching the readout register. 7 These holes, however, generate more clock-induced charges (CIC) during the vertical transfer of electrons to the readout register. This causes a significant decrease in the detector's dark current, but an increase in CIC. IMO was the previous mode of operation for this EMCCD. The iteration of tests in this mode is presented in an AO4ELT7 proceeding. 10 IMO has previously been found to cause charge diffusion due to its reduction in the potential barrier between adjacent pixels. 11 This reduction in the potential barrier between neighboring pixels caused a significant charge diffusion effect (blurring) in our images. The result also appeared to be wavelength-dependent, with shorter wavelengths showing slightly larger FWHMs in their PSFs. The EMCCD was sent back to Nuvu cameras so its readout mode could be changed to NIMO, which would increase the potential barrier between adjacent pixels and thus improve the resolution of the detector. This mode is expected to decrease CIC events. However, it would also significantly increase dark current, although such an effect should not be expected to significantly affect the pyramid wavefront sensor's performance since the frame rate of operations is quite high (2k FPS), and the camera is cooled to -45 C to mitigate dark current. This report presents the results of EMCCD performance after its change from IMO to NIMO. Here we also compare the results in both cases.", "pages": [ 4 ] }, { "title": "2. CAMERA RESOLUTION", "content": "Due to the charge diffusion effect caused by IMO, we assess the change in image quality of the camera when the camera's operation mode was changed to NIMO. In order to do this, we image a standard resolution card by Thorlabs (NBS 1952) and analyze the line resolutions for our corresponding pixel size. The camera possesses a 24 µ m pixel size. The image of our resolution card is shown in Figure 4 for the IMO and NIMO modes of operation. It is clear from the figure that changing the operation mode drastically changes the resolution capabilities of the camera. In order to verify that the camera is operating within the expected resolution, we must verify that the pixel size resolution is achieved. In order to do this, we use the set lines on the resolution grids, which have specific sizes in µ m. We focus on line 6.8 for our analysis, which has a linewidth of 73.5 µ m. Given the magnification factor of , these lines should have a size of 27.8 µ m, which is about 1.2 pixels. Our results of a cross cut for these lines are shown in Figure 5. We fit a curve to the image of the lines' cross cut and find that the actual width is closer to 1.3 pixels, which is slightly above but near the expected width given the camera's resolution.", "pages": [ 4, 5 ] }, { "title": "3. CONDUCTED TESTS", "content": "For all of our conducted tests, we separate our results into each of the 8 outputs of our EMCCD, as was done in the previous iterations of tests. 10 Separating the results allows for a better characterization of the individual", "pages": [ 5 ] }, { "title": "Line 6.8, width of 27.8 um", "content": "outputs of the EMCCD. We represent the 8 outputs for a median bias frame at -45 · C in Figure 6.", "pages": [ 6 ] }, { "title": "3.1 Readout Noise", "content": "We repeat our readout noise procedure described in the previous camera test report, where the camera was operating in its inverted mode (IMO), 10 but now we compare the results to the NIMO operating mode. For each output, we subtract the median of dark frames from the 1,000 dark frames and obtain the standard deviation of the 1,000 frames. We then multiply the standard deviation frame by the K-Gain set for each exposure series and divide by the EM gain of 5,000 to obtain units of electrons. We then obtain the median of these values for each output. We represent the median readout noise for each output in Table 1. We found that it the median readout noise does not significantly change from what it was when the camera was on IMO. Despite the requirement stating 0.1 e-, all of the contrast simulations were performed expecting 0.4 e- of readout noise. Therefore, all of the outputs are below the values used in simulations for the wavefront sensor's expected performance.", "pages": [ 6 ] }, { "title": "3.2 Dark Current", "content": "We also test the dark current present in our detector and compare it to our previous results when the camera was in the inverted mode of operation. We test this at EM gain of 5,000. We measure the dark current by changing the exposure time in our detector and taking dark frames. The dark frames are all bias subtracted with matching EM gain and all of the frames are taken at -45 · C. The exposure times used are 1, 2, 4, 8 and 10 seconds. We calculate the dark noise as follows: we first subtract bias frame from mean of each exposure time for each output, then take the mean for the pixels in each output, obtaining units of [ADU/pix/fr]. Finally we transform from ADU to e- using the EM and k-gain values. Our results are presented in Figure 7. We note that the dark current is about 3.8x (median) larger in NIMO than in IMO. A larger dark current in NIMO is expected. We plot the ratio for different exposure times in Figure 8. Finally, we test our dark current at exposure times for different temperatures (for -45 · C, we test in the second and millisecond regime). We verify that in the ms regime, at -45 · C, which is where we will be operating (a) our wavefront sensor, dark current is low even in NIMO mode (see Figure 9, top right panel). We note that the dark current is temperature dependent, as is expected, where we find saturating levels for high EM gain and exposure time for -25 C. We show our results in Figure 9.", "pages": [ 6, 7 ] }, { "title": "3.3 Clock-Induced Charges", "content": "We also test the clock-induced charges (CIC) of the EMCCD. The CIC is a source of noise in the EMCCD where false counts are created when the photoelectrons travel in the EM register. 7 We repeat the procedure performed in our previous tests. 10 We set the EMCCD to 'photon counting mode', where we take dark exposures at the max frame rate of 0.33 ms, and bias subtract them. Then, we obtain frames where all of the pixels should have '0' counts unless a CIC event occurs, in which case the pixel will have a '1' count. The CIC average for each output is calculated by summing the CIC events for every output over the number of frames and then dividing the sum by the number of frames and pixels in each output (120 x 60). Our results are shown in comparison to IMO in Table 2.", "pages": [ 7 ] }, { "title": "3.4 Flat Field Tests", "content": "We conduct flat fielding tests for our EMCCD. For this, we utilize the Newport 819D-SL-3.3 integrating sphere, used in conjunction with a Newport 6332 quartz tungsten halogen lamp, a white light lamp operated at 50 W and a Newport 60043 socket adaptor. The bulb allows for the change in light intensity using a Kikusui Stabilized power supply (Model PAB 8-2.5). We test the EM gain, exposure time and light level linearities for the EMCCD. For all of our tests, we subtract the bias frame with matching EM gain and use the median of a cube with 300 frames.", "pages": [ 7, 8 ] }, { "title": "3.4.1 EM Gain Linearity", "content": "We first test the linearity with changing EM gain for the camera at higher light levels (1.4 Amps). We test this at temperatures of -45, -40, -35 and -25 · C up to 1000 EM gain. We do not go further than that such that we do not saturate all the outputs in the EMCCD, as that can damage the detector. We plan on operating the PWFS at -45 · C, however this test allows us to characterize the camera's dependence on temperature. We subtract the bias with matching EM gain from each cube with 300 frames, then obtain the median of each output. We show our results in Figure 10. We find that for Outputs 7 and 8 there is a saturation 'cap' at higher temperatures (mainly -25 · C). The same behavior was also present in IMO. 10 Amp We then test the EM gain linearity at a lower light level, at 1.1 Amps and 1.2 Amps (up to 5,000 EM gain). We verify that at lower light levels we can obtain a linear behavior for the EM gain. We show our results in Figure 11.", "pages": [ 8, 9 ] }, { "title": "3.4.2 Exposure Time and Light Level Linearity", "content": "We test the exposure time linearity of our EMCCD. We perform this test for two light levels: by setting our flat lamp to 1.7 Amps and by keeping the level at 1.6 Amps but placing a Thorlabs Neutral Density (ND) 0.6 filter in front of the camera. This should decrease light levels by a factor of ≈ 4. Our exposure times used are 2, 4, 8, 12, 16, 18 and 20 ms. Results are presented in Figure 12. We find that for both light levels the behavior all outputs is linear with changing exposure time. Our results remain unchanged from the previous readout mode of the camera.", "pages": [ 10 ] }, { "title": "3.4.3 Light Level Ratios", "content": "We test that the counts obtained by the EMCCD correspond to the expected ratio given by our filter value (of 0.6, which corresponds to a decrease in light levels by a factor of 3.981). In order to do that, we plot the ratio of counts given by the two individual light levels at corresponding exposure times. Our results are presented in Figure 13. Our results remain unchanged from the previous readout mode of the camera.", "pages": [ 10 ] }, { "title": "3.5 Multiple Regions of Interest and Binning", "content": "We do not expect the multiple regions of interest (mROI) functionality and the binning of the EMCCD to be affected by the change in operating mode, but present the results for completeness. We repeat the test procedure from our previous iteration. 10 The results are shown in Figure 14 and 15. We use an image of the GPI 2.0 logo for facilitating the visualization of the features.", "pages": [ 11 ] }, { "title": "4. CONCLUSION & FUTURE WORK", "content": "Our final performance table for the EMCCD is presented in Table 3. We find that the camera's performance in readout noise, mROI, CIC, binning, EM Gain linearity, exposure time linearity remain the same as before the change to NIMO readout mode. The dark current is about 3.8x higher than before, as expected for NIMO. However, in our mode of operations (low temperature and low exposure time), the difference in dark current is essentially negligible. The image quality is improved and we are now achieving the expected resolution. This will allow us to obtain the required performance for the GPI 2.0 PyWFS.", "pages": [ 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "GPI 2.0 is funded by in part by the Heising-Simons Foundation through grant 2019-1582. The GPI project has been supported by Gemini Observatory, which is operated by AURA, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the NSF (USA), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), MCTI (Brazil) and MINCYT (Argentina). Portions of this work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "Encyclopedia of Astrophysics 1st Edition", "pages": [ 1 ] }, { "title": "Authors", "content": "Lena Noack, Freie Universität Berlin, Department of Earth Sciences, Malteserstr. 74-100, 12249 Berlin, Germany, [email protected], Corresponding author Caroline Dorn, ETH Zurich, Department of Physics, Wolfgang-Pauli-Str. 27, 8093 Zurich, Switzerland, [email protected] Philipp Baumeister , Freie Universität Berlin, Department of Earth Sciences, Malteserstr. 74100, 12249 Berlin, Germany, [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the last 15 years, since the discovery of the first low-mass planets beyond the solar system, there has been tremendous progress in understanding the diversity of (super-)Earth and sub-Neptune exoplanets. Especially the influence of the planetary interior on the surface evolution (including the atmosphere) of exoplanets has been studied in detail. The first studies focused on the characterization of planets, including their potential interior structure, using as key observables only mass and radius. Meanwhile, a new field of geosciences of exoplanets has emerged, linking the planet to its stellar environment, and by coupling interior chemistry and dynamics to surface regimes and atmospheric compositions. The new era of atmospheric characterization by JWST as well as the ELT will allow testing of these theoretical predictions of atmospheric diversity based on interior structure, evolution, and outgassing models.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the past thirty years, the number of known planets changed dramatically, from nine (eight) planets within our solar system to more than 5000 unambiguously detected worlds around other thousands of stars. What have we learned in these past decades, since the first confirmed discoveries of exoplanets around solar-like stars in the early 1990's (Mayor and Queloz, 1995)? The large sample size allows us to make statistical arguments and to put our own planetary system into a much larger context. The first discoveries of exoplanets were strongly biased towards close-in and massive planets (so-called 'Hot Jupiters') due to their improved detectability, but in recent years, mostly thanks to the Kepler survey, we have found many planetary systems that resemble our own, with rocky planets close-in and gas giants and Neptunes further out. At the same time, we have also discovered many systems that lack either small and potentially rocky planets (e.g. due to migration of gas giants) or in turn only consist of roughly Earth-size bodies (especially around M dwarfs, including for example the famous TRAPPIST-1 system, Gillon et al., 2017). Our own system therefore seems to be neither the rule nor the exception, but only one example of system architectures, that can be immensely diverse depending on the stellar properties. On the other hand, our current observations are still strongly limited and biased, and future long-term observational strategies from ground as well as new space telescopes such as PLATO will extend our current statistical view on exoplanets. One important finding of the past two decades, however, was that planets are not restricted to distinct planetary classes observed in our solar system (with a large gap in size/mass between rocky planets of maximum Earth' size in the inner solar system and the ice or gas giants in the outer solar system), but that the parameter space in between is filled with a large number of super-Earth and sub-Neptune planets, whose exact nature we still do not fully understand. We still do see a clustering of detected planets around low-mass (up to a few Earth masses) and high-mass planets, with a gap in-between in radius (Fulton et al., 2017), which, however, is much smaller than observed in the solar system and whose extent also strongly depends on the stellar type. By now an observational bias can be excluded and theories on planet formation and early evolution (especially atmosphere losses for subNeptune planets) have been developed that can explain the observed exoplanet populations (Owen and Schlichting, 2024). In addition to statistical arguments, selected multi-planetary systems also allow for a more elaborate characterization of planets, including for example their potential compositional variation within one system depending on their orbital distances (e.g. Acuña et al., 2022), including the previously mentioned TRAPPIST-1 system, but also TOI-178 (a 7 Gyr old system) or Kepler-80 (a 2 Gyr old system). Variations in density can then be attributed to differences in composition, which specifically would hint at either different accretion histories (especially in the case of super-Mercuries showing larger-than expected core-mass fractions, see Section 2.1), variations in ice/water fractions, or the existence of different types of atmospheres including extended, low-density atmospheres (such as primordial H2-He atmospheres). But data on exoplanets is not limited to mass and radius, or age of the system. Next to orbital information that can inform us about expected effective surface temperature (i.e. in the absence of greenhouse gases) and tidal forces leading to additional energy dissipation in the planet's interior, we can also obtain a first order estimate on a planet's composition from the stellar metallicity and chemical abundances in the stellar spectrum. Emission spectroscopy (either via direct imaging or as a complement to the stellar signal during a secondary eclipse) can give us valuable information about the surface or atmosphere of a body, which in the era of the JWST is allowing us a large step forward towards the characterization of planets beyond our solar system. The wealth of information that we can now gain about an exoplanet are (in an ideal scenario) for example comparable to the data obtained for the moons of Jupiter before they were visited by the first spacecraft Pioneer 10 : remote determination of orbital information as well as albedo measurements and spectroscopic data allowed for a first characterization of Io as a rocky moon including a sulfur-rich surface (Lee et al., 1972) in contrast to the other icy moons Europa, Ganymede and Callisto, where the brightness of these bodies already allowed for a first indication of the age of the ice (with Europa having a fresh water-ice crust and Callisto having an old crust composed of a mixture of ice and dust). While the age of spaceflight did allow for astonishing discoveries (including the previously underestimated strength of tidal heating inside of Io, as well as the discovery of subsurface water on Europa, Ganymede and maybe even Callisto), the first-order remote characterization matched our current knowledge of these moons surprisingly well, which is promising with respect to the interpretation of current and future exoplanet observations. These interpretations are aided by modeling approaches to better understand interior and surface processes or exoplanets, where both modeling and experimental advances in the past 20 years since the first discovery of low-mass rocky exoplanets (such as CoRot-7b and Kepler-10b) allowed characterization of exoplanets to grow out of its childhood into a mature research field. These advances were only possible due to interdisciplinary collaborations connecting the dots between the stellar environment of planets with state-of-the-art experimental data and looking-beyond-the-boundaries of solar system knowledge to begin to grasp the diversity of exoplanetary interiors that can be out there - but learning from previous experience it is clear, that many discoveries still await us!", "pages": [ 4, 5 ] }, { "title": "2.1. Star-planet connection", "content": "Numerical models of planet formation that examine equilibrium condensation sequences propose that planets inherit some chemical make-up of their host stars. For example, Thiabaud et al. (2015) illustrate that most planets exhibit a bulk refractory composition similar to their host star, specifically for the rock-building elements of Fe, Si, and Mg. They have high condensation temperatures (>1000 K), such that refractory species (e.g., oxide species) condense close to the host star in a protoplanetary disk. In consequence, planets tend to replicate the refractory element ratios of the protoplanetary disk. Elements like Mg, Si, and Fe are observable in stellar photospheres, and their ratios are useful constraints for planet interior modeling. When these constraints are applied to interior models, they suggest that the mantle typically forms the largest layer, as opposed to the iron core, in most rocky planets. However, there is still debate about the extent to which stellar abundance proxies can accurately inform planetary rock compositions (Dorn et al., 2015, Schulze et al., 2021, Plotnykov and Valencia, 2020). Adibekyan et al. (2021) found a correlation between the compositions of rocky planets and their host stars, indicating that the relationship for Fe/Mg is not exactly 1:1, and that planets can be richer in iron than would be expected from their host stars. However, for a final conclusion there is a need for robust and comparable stellar abundance estimates. Spectra of white dwarfs that were polluted by recently accreted materials (broken-up parts of asteroids, in other words planetary building materials, or even remnants of planetary bodies themselves) can give additional chemical constraints on the compositional variety of exoplanets. While the main rock-forming elements Mg, Si, and Fe show similar condensation temperatures (Lodders, 2003), other refractory as well as volatile elements condense from the nebula at very different temperatures, leading to various different condensation or ice lines and therefore compositional variations in the planetary building blocks depending on the distance to the host star (devolatilization trend, Wang et al, 2019). However, especially for volatile species, a direct link between the condensation ice lines and later planetary compositions is not straight-forward due to several secondary processes including pebble migration, disk evolution and instabilities, devolatilization during planetesimal accretion, and outgassing during early melting events in proto-planetary bodies. Observational constraints on the link between stellar and planetary composition therefore focuses on close-in planets, including extreme cases such as super-Mercuries as well as strongly heated magma ocean planets. Adibekyan et al. (2021) showed, for example, that super-Earths and super-Mercuries might be distinct populations, suggesting that the latter may not be formed by giant impacts as often proposed. Apparently, the collisional history of planet formation does not explain the observed diversity in planet density. While giant impacts are one possible component to form super-Mercuries, there are other possibilities which have been explored on how to form super-Mercuries. These include, for example, nucleation and growth processes of iron pebbles (Johansen & Dorn, 2022). Mercury in the solar system shows anomalous characteristics. Its origin is still debated and so far, no single process (e.g., condensation sequence, giant impact accretion processes) has been identified to explain all the observed features (e.g., lack of FeO, reduced oxidation state of crust and mantle, moderately volatile elements present on surface). For exoplanets, as we are probing predominantly close-in planets, compositionally extreme worlds may be found that form in high-temperature regimes. Using the condensation sequence of proto-planetary gas disks (Dorn et al., 2019) have identified a potential class of exoplanets that forms from high-temperature condensates (iron-poor and rich in Ca- and Al-oxides) and whose bulk densities are lower compared to Earth-like compositions. Their existence may be verified by atmospheric characterization. Similarly, Plotnykov and Valencia (2020) have shown that the statistical range of possible abundances of rocky planets scatters wider than host star abundances, including potentially Fe-Si-depleted rocky planets. On the other hand, close-in low-density planets may also be explained by large fractions of melt being less dense than solid rock. Most super-Earths are hot worlds which reside within the moist greenhouse radiation limit (> 400 K equilibrium temperature). This implies that any atmosphere may increase surface temperatures drastically to allow for molten silicates, i.e., a magma ocean. For a hand-full of super-Earths this is even true without any atmosphere where equilibrium temperatures are above ~1800 K. Hence, the majority of the observed super-Earth population is dominated by long-lived magma oceans. The boundary between magma ocean and atmosphere is compositionally coupled, chemically reactive, and thermally active (e.g. Kite et al., 2020). Future detections of rocky exoplanets on longer orbits around F- and G-type stars with PLATO will allow us to obtain an improved view on potential compositional links between star and planet at different condensation regimes and deviations from the current predictions, since our current view is heavily biased towards M-dwarf systems. Comparative planetology in multi-planet systems will allow for the constraint or refutation of density trends within planetary systems, that can then be linked with theories on planet formation and migration.", "pages": [ 5, 6 ] }, { "title": "2.2. Equations of state", "content": "Building upon the studies of the interior structure of planets in the Solar System, models aimed at characterizing the interior of low-mass exoplanets generally assume that a planet consists of layers with physical and chemical properties. Rocky (terrestrial) planets are dominated by silicates and iron-rich cores. Planets with lower densities likely contain significant amounts of volatile elements, such as hydrogen-rich atmospheres or water. To first order, planets are spherically symmetric and in hydrostatic equilibrium. Under this assumption, the interior structure of a planet can be characterized by a set of 1D fundamental structural equations which link mass 𝑚𝑚 , radius 𝑟𝑟 , density 𝜌𝜌 , pressure 𝑃𝑃 , and temperature 𝑇𝑇 inside the planet, depending on the gravitational constant 𝐺𝐺 and composition 𝑐𝑐 : Central to the modeling of the interior structure of planets are the Equations of State (EoS, equation 3), that describe the relationship between thermodynamic parameters such as density with pressure and temperature for a given material via a function 𝑓𝑓 . In the context of rocky planets, equations of state typically stem from thermodynamic theoretical models, which are fitted to experimental data, for example from high-pressure diamond anvil cells and laser shock compression experiments. Numerical models, e.g. ab-initio calculations are employed for extreme conditions unachievable in the laboratory. EoS are usually valid only over the specific range of the experimental data, which is often exceeded in planetary conditions. This necessitates the extrapolation of the EoS to extreme conditions, where predicted material properties may not be reliable anymore. This leads to typically small, but non-negligible uncertainties in the characterization of interior properties. Hakim et al. (2018) demonstrate that in the TPa range, the extrapolation of Fe EoS can lead to differences in inferred iron density of up to 20%, which results in modeled radius uncertainties of several percent. However, overall, these effects are generally minor compared to the uncertainties arising from compositional unknowns. Nevertheless, special care should be taken to select EoS appropriate to the pressure and temperature conditions in the planet. Aguichine et al. (2021) find similar results for water-rich planets. They find that extrapolating water EoS outside their defined range can lead to an overestimation of the planetary radius by up to 10%. These conditions occur for water mass fractions above 5%, which necessitates the use of a water EoS which holds up to a few TPa and several thousand Kelvin. One influential compositional unknown is the amount of light elements in the iron core. It is well known that the density of Earth's core is too low for it to be pure iron. This density deficit can be explained by the presence of lighter elements, such as hydrogen, sulfur, or carbon, but the exact nature of the density deficit is still unknown even with the wealth of information available on Earth. It is likely that the presence of lighter elements in exoplanet cores is the norm. This introduces a significant compositional degeneracy in the interior structures. Hakim et al. (2018) show that the density deficit arising from lighter elements can significantly impact the interpretation of the retrieved interior structure. A 20% reduction in core density decreases the modeled mass of a planet by 10-30%, depending on core size. The appropriate choice of EoS is particularly important for volatile phases, such as water layers or atmospheres. Depending on the irradiation received from the host star, massive water-rich planets may display supercritical water layers surrounded by thick steam atmospheres instead of high-pressure ice phases (Aguichine et al. 2021), which offers an additional explanation for the radius gap. Given the uncertainties regarding mass, radius and composition of a planet, the temperature profiles play only a minor role in silicate and iron layers, but can be significant for volatile-rich layers such as water layers (Thomas and Madhusudhan, 2016) and the atmosphere (Turbet et al., 2020).", "pages": [ 7, 8 ] }, { "title": "2.3. Interior structure", "content": "Super-Earths and sub-Neptunes are the most common observed exoplanets. They show significant variability in their mass and radius. Super-Earths, with radii smaller than about 1.5 Earth radii, are thought to have rocky compositions, but they may also contain low amounts of water or other volatiles. Sub-Neptunes, with radii between approximately 2 and 3 Earth radii, likely have thick hydrogen/helium envelopes and may also host extensive water layers mixed within their envelopes (e.g., Kite et al., 2020, Schlichting and Young, 2022). There is an inherent degeneracy in interior modeling, meaning that a given set of observables (mass and radius) can correspond to multiple different interior compositions and structures, as illustrated in Figure 1. To reduce this degeneracy, additional constraints and data are necessary that come from a large variety of sources, including first principle considerations on the general composition of planets, lab experiments, ab-initio calculations, stellar properties, planet system architecture and tidal dissipation considerations, or planet formation. Among the most important constraints on the interiors of super-Earths and subNeptunes is the bimodal distribution of planet sizes (Fulton et al., 2017). Between around 1.5-2 Earth radii, there is a clear scarcity of planets, which has been interpreted to be due to evolution processes: sub-Neptunes have thick H/He atmospheres, while super-Earths have lost them and represent the stripped interiors. Both core-powered mass loss and evaporative loss shape the super-Earths population, while the radius valley itself is carved by photoevaporative loss (Owen & Schlichting, 2023). The interiors of super-Earths are not necessarily just the H/He-stripped counterparts of sub-Neptunes. Super-Earths and subNeptunes may differ also in terms of their water budgets by formation (Venturini et al., 2020). In fact, formation models predict that water-rich worlds are efficiently migrating inwards to locations where we observe them today. That said, the radius valley as the most important constraint on the planet interiors is (not yet) imposing strong enough constraints to reduce the inherent degeneracy. Recent advancements in exoplanet science have emphasized the importance of considering complex interactions between a planet's interior and its atmosphere. Many super-Earths and sub-Neptunes are likely to have magma oceans even at their evolved stages. In addition, even temperate rock-dominated planets start their evolution hot, mainly because of the release of gravitational potential energy from accretion, radiogenic heating and differentiation. Several processes act in magma oceans with implications on the observable atmosphere of super-Earths. Melting and solidification, outgassing, redox-reactions, core differentiation and the loss of atmospheres to space are governing processes in this early stage of a planet's evolution. To address these challenges, interior models are developed that account for chemical and compositional coupling between atmosphere and deeper interior (e.g. Kite et al., 2020; Schlichting and Young, 2022; Baumeister et al., 2023). Such coupling is crucial for accurately estimating volatile inventories, which inform us about a planet's formation environment, evolutionary history, redox state, and potential habitability. For instance, the presence of water and other volatiles can significantly affect melting temperatures of silicates, interior structure and atmospheric structure. As mentioned above, one of the most intriguing systems for studying exoplanet interiors is the TRAPPIST-1 system, which contains seven Earth-sized planets orbiting an ultracool dwarf star. The TRAPPIST-1 planets offer a unique opportunity to study a diverse set of planetary interiors within the same system. Observations suggest that these planets have densities lower than Earth, indicating the presence of volatiles or less dense refractory materials. The exact compositions of these planets remain uncertain, but models suggest they could have a range of water content, from dry, rocky planets to those with substantial water envelopes (Agol et al., 2021). For instance, TRAPPIST-1b, the inner-most planet of the system, is thought to have a rocky composition with no clear evidence of an atmosphere. TRAPPIST-1c may have a thin secondary outgassed atmosphere but otherwise does not seem to contain many volatile elements (Zieba et al., 2023). TRAPPIST-1f and TRAPPIST1g might have thicker atmospheres and higher water contents. The varying compositions among the TRAPPIST-1 planets illustrate the diversity of interior structures within a planetary system.", "pages": [ 8, 9 ] }, { "title": "3. Feedback between interior and surface", "content": "One of the main factors influencing the surface conditions of a planet (including the atmosphere) is the interior of the planet. At least for low-mass planets, the long-term evolution of the atmosphere is driven by volcanic outgassing as well as (at least in the case of Earth) recycling of volatiles back into the interior via plate tectonics. The composition of the interior as well as remelting processes of crustal material (for example via subduction) define the composition of the surface material - which is what we may, at least for some exoplanets, observe remotely. But also distinct features at the surface such as a dichotomy between lowlands and highlands as on Mars, crustal compositional variations as on Venus, or the possibility to have a partial coverage of the surface by water depending on the topography and the global carbon cycle as on Earth, are all influenced by interior processes. For a correct interpretation of observational features in the emission or transmission spectrum, or in the phase curve of an exoplanet, a good understanding of different interior processes is inevitable.", "pages": [ 9 ] }, { "title": "3.1. Interior as global heat engine", "content": "The main planetary processes that are driven by the interior include, but are not limited to, mantle convection, surface movement (for example via plate tectonics or convective mobilization), surface lava flows, volcanic outgassing, recycling of volatiles, and generation or maintenance of a magnetic field. All of these processes are directly linked to the heat sources available in the interior of a planet. We distinguish between several different sources of internal energy:", "pages": [ 10 ] }, { "title": "1) Accretional energy", "content": "This term describes the energy increase of the planet by transferring kinetic energy of accreted material to the planet. The energy is often described with the following formula, which describes the gravitational energy that is released when bringing a particle 𝑖𝑖 to a planet of mass 𝑀𝑀 and radius 𝑅𝑅 ,", "pages": [ 10 ] }, { "title": "2) Gravitational energy due to core formation", "content": "where 𝜇𝜇 𝑖𝑖 is the mass of the particle, 𝐺𝐺 is the universal gravitational constant, and 𝐶𝐶 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 is a constant describing energy loss processes. When the heavy iron separates from the molten silicate rock to combine into iron droplets that sink towards the core due to gravitational forces, energy is released and contributes to both heating the mantle as well as the core. Following Foley et al. (2020), the release of gravitational potential energy release due to core formation alone was sufficient to increase the temperature at the core-mantle boundary of Earth by 4000 K.", "pages": [ 10 ] }, { "title": "3) Latent heat release and gravitational energy upon core crystallization", "content": "For an Earth-size or super-Earth-size body, the metal core is expected to crystallize from the interior out, leading to an inner solid core of almost pure metals and a liquid outer core enriched in lighter elements (such as sulfur). The phase transition from liquid to solid releases latent heat, which can be approximated as where is the latent heat per mass and 𝑚𝑚 is the increasing mass of the inner core. Due to the release of the lighter elements from the freezing inner core into the outer liquid core, additional gravitational energy 𝐸𝐸 𝐺𝐺 is released into the core and depends on the density difference between pure and enriched metals as well as the radius of inner and outer core.", "pages": [ 10 ] }, { "title": "4) Radiogenic heating", "content": "One of the main heat sources inside rocky planets after the accretion is heating by release of energy during radioactive decay, especially of the isotopes 𝐴𝐴𝑙𝑙 26 , 𝐹𝐹𝐹𝐹 60 , 𝑈𝑈 235 , 𝑈𝑈 238 , 𝑇𝑇ℎ 232 and 𝐾𝐾 40 . An overview of radiogenic heat generation for each isotope as well as compositional variation between known bodies (i.e. in the solar system) is summarized in Foley et al. (2020). The first two isotopes are very short-lived and become extinct during only a few Myrs, but they may have a strong influence on potential melting processes inside planetesimals. The remaining heat sources are long-lived isotopes with half-life times between several hundreds of Myrs and more than ten Gyrs. As a result, the total radiogenic heating for an Earth-like mantle decreases by a factor of about 5 over 4.5 Gyr of evolution, see Figure 2, with heating initially being dominated by the U and K radiogenic isotopes, and long-term heating being attributed to the more stable Th isotopes due to their slow decay time.", "pages": [ 10, 11 ] }, { "title": "5) Tidal heating", "content": "For close-in exoplanets or exomoons orbiting gas giants (similar to the close-in moons of Jupiter), tidal heating due to the interplay between gravitational pulls of the other bodies in the system can lead to strong tidal deformations of the body and frictional heating in the interior, leading potentially to extremely high interior temperatures and even subsurface magma oceans (as suggested for Io or the innermost TRAPPIST-1 planets, for example). The total dissipated power of tidal heating for a body in synchronous rotation can be calculated from where | 𝑘𝑘 2 | is the absolute value of the Love number 𝑘𝑘 2 , 𝑄𝑄 is the dissipation factor, 𝜛𝜛 is the orbital period, 𝐺𝐺 the gravitational constant, 𝐹𝐹 the eccentricity, and 𝐼𝐼 is the obliquity. The dissipation factor 𝑄𝑄 is a few hundred for Earth, about 100 for Mars, and a few tens for the Moon. For Io, on the other hand, 𝑄𝑄 may reach values of about 10 6 . This comparison shows that there is no linear relationship between 𝑄𝑄 and body size, since the interior state influences the value of 𝑄𝑄 as well. Figure 3 shows an estimate of the variation in tidal energy released inside of the TRAPPIST-1 planets depending on different assumptions on mantle viscosity (for a fixed mantle shear modulus), leading to a large range of dissipation factors and tidal heating values. 108", "pages": [ 11, 12 ] }, { "title": "6) Induction heating", "content": "For close-in planets around stars with strong magnetic fields, near-surface rocks can be heated by magnetic induction heating depending on the exact orbital configuration of the planet including eccentricity and obliquity, magnetic field axis in alignment with stellar rotational axis, and orbital inclination. Electrically conductive materials in the rocky crust and lithosphere of a planet (mainly referring to hydrated crust and iron-rich minerals) can then be heated by induction when exposed to variations in the stellar magnetic field (Kislyakova et al., 2017) - similar to the principle of induction ovens on Earth which are used to melt metals.", "pages": [ 12 ] }, { "title": "7) Irradiation", "content": "Depending on the distance to the star, a non-negligible heat source may be the irradiation of the surface of the planet from its star. For exoplanets, a first indication of the resulting surface temperature is the effective temperature depending on the solar flux and albedo of the planet. A thick greenhouse atmosphere can strongly amplify the heating effect at the surface. For close-in planets, the surface temperature may thus exceed even the melting temperature of rocks, which would lead to a hemispherical magma ocean (since close-in planets would be assumed to be tidally locked, hence with a fixed hemisphere exposed to the stellar irradiation). If no atmosphere persists that transports the heat from the day-side of the planet to the night-side, the hot day-side would reduce efficient cooling from the interior of that hemisphere. Considerable temperature variations would therefore also be expected in the deep-interior, as interior mantle convection may not be fast enough to homogenize interior temperatures from day- to night-side of the planet. All heat sources are summarized again in Figure 4.", "pages": [ 12 ] }, { "title": "Planetary heat sources", "content": "Cooling of planetary bodies For a solid surface, cooling of the planet is mostly limited to conductive heat flow 𝑞𝑞 at the surface through the lithosphere and crust, with the exception of direct transport of lava flows to the surface (in the case of volcanic activity, or, in the more extreme case, heat-pipe mechanism), which can lead to high local heat fluxes as seen on Io. The heat flux depends on the local thermal conductivity 𝑘𝑘 as well as the temperature gradient through the lithosphere and crust. On Earth, the heat flux varies between ~65 mW/m² (continental crust) to ~105 mW/m² (oceanic crust). For the moon, Apollo measurements showed a much lower heat flux in the range of 15-20 mW/m². In contrast, Io's heat flux has been measured to be about 2000 mW/m² due to the strong interior heating by tidal dissipation. For the general efficiency of cooling of the interior (including secular cooling, i.e. the cooling initial accretional and gravitational heat sources, but also cooling of produced internal energy from tidal heating, radiogenic heating, or induction heating), smaller bodies should cool much faster than more massive planets, since the cooling occurs over the planetary surface, and the ratio of planet surface to planet volume (i.e. heat) decreases with increasing radius.", "pages": [ 13 ] }, { "title": "3.2. Interior as driver of planetary dynamics", "content": "Heat sources as described above are the main driver of planetary geodynamics, where thermally-induced buoyancy in core or mantle leads to the initiation of convective currents in the core one of the necessary ingredients to drive a magnetic field. A planetary dynamo is typically only expected for cooling cores. For a strongly cooling core (typically assumed for the early evolution of low-mass rocky planets), the top of the liquid metal core will become denser than the underlying, warmer core, and convective currents are triggered to establish again a gravitationally stable field, i.e. lighter material lying on top of heavier material. A dynamo created by such a strong heat flux at the core-mantle boundary is termed a thermal dynamo. On Earth, today the heat flux at the core-mantle boundary would not be sufficient anymore to maintain a magnetic field. However, cooling of the core over several Gyr led to the freezing of an inner, solid core. Since lighter elements in the core (such as various volatile elements) remain preferentially in the liquid upon solidification of iron and other metals, the layer above the solid inner core is enriched in lighter elements, hence the layer is lighter than the rest overlying liquid core, again triggering convection. A dynamo driven by the chemical variation of different core layers is accordingly referred to as chemical dynamo. Other types of chemically-driven dynamos exist, such as the iron-snow regime (especially relevant for lowpressure environments, e.g. for the core of Ganymede), and depend on the chemical composition and pressure-dependence of the adiabatic temperature profile in comparison to the melting temperature (Breuer et al., 2015). We also expect convection in the solid, rocky mantle, which is then similarly driven by thermal or chemical density variations, but convection in the mantle occurs on much longer, geological timescales. Indications for the existence of mantle convection (i.e. that the mantle indeed behaves as a viscous fluid on geological timescales) go back to the observation of plate motion at the surface of Earth as well as seismic tomography indicating upwelling mantle plumes. Experimental rheological studies apply large stresses to upper mantle rocks or analogue materials and can be used to derive the viscosity of different materials. For rocks under higher pressure, theoretical studies can investigate first-order deformation mechanisms and derive rheological laws applicable to the deep interior of Earth, and potentially of super-Earths, as well. Convection in a compressible mantle is typically described with the following so-called truncated anelastic liquid approximation, describing the conservation of mass, momentum and energy. (8) ∇ · ( 𝜌𝜌𝜌𝜌 ) = 0 (9) -∇𝑝𝑝 + ∇ · 𝜎𝜎 = 𝑅𝑅𝑅𝑅 𝜌𝜌 𝑔𝑔 𝛼𝛼�𝑇𝑇 - 𝑇𝑇 𝑟𝑟𝑟𝑟𝑟𝑟 �𝐹𝐹 𝑟𝑟 𝜎𝜎 = 𝜂𝜂 �∇ v + ∇ v T -2 3 ∇ · 𝜌𝜌 𝐼𝐼� These equations depend on several thermodynamic parameters (such as density 𝜌𝜌 , heat capacity 𝐶𝐶 𝑑𝑑 and thermal expansion coefficient 𝛼𝛼 ), transport properties (such as viscosity 𝜂𝜂 and thermal conductivity 𝑘𝑘 ), and variables of interest (temperature 𝑇𝑇 , velocity vector 𝜌𝜌 , and convective pressure 𝑝𝑝 ), as well as various heat sources 𝜌𝜌 as described above and geometric factors such as the radius union vector 𝐹𝐹 𝑟𝑟 and union matrix 𝐼𝐼 . There are well-established numerical routines to solve such a system of equations. However, parameters and variables are defined for very different orders of magnitude (with viscosities in the order of 10 20 Pas and velocities in the order of 10 -12 m/s. It is therefore an established approach to nondimensionalize the conservation equations, as was done already in the equations (8)-(10) above, i.e. dividing each quantity by a reference value to yield non-dimensional values in the order of 1 . With this approach, new quantities appear as left-overs of the nondimensionalization process, that can give a first-order characterization of the overall strength of convection (as in the case of the Rayleigh number 𝑅𝑅𝑅𝑅 ) or of the compressibility of the mantle (expressed via the Dissipation number 𝐷𝐷𝑖𝑖 ): where 𝑔𝑔 0 is the surface gravitational acceleration, ∆𝑇𝑇 a reference temperature contrast (such as initial CMB temperature minus surface temperature), 𝐷𝐷 is a measure of convection length (such as difference between planet and core radius), and 𝜅𝜅 = 𝑘𝑘 /( 𝜌𝜌 𝐶𝐶 𝑑𝑑 ) is the thermal expansivity. Especially the Rayleigh number is of high importance to obtain first estimates on the strength of convection, where (12) 𝜌𝜌 ≈ 𝑅𝑅𝑅𝑅 2𝛽𝛽 , or in an extended definition (Valencia et al., 2007), as well as for the strength of the surface heat flow, expressed as Nusselt number, indicating the strength of convective heat flow over convective and conductive heat flow at the planet's surface, (14) 𝑁𝑁𝑁𝑁 ≈ 𝑅𝑅𝑅𝑅 𝛽𝛽 . The exponent 𝛽𝛽 has been derived in various laboratory, theoretical and numerical studies and lies approximately between 1/4 and 1/3 (depending on convection regime and geometry of the model domain). The above defined dimensionless quantities (Rayleigh number, Dissipation number, and Nusselt number) are commonly used in fluid dynamics to characterize fluid properties and are used to derive scaling relationships describing the general dynamical behaviour of a convective medium. Since rocks behave similarly to fluids on geological time scales, these scaling laws allow to compare convection inside of a rocky mantle to small-scale laboratory experiments using fluids with different rheological properties. (15) 𝛿𝛿 ≈ 𝐷𝐷 · 𝑅𝑅𝑅𝑅 -𝛽𝛽 . Following Valencia et al. (2007), the total stress underneath the lithosphere of a planet can be approximated with (16) 𝜏𝜏 ≈ 𝜂𝜂𝜌𝜌 / 𝐷𝐷 , which is related to the horizontal normal stress, that is needed for the lithosphere to break, (17) 𝜎𝜎 ≈ 𝜏𝜏𝛿𝛿𝜌𝜌 / 𝜅𝜅 . If the convecting mantle 𝐷𝐷 as well as convecting velocities 𝜌𝜌 increase with planet size, this would imply an increase in the likelihood of plate tectonics with planet size. However, other studies (summarized in Ballmer and Noack, 2021) have shown, that for increasing internal temperatures (as expected with increasing planetary mass), the likelihood of plate tectonics An additional relationship has been observed between the Rayleigh number and the thickness of the thermal boundaries 𝛿𝛿 of a convecting layer, which are the layers forming between the actively convective region and the top or bottom boundary of the layer, and where thermal instabilities occur that lead to local overturns, driving convection in the layer, may decrease. On the other hand, such scaling laws as described above depend strongly on the thickness of the convecting layer, which is not necessarily the mantle thickness. For strongly increasing viscosities with pressure, the convecting layer of the mantle may actually decrease with increasing planet mass, leading to thicker lithospheres, lower mantle velocities and lower stresses, favouring a stagnant-lid scenario for super-Earth planets. Plate tectonics may therefore be more or less likely for increasing planetary mass, depending on several factors such as heat sources or pressure-dependence of the viscosity, and no linear trend can be observed. The thickness of a stagnant upper layer (i.e. the lithosphere) is not only relevant for the prediction of the surface regime (plate tectonics vs stagnant lid), but also influences if melt can reach the surface though a stagnant lid or remains trapped at the base or inside the lithosphere. Depending on the assumed rheology in the interior, the predictions of melt volumes can therefore also be higher or lower for more massive planets. High-pressure rheological studies for various mantle compositions may therefore give a hint at what to expect for massive super-Earths - bare-rock, stagnant-lid surfaces without strong volcanic activity, or Earth-like worlds with diverse atmospheres fed by continuous plate-tectonicsdriven volcanic outgassing. In the end, observations of active volcanism on rocky exoplanets (i.e. by measuring traces of recent, instable volcanic compounds in the atmosphere or by observing bare-rock planets with strong interior heating) for planets of various planet masses may tell us in the future how the interior state as well as outgassing strength varies with planet mass, composition and orbital configuration (influencing e.g. tidal heating and induction heating of the planet).", "pages": [ 14, 15, 16 ] }, { "title": "3.3. Interior as source of crust and atmosphere", "content": "Such observations - especially of bare-rock planetary surfaces - can also give us additional insights on the interior of selected rocky exoplanets. Specifically, the composition of the crust may reflect at least to some part the composition and therefore mineralogy of the upper mantle of these bodies, where melt is formed by partial melting of rocks. On Earth, we distinguish rocks (mantle and crust) in four basic categories: ultramafic, mafic, intermediate, and felsic crust. An ultramafic composition is assigned to primitive mantle rocks, whereas mafic (basaltic) crust stands for a rock that is magnesium-rich and silicon-poor, i.e. composed of minerals that have a lower melting temperature in the mantle, therefore accumulating first in partial melt. Felsic crust describes a high rock with high feldspar and silicon contents, being therefore relatively magnesium-poor. Such a crust can contain large quantities of quartz (SiO2) and is in general less dense than a mafic crust. Intermediate rocks contain a silica fraction between mafic and felsic crust. Prominent examples of the different rock types are peridotites (ultramafic mantle rocks), basalts (mafic crust, which is the dominant crustal rock of rocky planet surfaces in the solar system), as well as granites (felsic crust). Intermediate and felsic crusts are typically thought to be formed by re-melting of crust, driven by plate tectonics and subduction of crust and volatiles in the mantle. The oceanic crust of Earth actually consists of basalts, and the continental crust is mostly made of felsic material. However, the same analogy cannot be transferred to our neighbor planets, as SiO2-rich rocks have been found embedded in basaltic crust on both Venus and Mars, with no clear link to a continental crust, plate tectonics or recycling and remelting of crust. Similarly, our main classification of rocks (from ultramafic to felsic) may fail on planets with different initial compositions around stars that are for example by nature magnesium-rich or silicon-rich. In the light of new observational capabilities for exoplanet surface rocks, more experiments for various mantle compositions as well as pressure- and temperature conditions are needed to understand the link between mantle and crust for non-Earth-like compositions, as well as to constrain their spectral features. In general, depending on the exact melt composition, structure, porosity and various weathering processes, crustal rocks can come in a rich variety of colors and major components. Care should therefore be taken when trying to characterize the planetary surface of a bare-rock exoplanet from spectral analysis, and especially when interpreting observations with respect to crust heritage and implications for surface recycling. The same caution should be applied for the interpretation of atmospheric gases and their potential link to the interior of a planet. First, there is need to distinguish different types of atmospheres, varying mostly with planet mass and age. Second, all processes shaping a planetary atmosphere need to be considered, including internal and external sources as well as sinks of atmospheric gases. We can broadly distinguish three types of atmospheres: The first type is a primordial atmosphere , that is formed by gravitational attraction from the planetary disk if the planet reaches a critical mass during the lifetime of the disk. This atmosphere should roughly resemble the stellar atmospheric composition (as is the case for the gas and ice giants in the solar system) with some variations with orbital distance and hence local temperature. These atmospheres are dominated by hydrogen and helium, but can also contain other gases including for example water, ammonia and methane (methane being the gas that gives Neptune and Uranus their blue color). For young, rocky planets, and potentially also for sub-Neptune planets, the atmospheric composition is influenced by interactions with the interior, at least as long as the rocky mantle is molten at the surface and allows for volatile exchange with the atmosphere. This interaction may be a mixture of first-order chemical equilibrium, kinetic effects, saturation limits in the atmosphere, combined with solubility constraints for volatiles in the deep magma ocean. Due to a continuous exchange of volatiles between interior and atmosphere, especially during a potential solidification of the magma ocean leading to an enrichment of volatiles in the remaining magma ocean, a primary outgassed atmosphere forms, which can potentially also contain still large fraction of the primordial atmosphere - depending on the efficiency of atmospheric escape to space. It should be noted that almost none of the primordial hydrogen would be expected to be dissolved in the magma ocean due to the low solubility of hydrogen compared to other volatiles. Similarly, helium is incompatible with melts and solidified rocks and can only be a trace element in a planetary mantle and trace gas in planetary atmospheres that lost their primordial atmosphere. For low-mass rocky planets, atmospheric escape may lead to the removal of not only the primordial, but also of the primary outgassed atmosphere after magma ocean solidification depending on the surface gravity, stellar activity, and atmospheric composition. While Mars and Earth do not show any survival of their earliest atmospheres, and Mercury as well as the Moon do not contain any considerable atmosphere at all, anymore, for Venus the debate is ongoing. Was its current atmosphere mostly or entirely built by secondary outgassing , i.e. volcanic activity, or is some of it a relic of the primary outgassed atmosphere? Better atmospheric measurement of several trace gases in the next decade with the upcoming new Venus missions may shed a better light on the origin of Venus' atmosphere - and depending on the outcome, may finally help to answer the question if Venus was ever in a habitable phase or was always a hellish world, with CO2 levels always beyond the runaway greenhouse point. The different stages of atmospheric evolution are depicted with the example of our Earth in Fig. 4, where the early stages of a possible primordial atmosphere as well as the primary outgassed atmosphere on top of a magma ocean spanned in reality a much shorter time frame than depicted in the figure (for better visibility). The long-term atmospheric evolution was shaped first by volcanic outgassing (secondary outgassed atmosphere), followed in the case of Earth by a fourth atmospheric type - the tertiary atmosphere shaped by photosynthetic life leading to the great oxidation event. From an observational point-of-view, however, it is not an easy task to differentiate between primordial or outgassed atmospheres, or any hybrid atmosphere stage in-between. One of the key elements in primordial atmospheres, however, is helium. In principle, the detection or non-detection of He in an exoplanet atmosphere would indicate if it is of primordial origin or not. However, detection of helium is highly challenging and currently restricted to exoplanets with an escaping atmosphere around stars with a specific radiative signature to make the helium visible. For now, theory therefore needs to predict how atmospheric types should differ for various planet masses and orbital configurations, including predictions on outgassing strengths from the interior of the planets (either during or after the magma ocean stage). For both the primary and secondary outgassed atmospheres, the interior dynamics, energy budget, and chemistry indeed play major roles in shaping the final atmospheric composition as well as atmospheric pressure. However, several external factors further influence the long-term evolution of the atmosphere, including the stellar insolation (influencing the temperature and extend of the atmosphere, and therefore thermally-driven escape processes), stellar flares and CMEs stripping part of the atmosphere (decreasing strongly with distance), as well as the twofold effect of impacts - ranging from late veneer addition of volatiles to an atmosphere (potentially strongly changing the composition of the atmosphere) to the destructive potential of a larger impactor, stripping part of the atmosphere from the planet rather than increasing the atmospheric volatile content. The main principles of atmospheric losses (including the importance of a magnetic field for some of the non-thermal loss processes) are a complex topic deserving their own review chapter. Here it should suffice to say that one of the main components for the survivability of an atmosphere is the composition of the atmosphere, where atmospheres with higher mean molecular weights (such as Venus' CO2-dominated atmosphere) are more resistant against atmospheric escape, whereas hot, extended atmospheres (such as H2-dominated atmospheres) are prone to atmospheric escape independent of the existence of a magnetic field due to thermal escape processes. The main factor influencing, however, the dominant species in the atmosphere (as long as it is not of primordial origin), is volcanic outgassing from the interior - depending not only on the volatile composition of the mantle, but also the redox state of the mantle rocks and melts from which volatiles are degassed at the planetary surface. The redox state strongly influences the gas speciation and solubility of volatiles in the mantle or melt. In addition, outgassing from surface magma is also limited by the atmosphere itself, more precisely by the partial pressures of individual gases in the atmosphere. For dense atmospheres, water and sulfur species may remain dissolved in the magma and not further contribute to the atmosphere, whereas other species such as CO2 or H2 degas easily and can build up very dense atmospheres. Which gases ultimately enrich an atmosphere is therefore not directly linked to the redox state of the melt (or at least not only), and measurements of atmospheric compositions therefore do not allow for a linear link to interior chemistry - though endmember atmospheres (e.g. very reducing or very oxidizing atmospheres) may shed a first light on the mantle chemistry and composition. It should also be mentioned that the atmosphere is of course also prone to changes by several additional surface and atmospheric processes, including atmosphere losses (e.g. loss of H2 oxidizing the remaining atmosphere), various chemical pathways in the atmosphere, equilibrium vs. disequilibrium considerations, condensation and formation of clouds or water oceans, weathering and chemical reactions at the surface, recycling of surface reservoirs, global feedback cycles such as the carbon-silicate cycle, and last but not least, potentially, due to the influence of life.", "pages": [ 16, 17, 18, 19 ] }, { "title": "Conclusion", "content": "The field of exoplanetary research has seen a tremendous change from first-order characterization of planets based on mass and radius measurements to the investigation of the complex and strongly interlinked evolutionary pathways of planetary interiors, surfaces and atmospheres by studying planets in the context of their environment (especially with respect to composition), interior dynamics (from accretion to long-term evolution of planets) and feedback links between the interior and the atmosphere. Future, more detailed atmospheric characterization surveys of planets in multiplanetary systems, as well as planets over large parameter spaces including stellar diversity, variable ages and different orbital configurations, will allow us to test our theoretical predictions and better understand the place of our own solar system planets within the exoplanetary context.", "pages": [ 19 ] }, { "title": "Acknowledgements", "content": "Funded by the European Union (ERC, DIVERSE, 101087755). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. C.D. acknowledges support from the Swiss National Science Foundation under grant TMSGI2_211313.", "pages": [ 19 ] } ]

[ { "title": "ABSTRACT", "content": "XXXII IAU General Assembly Proceedings IAU Symposium No. , eds. doi:10.1017/xxxxx", "pages": [ 1 ] }, { "title": "Revealing EMRI/IMRI candidates with quasiperiodic ultrafast outflows", "content": "Michal Zajaˇcek 1 , Petra Sukov'a 2 , Vladim'ır Karas 2 , Dheeraj R. Pasham 3 , Francesco Tombesi 4 , 5 , 6 , 7 , 8 , Petr Kurfurst 1 , Henry Best 1 , Izzy Garland 1 , Mat'uˇs Labaj 1 , Monika Pikhartov'a 1 1 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotl'aˇrsk'a 2, 611 37 Brno, Czech Republic 4 Physics Department, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133 Rome, Italy 5 INAF Astronomical Observatory of Rome, Via Frascati 33, 00040 Monte Porzio Catone, Italy 6 INFN-Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 7 Department of Astronomy, University of Maryland, College Park, MD 20742, USA 8 NASA Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA Abstract. The first detection of the quasiperiodic ultrafast outflow in the ASASSN-20qc system was reported by Pasham et al. (2024). The outflow is revealed in the soft X-ray spectra as an absorption feature, which is periodically enhanced every ∼ 8 . 3 days. The periodic nature of the ultrafast outflow is best explained by an orbiting massive perturber, most likely an intermediate-mass black hole (IMBH), that is inclined with respect to the accretion flow around the primary supermassive black hole (SMBH). In this way, the orbiting body pushes the disc gas into the outflow funnel, where it is accelerated by the ordered magnetic field (Sukov'a et al. 2021). Quasiperiodic ultrafast outflows (QPOuts) are thus a novel phenomenon that can help reveal new extreme-/intemediate-mass ratio inspiral (EMRI/IMRI) candidates that otherwise do not exhibit significant periodic patterns in the continuum flux density. Keywords. Supermassive black holes, Nuclear transients, Quasiperiodic ultrafast outflow, Extrememass/Intermediate-mass ratio inspirals", "pages": [ 1 ] }, { "title": "1. Quasiperiodic ultrafast outflows (QPOuts) in galactic nuclei", "content": "When searching for electromagnetic counterparts of large mass-ratio binary systems, in which a smaller body (a star or a compact object) orbits the supermassive black hole (hereafter SMBH), the focus has mainly been on quasiperiodic flares or outbursts (Miniutti et al. 2019; Linial & Metzger 2023; Zajaˇcek et al. 2024; Linial & Metzger 2024), hence significant increases in the continuum emission of the accreting nuclear sources that are of a deterministic rather than a stochastic variability nature. One of the most monitored candidate SMBH-binary systems is the blazar OJ287, which is characterized by optical flares repeating every ∼ 11 -12 years (Sillanpaa et al. 1988) and potentially also radio flares repeating every ∼ 20 -30 years (Britzen et al. 2018, 2023). Recently, quasiperiodic X-ray eruption sources (QPEs; Arcodia et al. 2021) turned out to be promising electromagnetic counterparts of extreme-mass/intermediate-mass ratio inspiral systems (EMRIs/IMRIs; Franchini et al. 2023; Kejriwal et al. 2024) that are expected to be detected using low-frequency gravitational-wave observations (between ∼ 0 . 1 mHz and ∼ 1 Hz), such as Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2012). Recently, Sukov'a et al. (2021) pointed out that when a secondary, smaller body with an influence radius of about a gravitational radius ( r g = GM · / c 2 ) traverses through the accretion flow at a higher inclination, not only is the accretion rate periodically perturbed but also the outflow rate. This is caused by the fact that the gas around the perturber within the influence radius is dragged along its orbit and starts comoving with the body. As the perturber rises above the accretion flow, so does a fraction of the matter from the disc. Due to the ordered poloidal magnetic field, this matter is accelerated away from the SMBH and eventually moves at relativistic velocities of ∼ 0 . 2 -0 . 3 of the light speed. Moreover, the periodicity in the outflow rate was found to be significant even if the periodicity in the inflow rate was rather weak or insignificant, for instance due to a small cross-section of the perturber or its larger distance from the SMBH. If the increase in the outflow rate is associated with a partially ionized gas, this can observationally be manifested by quasiperiodic enhancements in the absorption of the underlying accretion-disc continuum emission. Hence, instead of recurrent peaks in the continuum, one would detect repetitive drops at specific energies. This is quite a different, yet realistic, concept in comparison to the canonical view of how secondary bodies surrounding SMBHs can interact with the circumnuclear gaseous-dusty medium. The series of perturber-accretion disc interactions studied by Sukov'a et al. (2021) using 2D and 3D GRMHD simulations turned out to be useful for the interpretation of the peculiar source associated with the tidal disruption event (TDE) revealed by the optical flare ASASSN20qc (Pasham et al. 2024a). The optical flare ASASSN-20qc/Gaia21alu/AT2020adgm detected on December 20, 2020 has been associated with the galaxy at z = 0 . 056 (250 Mpc). For this galaxy, the spectroscopic measurements using broad emission lines as well as the broadband photometry imply the SMBH mass of M · = 3 + 5 -2 × 10 7 M ⊙ , i.e. we consider the range of M · = 10 7 -10 8 M ⊙ for further estimates. Based on eROSITA non-detections of the source in January and July 2020, the upper limit on the X-ray luminosity L X < ∼ 6 × 10 40 erg s -1 implies the Eddington ratio (relative accretion rate) of ˙ m = ˙ M / ˙ M Edd < ∼ 5 × 10 -4 before the optical outburst, where we assumed the radiative efficiency of η rad = 0 . 1 and the bolometric correction of κ bol = 10. Therefore, the inner accretion flow was initially optically thin and geometrically thick hot flow dominated by advection (advection-dominated accretion flow or ADAF), which transitioned into the thin, optically thick disc at r ADAF > ∼ 2200 α 0 . 8 0 . 1 β -1 . 08 ˙ m -0 . 53 0 . 0005 r g , where the limiting radius is set by evaporation due to heat conduction between the standard cold disc and hot two-temperature corona (Czerny et al. 2004). The quantities involved in r ADAF include the viscosity parameter α (scaled to 0.1), the magnetization parameter β ≡ P g / ( P g + P m ) , which stands for the ratio between the gas pressure and the total pressure ( β = 1 for the negligible magnetic field). The presence of the standard disc on larger scales is revealed by the detection of broad emission lines of H α and H β , which are dynamically related to the thin disc (broadline emitting clouds form a flattened structure; there is a significant correlation between the broad-line radius and the disc monochromatic luminosity; Czerny & Hryniewicz 2011). About 52 days after the optical outburst, the prominent increase in the X-ray emission was detected by the Swift telescope , reaching the 0.3-1.1 keV luminosity of 5 × 10 43 erg s -1 . Hence, the source soft X-ray luminosity increased by more than three orders of magnitude. At this time, the source started to be intensively monitored by the NICER and the spectra were also obtained using the XMM-Newton telescope . After about 100 days, the X-ray luminosity of the source dropped by two orders of magnitude to ∼ 3 × 10 41 erg s -1 . During the X-ray emission peak, the X-ray spectra were characterized by a thermal accretion-disc emission between 0.30 and 0.55 keV with kT disc ∼ 85 eV ( T disc ∼ 10 6 K) with a broad absorption feature between 0.75 and 1.00 keV. This broad absorption feature is consistent with an ultrafast outflow moving at the velocity of ∼ 0 . 3 c . While the thermal X-ray continuum variability (0 . 30 -0 . 55 keV) is clearly stochastic, the variability of the absorption band (0 . 75 -1 . 00 keV) or rather the ratio of the flux at 0 . 75 -1 . 00 keV to the flux at 0 . 30 -0 . 55 keV, which is denoted as the outflow deficit ratio (ODR ≡ F 0 . 75 -1 . 00 / F 0 . 30 -0 . 55 )†", "pages": [ 1, 2 ] }, { "title": "Weighted Wavelet z -transform", "content": "shows a significant periodic behaviour with the period of 8 . 3 days (Pasham et al. 2024a). This period is recovered when applying the Lomb-Scargle periodogram, the phase-dispersion minimization, and the weighted wavelet z -transform (see Fig. 1) to the temporal evolution of the ODR during 12 cycles, which confirms the robustness of the QuasiPeriodic ultrafast Outflow or the QPOut feature in this source.", "pages": [ 3 ] }, { "title": "2. Dynamical interpretation", "content": "To interpret the periodic ultrafast outflow that is responsible for the absorption at 0.751.00 keV, different scenarios are considered, ranging from purely geometric effects, accretiondisc instabilities, to orbiting bodies. In addition to the periodicity, one has to consider basic properties of the outflow as inferred from photoionization models of the X-ray spectra. During the ODR minima, the column density is larger (log [ N h ( cm -2 )] ∼ 22) in comparison with the ODR maxima (log [ N h ( cm -2 )] ∼ 21). The ODR minima are also characterized by a larger ionization parameter. In contrast, there is no significant difference in terms of the outflow velocity, which remains close to ∼ 0 . 3 c . We provide here a brief overview including the notes and the references as well as the information whether the model is (dis)favoured considering all the observables: · inner disc precession: we would expect modulations of the continuum flux with the changes of the viewing angle during the precession period, i.e. F X ∝ cos ι , see also Pasham et al. (2024b); disfavoured ✗ ,", "pages": [ 3 ] }, { "title": "3. QPOuts induced by an orbiting body: black holes vs. stars", "content": "In the model setup, we assume that the clumps that are further accelerated by the ordered, poloidal magnetic field can cause significant absorption of the underlying thermal continuum only when the orbiting body rises above the thermally emitting disc. Along with the perturber, the gas is pushed towards the observer and blocks a fraction of the thermal continuum, which presumably originates in the compact disc formed due to the TDE (see Fig. 2 for an artistic illustration of the setup). In this way, the detected QPOut period approximately coincides with the orbital period of the body, P QPOut ≃ P orb ∼ 8 . 0 days, where we took into account the redshift of the source to get the rest-frame period of the QPOuts/perturber. Scaling the primary SMBH mass to 10 7 . 5 M ⊙ , we can constrain the expected distance/semi-major axis of the perturbing body a per in gravitational radii of the primary SMBH, Since before the TDE-like optical outburst the ASASS-20qc host was low-luminous, the inner accretion flow was most likely a hot ADAF up to the distance of a few 1000 r g. The TDE brought stellar material, from which a compact, thermally emitting disc formed on the scale of a tidal radius, which is about one order of magnitude less than the orbital distance of the perturber, assuming the stellar mass of m ⋆ = 1 M ⊙ . Hence, it is quite plausible that the orbiting body crosses a hot flow rather than the thin disc, and the ejected gas blobs periodically obscure the thermal continuum emitted by the inner post-TDE disc as is shown in Fig. 3, which illustrates the likely distribution of the circumnuclear material within the composite accretion flow in the ASASS-20qc host. Different options have been discussed in the literature regarding the nature of the orbiter: a giant star with an extended atmosphere that becomes gradually ablated by the surrounding environment (Zurek et al. 1992; ˇ Subr et al. 2004; Linial & Metzger 2023), confined core-less clouds (Mathews 1993), a magnetic star (e.g. a strongly magnetized neutron star or Ap star), where the interaction radius is defined by the extent of its magnetosphere that can reach well above the direct geometrical cross-section (Lipunov 1992; Toropina et al. 2012; Zajaˇcek et al. 2015; Karas et al. 2017; Kop'aˇcek et al. 2018), and an intermediate-mass black hole (IMBH) that is the most compact possibility, whereas the interaction cross-section is sufficiently large due to its large mass (i.e. super-solar, albeit less than SMBH; Pasham et al. 2024a). The effective cross-sectional area for the interaction between the body of a tentative perturber and the gaseous/dusty medium is the crucial aspect that defines the efficiency of the mutual interaction between the accretion flow and the transiting body. In fact, the effects of magnetic fields with an organized (large-scale) component permeating the accretion disc corona or a jet play an important role (as suggested in various simulations and indicated by polarimetric observations). Until the imaging resolution reaches ∼ microarcsecond level (as in e.g. GRAVITY Collaboration et al. 2018; Event Horizon Telescope Collaboration et al. 2021, 2024), the most likely scenario has to be inferred from supplementary evidence consistent with the timing and spectral properties of the system under discussion. In the context of ASASSN-20qc we have pursued the latter cited scheme (IMBH perturber) as the most appealing possibility that is in agreement with currently available observational evidence. To infer the required influence radius of the perturber, we consider the column density of the outflow N h ∼ 2 × 10 22 cm -2 during the ODR minima. Here we consider the interaction of the orbiter with the ADAF flow with the number density n ADAF. The length-scale of the ejected clumps can be estimated as R clump ≈ f g N h / n ADAF, where f g is the geometrical/dynamical factor related to the clump shape and evolution shortly after the ejection. Considering the number density for the ADAF at a given distance, n ADAF ≃ 5 . 02 × 10 9 ( α / 0 . 1 ) -1 ( M · / 10 7 . 5 M ⊙ ) -1 ( ˙ m / 0 . 01 )( r / 79 . 4 r g ) -1 cm -3 (Yuan & Narayan 2014), we can estimate the typical clump radius in gravitational radii, In order to eject clumps of the size R clump , the influence radius of the perturber should be comparable. Therefore, for the following estimates, we set R inf ≈ 1 r g. The mass of the compact (inert) orbiter can be constrained by the tidal (Hill) radius, R Hill = r per [ m per / ( 3 M · )] 1 / 3 which stands for the distance range from the body where it dominates over the SMBH gravitational influence. This yields the Hill mass of, Another constraint can be obtained from the Bondi (gravitational capture) radius, R B = 2 Gm per / ( v 2 rel + c 2 s ) , which expresses the distance from the perturber, where it dominates over the thermal motion of the surrounding ADAF, where f s ≈ 2 . 8 × 10 20 cm 2 s -2 is the normalization constant related to the sound speed in the ADAF (Yuan & Narayan 2014). Hence, the length-scales associated with the gravitational attraction of the orbiting body imply that it is a black hole in the intermediate-mass range between ∼ 10 2 M ⊙ and ∼ 10 5 M ⊙ . An independent constraint or rather an upper limit on the IMBH mass is given by the gravitational-merger timescale τ gw (Peters 1964). It can be argued that τ gw should be at least comparable to or longer than the timescale related to the average TDE rate per galaxy, which is τ TDE > ∼ 10 4 years (Stone & Metzger 2016). Otherwise the occurrence of the TDE during the lifetime of the SMBH-IMBH system would be very unlikely. This statistical argument yields the upper limit on the IMBH mass, The constraints on τ TDE for the ASASSN-20qc host galaxy, which can be inferred from the photomery and the surface-brightness distribution of the stellar population (recent starburst; a core-like or a cusp-like nuclear stellar cluster; Stone & van Velzen 2016), could thus help constrain the IMBH mass range. In particular, the indication for a longer τ TDE would result in narrowing down the mass range for the IMBH. The constraints given by the required influence radius in combination with the gravitational effect on the surrounding gas, Eq. (4), and the long enough gravitational merger timescale of the SMBH-IMBH system, Eq. (6), yield the IMBH mass in the range between ∼ 10 2 M ⊙ and ∼ 10 4 M ⊙ . Concerning an alternative explanation involving a normal star, it would need to reach a stellar radius of R ⋆ ≈ R inf ≃ 1 r g ∼ 67 ( M · / 10 7 . 5 M ⊙ ) R ⊙ , hence it would need to be an evolved, late-type star. However, given the tidal (Hill) radius for 1 M ⊙ star orbiting the SMBH of M · = 10 7 . 5 M ⊙ every 8 days, such a large star would get disrupted during one orbital period. However, for lighter SMBHs in other galaxies, the star could be tidally stable and at the same it would have the influence radius of ∼ 1 r g. The condition R Hill > ∼ R ⋆ ∼ 1 r g yields the limiting SMBH mass M lim · , below which stellar and black-hole perturber regimes are both plausible, while above it, black-hole orbiters perturbing the accretion flow in a significant manner are more plausible. Considering the comparable periodicity as in ASASSN-20qc, we can estimate M lim · as For M lim · , the star with the influence radius of 1 r g would have the physical radius of R ⋆ ∼ 11 . 6 R ⊙ . In these estimates, we have not considered hydrodynamical effects, such as strong stellar winds, however, for the accretion rate of ˙ m ∼ 0 . 05, they do not yield large enough kinetic pressure to increase the influence radius of stars above their physical radii. In Fig. 4, we plot the influence radius in Solar radii as a function of the SMBH mass, considering R inf = 1 r g. The separation of stellar and black-hole perturber regimes around 10 6 . 74 M ⊙ is depicted by plotting tidal (Hill) radius for m ⋆ = 1 M ⊙ and at the same time tidal (Hill)/Bondi radii for black-hole perturbers with m per = 10 -10 5 M ⊙ (gray-shaded region and black lines); see also Pasham et al. (2024a) for details. For completeness, we also show expected stellar bow-shock sizes in both the ADAF and the standard disc for the parameters listed in the legend.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "4. Implications for EMRI/IMRI systems", "content": "So far the studies of electromagnetic counterparts of EMRI/IMRI systems focused mostly on the flux density enhancements (flares, eruptions) due to inclined smaller bodies crossing an accretion disc, producing shocks emitting mostly in soft X-ray/UV domains. The seminal study of Sukov'a et al. (2021) showed that perturbing bodies with a sufficient cross-section (compact remnants or stars) could actually affect the nuclear outflow rate more visibly than the inflow rate, especially if there are located at a relatively larger distance from the primary massive black hole. Observationally, such an effect can be revealed by the periodically launched ultrafast outflow (QPOut) resulting in the enhanced absorption of the underlying disc continuum, as was shown to be the likely cause of the observed spectral changes in the soft X-ray domain in the ASASSN-20qc host (Pasham et al. 2024a). The tidal (Hill) radius of R inf ≈ 1 r g puts a lower limit on the perturbing body mass of m per > ∼ 100 M ⊙ . The statistical argument that the merger timescale of the system should be longer than the timescale associated with the typical TDE rate per galaxy put an upper limit on the perturber mass of m per < ∼ 10 4 M ⊙ . Hence, the best dynamical explanation of the QPOut in the ASASSN-20qc host appears to be an IMBH orbiting close to ∼ 100 r g on a moderately eccentric orbit. This can address (i) the QPOut as well as (ii) no significant periodic variability of the inflow/accretion rate. The constraints on the perturber mass depend on the primary SMBH mass. It can be shown that for the required influence radius of the perturber of ∼ 1 r g, perturbers can also be normal stars if the primary SMBH mass is < ∼ 10 6 . 74 M ⊙ . This has consequences for the interpretation of nuclear quasiperiodic photometric and spectral changes in other hosts. Another relevant aspect of the study of the ASASSN-20qc event are the implications for the origin of IMBHs. Since they are only very few confirmed cases of IMBHs (Greene et al. 2020), the QPOuts provide a novel observational way to look for their fingerprints. The broad mass range for the putative IMBH in the ASASSN-20qc host (10 2 M ⊙ < ∼ m per < ∼ 10 4 M ⊙ ) is consistent with the following three proposed scenarios for the origin of SMBH-IMBH binaries: · infall of a globular/star-forming cluster hosting an IMBH with a certain occupancy The merger rates of stellar black holes (involving both compact remnants and stars) are expected to be greatly enhanced in the zones called migration traps where the smaller objects tend to accumulate due to the boundary between the inward and the outward migration of objects due to gas torques within the accretion disc (McKernan et al. 2012, 2014). The inferred distance of the IMBH in ASASSN-20qc ( ∼ 100 r g) roughly coincides with the distance range for such a migration trap ( ∼ 40 -600 r g; Bellovary et al. 2016). The interaction of an orbiting body with the standard accretion disc will make it aligned with the disc plane due to efficient hydrodynamical drag on the timescale of ∼ 10 4 years (Syer et al. 1991; Pasham et al. 2024a), assuming the parameters comparable to the ASASSN-20qc system. If theperturber embedded within the disc is massive enough, of the order of 10 -2 of the primary SMBH mass, it opens a gap in the disc (Gultekin & Miller 2012; ˇ Stolc et al. 2023). The gap presence is imprinted in the spectral energy distribution (SED) of the nuclear source, specifically it leads to flux density depressions at specific wavelengths (from the UV to the optical bands) corresponding to the distance of the perturber from the SMBH ( ˇ Stolc et al. 2023). In comparison with inclined orbiting bodies, the aligned, embedded perturber thus creates a quasistationary effect, which can be revealed in one single-epoch SED with the measurements spaced closely in time.", "pages": [ 9, 10 ] } ]

[ { "title": "Magnetic massive stars: confirming the merger scenario for the magnetic field generation", "content": "Swetlana Hubrig 1 , Markus Scholler 2 , Silva P. Jarvinen 1 , Aleksandar Cikota 3 , Michael Abdul-Masih 4 , Ana Escorza 4 , Ilya Ilyin 1 1 Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany email: [email protected], [email protected], [email protected] 2 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany email: [email protected] 3 Gemini Observatory / NSF's NOIRLab, Casilla 603, La Serena, Chile email: [email protected] 4 Instituto de Astrof´ısica de Canarias, C. V´ıa L´actea s/n, 38205 La Laguna, Santa Cruz de Tenerife, Spain email: [email protected], [email protected] Abstract. Magnetic fields are considered to be key components of massive stars, with a farreaching impact on their evolution and ultimate fate. A magnetic mechanism was suggested for the collimated explosion of massive stars, relevant for long-duration gamma-ray bursts, X-ray flashes, and asymmetric core collapse supernovae. However, the origin of the observed stable, globally organized magnetic fields in massive stars is still a matter of debate: it has been argued that they can be fossil, dynamo generated, or generated by strong binary interactions or merging events. Taking into account that multiplicity is a fundamental characteristic of massive stars, observational evidence is accumulating that the magnetism originates through interaction between the system components, both during the initial mass transfer or when the stellar cores merge. Keywords. stars: atmospheres, stars: early-type, stars: magnetic field, stars: winds", "pages": [ 1 ] }, { "title": "1. Introduction: Scientific background", "content": "Unlike Sun-like stars, which possess magnetic fields with complex topologies generated through a dynamo process sustained by a convective dynamo operating close to the surface of the star, magnetic massive stars host large-scale organized fields with simple topologies usually described by a magnetic dipole tilted to the rotation axis. Stellar winds in massive magnetic stars are confined by the magnetic field building dynamical or centrifugal magnetospheres: slowly rotating stars for which the Alfv'en radius R A is smaller than the Kepler radius R K build dynamical magnetospheres and more rapidly rotating stars with R A > R K build centrifugal magnetospheres (Petit et al. 2013). In Fig. 1 we present the first image of the dynamical magnetosphere around a typical magnetic O star, the slowly rotating ( P rot = 7 . 4yr) O star HD 54879 with a dipolar magnetic field of kG-order. This image is produced using monitoring of the H α profile over 9.7 yr (Kuker et al. 2024). Studies of the magnetic characteristics of massive stars have recently received significant attention because these stars are progenitors of highly magnetised compact objects. Stars initially more massive than about 8 M /circledot leave behind neutron stars and black holes by the end of their evolution. The merging of binary compact remnant systems produces astrophysical transients detectable by the gravitational wave observatories LIGO, Virgo, and KAGRA. Also studies of magnetic fields in massive stars in a low metallicity environment are of particular interest because they provide important information on the role of magnetic fields in the star formation of the early Universe. The recent discovery of magnetic fields of kG-order in three massive O-type targets in the Magellanic Clouds, two stars with spectral classification Of?p and one overcontact binary, suggests that the impact of low metallicity on the occurrence and strength of magnetic fields in massive stars cannot be strong (Hubrig et al. 2024). Despite the progress achieved in previous surveys of the magnetism in massive stars, the origin of their magnetic fields remains to be the least understood topic. It has been argued that magnetic fields could be fossil, dynamo generated, or generated by strong binary interaction. The currently most popular theoretical scenarios developed to explain the origin of magnetic fields in massive stars involve a merger event or a common envelope evolution (e.g. Tout et al. 2008; Ferrario et al. 2009; Schneider et al. 2016). Mass transfer or stellar merging may rejuvenate the mass gaining star, while the induced differential rotation is thought to be the key ingredient to generate a magnetic field (e.g. Wickramasinghe et al. 2014). Such interaction can take place already in the early evolutionary stages in star forming regions where accretion-driven migration as well as tides induced by the dense circumbinary material can lead to shrinking orbits and merging. In the context of this scenario, O stars are of special interest to support these theoretical considerations as they form nearly exclusively in multiple systems, with more than 90% born and living in such systems (Offner et al. 2023). Moe & di Stefano (2017) reported that the O-type multiplicity fraction is 6 +6 -3 % for single stars, 21 ± 7% for binary systems, 35 ± 3% for triple systems, and 38 ± 11% for quadruple systems, meaning that about 73% of O stars are members of multiple systems. According to these authors, the mechanisms required to bring the inner binary systems in multiple systems to shorter periods include migration through a circumbinary disk due to hydrodynamical forces, dynamical interactions in an initially unstable hierarchical multiple system, or secular evolution caused by the so-called Kozai-Lidov mechanism (Kozai 1962; Lidov 1962). Sana et al. (2012) reported that in the course of the stellar evolution 71 ± 8% of O-type stars will interact with companions in the systems with mass ratios q > 0.1 via Roche lobe overflow. This fraction should be even larger if the variations between orbital periods, q, and eccentricities are taken into account (Moe & di Stefano 2017).", "pages": [ 1, 2, 3 ] }, { "title": "2. Recent developments", "content": "Curiously, in spite of the fact that the evolution of massive stars is highly affected by interactions in binary and multiple systems, previous surveys of the magnetism in massive stars indicated that the incidence rate of magnetic components in close binaries with P orb < 20d is very low: of the dozen O stars with confirmed magnetic fields, only one short-period binary system, Plaskett's star, contains a hot, massive, magnetic star (Grunhut et al. 2013). However, it must be admitted that several factors hinder magnetic field detections in multiple systems and should be taken into account: the amplitudes of the Zeeman features (features appearing in the Stokes V spectra of magnetic stars) are much lower in multiple systems in comparison to their size in single stars. These features also appear blended in composite spectra and show severe shape distortions. Also the shapes of blended spectral lines in the Stokes I spectra look different depending on the visibility of each system component at different orbital phases. Given the much lower number of spectral lines in O stars in comparison to less massive stars, to measure their magnetic fields, the least-squares deconvolution (LSD) technique (Donati et al. 1997) is frequently applied. However, special care has to be taken to populate the LSD line masks for each system because the composite spectra of multiple systems usually show very different spectral signatures corresponding to the different spectral classification of the individual components. Importantly, the first analysis of spectropolarimetric ESO and CFHT archival observations of 36 binary and multiple systems with O-type primaries using a special procedure involving different line masks populated for each element separately showed encouraging results (Hubrig et al. 2023): out of the 36 systems, 22 exhibited in their least-squares deconvolution Stokes V profiles definitely detected Zeeman features. For 14 systems, the detected Zeeman features have been associated with O-type components, whereas for 3 systems they were associated with an early B-type component. This survey included systems at very different evolutionary stages, from young main-sequence systems to a few evolved systems with blue supergiants, Wolf-Rayet stars, one system with a Luminous Blue Variable candidate, and one post-supernova X-ray binary. Seven systems with definite field detections are known as particle-accelerating colliding-wind binaries exhibiting synchrotron radio emission. A few examples of binary and multiple systems with discovered magnetic fields are presented in Fig. 2. Strong indications that interaction between the components in binary and multiple systems can be important for the generation of magnetic fields comes also from a recent study of the massive Of?p binary HD 148937 (Frost et al. 2024), in which the primary is suggested to be a merger product. Numerical simulations also indicate the important role of stripped-envelope stars formed in systems with interacting components through one or multiple phases of Roche-lobe overflow (e.g. Yoon et al. 2017). Magnetic studies of such systems are of special interest because these systems occupy the mass range that has been predicted to produce most stripped-envelope supernovae or neutron star mergers like those that emit gravitational waves (Tauris et al. 2017; Gotberg et al. 2023). Recent detections of magnetic fields in stripped stars have been reported by Hubrig et al. (2022) for υ Sgr and by Shenar et al. (2023) for HD 45166. Perhaps the most interesting piece of observational evidence for the important role of component interaction for the generation of magnetic fields comes from the detection of rather strong longitudinal magnetic fields in three massive overcontact binaries. In our Galaxy, magnetic fields have been detected in HD 35921 (=LY Aur) and HD 167971 (=MYSer) (Hubrig et al. 2023). Most recently, a magnetic field of kG-order has been reported for Cl* NGC 346SSN 7 in the Small Magellanic Clouds (Hubrig et al. 2024). This was the first reported magnetic field in an extragalactic overcontact binary sys- tem. In Fig. 3 we present the recent discovery of the magnetic field in the system Cl*NGC346SSN7 based on low-resolution spectropolarimetric observations using the FORS2 instrument attached to an ESO VLT 8m telescope. The mean longitudinal magnetic field from the FORS 2 observations is usually determined as the slope of a weighted linear regression through the measured Stokes V values. The detection of magnetic fields in a representative number of overcontact systems, which directly precede the merger event, would give further credence to the theoretical scenario presented by Schneider et al. (2019), who carried out three-dimensional magnetohydrodynamical simulations of the coalescence of two massive stars.", "pages": [ 3, 4, 5 ] }, { "title": "3. New directions", "content": "The gradually increasing number of systems with definitely detected magnetic fields indicates that multiplicity plays a crucial role in the generation of magnetic fields in massive stars. A few systems with magnetic components studied by Hubrig et al. (2023) have recently been re-observed with HARPSpol confirming their magnetic nature. In Fig. 4 we show as an example previous and new observations of the triple system HD 167971. As most of the targets have been observed with spectropolarimetry only once or twice, the newly found magnetic systems are supreme candidates for spectropolarimetric monitoring over their orbital and rotation periods. The number of spectropolarimetric observations of massive stars is still too low to allow us to confirm current theories and simulations developed to explain the origin of magnetic fields in massive stars. In view of the mounting evidence for the importance of studying magnetic fields in massive binary and multiple systems, it is crucial to obtain trustworthy statistics on the magnetic field incidence, the magnetic field structure, and the field strength distribution in systems with magnetic components. Because not all magnetic components have known rotation periods, spectropolarimetric time-series are needed to determine the field structure. We also need to know what fraction of O stars are magnetic and what is the field structure across the different evolutionary stages. On the other hand, this requires the determination of fundamental properties and the evolutionary history of the magnetic components in binary and multiple systems. Only then can we try to understand the impact of the generated magnetic fields on the further evolution and the final fate of massive stars as supernovae and compact objects.", "pages": [ 5 ] }, { "title": "References", "content": "Donati, J.-F., Semel, M., Carter, B.D., Rees, D.E., & Collier Cameron, A. 1997, MNRAS , 291, 658 Ferrario, L., Pringle, J.E., Tout, C.A., & Wickramasinghe, D.T. 2009, MNRAS , 400, L71 Frost, A.J., Sana, H., Mahy, L., et al. 2024, Science , 384, 214 Gotberg, Y., Drout, M.R., Ji, A.P., et al. 2023, ApJ , 959, 125 Grunhut, J.H., Wade, G.A., Leutenegger, M., et al. 2013, MNRAS , 428, 1686 Hubrig, S., Jarvinen, S.P., Ilyin, I., & Scholler, M. 2022, ApJ , 933, 27 Hubrig, S., Jarvinen, S.P., Ilyin, I., Scholler, M., & Jayaraman, R. 2023, MNRAS , 521, 6228 Hubrig, S., Jarvinen, S.P., Scholler, M., et al. 2024, A&A , 684, L4 Ibanoglu, C., C¸ akırlı, O., & Sipahi, E. 2013, MNRAS , 436, 750 Jarvinen, S.P., Hubrig, S., Scholler, M., et al. 2022, MNRAS , 510, 4405 Kozai, Y. 1962, AJ , 67, 591 Kuker, M., Jarvinen, S.P., Hubrig, S., Ilyin, I., & Scholler, M. 2024, Astr. Nachr. , 345, e20230169 Lidov, M.L. 1962, Planetary & Sp. Sci. , 9, 791 Mahy, L., Rauw, G., Martins, F., et al. 2010, ApJ , 708, 1537 Mayer, P., Harmanec, P., Chini, R., et al. 2017, A&A , 600, A33 Moe, M., & di Stefano, R. 2017, ApJS , 230, 15 Offner, S.S.R., Moe, M., Kratter, K.M., et al. 2023, ASPC , 534, 275 Petit, V., Owocki, S.P., Wade, G.A., et al. 2013, MNRAS , 429, 398 Rosu, S., Rauw, G., Naz'e, Y., Gosset, E., & Sterken, C. 2022, A&A , 664, A98 Sana, H., de Mink, S.E., de Koter, A., et al. 2012, Science , 337, 444 Schneider, F.R.N., Podsiadlowski, Ph., Langer, N., Castro, N., & Fossati, L. 2016, MNRAS , 457, 2355 Schneider, F.R.N., Ohlmann, S.T., Podsiadlowski, Ph., et al. 2019, Nature , 574, 211 Shenar, T., Wade, G.A., Marchant, P., et al. 2023, Science , 381, 761 Tauris, T.M., Kramer, M., Freire, P.C.C., et al. 2017, ApJ , 846, 170 Tout, C.A., Wickramasinghe, D.T., Liebert, J., Ferrario, L., & Pringle, J.E. 2008, MNRAS , 387, 897 Wickramasinghe, D.T., Tout, C.A., & Ferrario, L. 2014, MNRAS , 437, 675 Yoon, S.-C., Dessart, L., & Clocchiatti, A. 2017, ApJ , 840, 10", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Although connections between flaring blazars and some IceCube neutrinos have been established, the dominant sources for the bulk extragalactic neutrino emissions are still unclear and one widely suggested candidate is a population of radio galaxies. Because of their relatively low gamma-ray radiation luminosities ( L γ ), it is rather challenging to confirm such a hypothesis with the neutrino/GeV flare association. Here we report on the search for GeV gamma-ray counterpart of the neutrino IC-180213A and show that the nearby ( z = 0.03) broad line radio galaxy 3C 120 is the unique co-spatial GeV γ -ray source in a half-year epoch around the neutrino detection. Particularly, an intense γ -ray flare, the second strongest one among the entire 16-year period, is temporally coincident with the detection of IC-180213A. Accompanying optical flare is observed, too. We also find that the IC-180213A/3C 120 association well follows the L γ -D 2 L correlation for the (candidate) neutrino sources including NGC 1068 and some blazars. These facts are strongly in favor of 3C 120 as a high energy neutrino emitter and provide the first piece of evidence for the radio galaxy origin of some PeV neutrinos. Keywords: neutrino astronomy; active galactic nuclei; Gamma-ray sources", "pages": [ 1 ] }, { "title": "A nearby FR I type radio galaxy 3C 120 as the PeV neutrino emitter", "content": "Rong-Qing Chen, 1 Neng-Hui Liao, 1 Xiong Jiang, 2, 3 and Yi-Zhong Fan 2, 3 1 Department of Physics and Astronomy, College of Physics, Guizhou University, Guiyang 550025, China Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, 3 School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Neutrinos, which interact weakly with matter, can escape from extreme astrophysical environments that are impenetrable to electromagnetic radiations and hence provide unique insights into these intriguing astrophysical environments. The IceCube neutrino observatory at the South Pole 1 has discovered some high-energy neutrinos of astrophysical origin (e.g., IceCube Collaboration 2013; Aartsen et al. 2015; Icecube Collaboration et al. 2023). These neutrinos were produced through the interaction of ultra-high energy cosmic rays (UHECRs) with ambient matter (i.e., p -p ) or radiation fields (i.e., p -γ ), hence are crucial for revealing the origin of the UHECRs. The production of such neutrinos will generate electromagnetic radiation, in particular γ -ray photons, as well. The proposed Corresponding author: Neng-Hui Liao, Yi-Zhong Fan [email protected], [email protected] 1 http://icecube.wisc.edu gamma-ray emitters for extragalactic high energy neutrinos include the blazars (Atoyan & Dermer 2001), starburst galaxies (Loeb & Waxman 2006), radio galaxies (Becker Tjus et al. 2014), and galaxy clusters (Murase et al. 2008). The neutrino/gamma-ray association is a powerful tool to probe the origin of the IceCube events. So far, the blazar as well as the very nearby Seyfert II galaxy have been confirmed to be the sources of the PeV neutrinos (IceCube Collaboration et al. 2018a, 2022). The advantage of establishment of a connection between a flaring blazar and an incoming neutrino is that both the spatial and temporal information can be utilized. Cases similar with TXS 0506+056/IceCube-170922A are mounting (e.g., Kadler et al. 2016; Garrappa et al. 2019; Giommi et al. 2020; Franckowiak et al. 2020; Liao et al. 2022; Sahakyan et al. 2023; Jiang et al. 2024). However, the fraction of contribution from blazars to the total observed neutrino flux is proposed to be limited, since no significant cumulative neutrino excess are found from the Fermi blazar directions (Aartsen et al. 2017). Recently, the IceCube Event Catalog of Alert Tracks (i.e., ICECAT-1) has been released (Abbasi et al. 2023). A 0.111 PeV track-like neutrino event (i.e., IC-180213A, a so-called bronze event), in an arrival directi eventon of R.A. 66 . 97 +2 . 46 -2 . 59 · and decl. 6 . 09 +1 . 95 -1 . 72 · (at a 90% confidence level), is reprorted in ICECAT-1. Interestingly, a nearby Fanaroff-Riley (FR, Fanaroff & Riley 1974) I broad line radio galaxy (BLRG) 3C 120 falls into the localization uncertainty region of the neutrino. It is a well-studied nearby source with z = 0.033 (Burbidge 1967), ≃ 145 Mpc away, hosting an efficiently accreting black hole with a mass of 5 . 5 × 10 7 M ⊙ by the reverberation mapping approach (Peterson et al. 2004). Radio Observations with the Very Long Baseline Array (VLBA) reveal superluminal components with apparent speed up to > 10c (Lister et al. 2016) and the jet inclination angle to the line of sight is constrained to be ∼ 10 -20 · , thus a Doppler factor of ∼ 2.4 as well as a bulk Lorentz factor of ∼ 5 are suggested (Jorstad et al. 2005; Hovatta et al. 2009). 3C 120 is famous for the strong variability from radio to GeV γ rays, with which an accretion-disk-jet connection has been established (Marscher et al. 2002; Abdo et al. 2010a; Kataoka et al. 2011). In this letter, we thoroughly investigate the multiwavelength data of 3C 120, to explore its potential association with the neutrino event IC-180213A. Data Analyses are presented in Section 2; while discussions and a short summary are given in Section 3. We adopt a ΛCDM cosmology with Ω M = 0.32, Ω Λ = 0.68, and a Hubble constant of H 0 = 67 km -1 s -1 Mpc -1 (Planck Collaboration et al. 2014).", "pages": [ 1, 2 ] }, { "title": "2.1. Fermi-LAT Data", "content": "We collected the first 16-year (i.e., MJD 54683 - 60443, or from 2008 August 4 to 2024 August 4) Fermi-LAT Pass 8 SOURCE data ( evclass = 128 and evtype = 3). Energy range of the data was selected between 100 MeV and 500 GeV. The Fermitools software version 2.2.0 was adopted for the data analysis, along with Fermitools-data version 0.18. In the initial data filtering procedure, a zenith angle cut (i.e., < 90 · ) was set to avoid significant contaminations from the earth limb, meanwhile, the recommended quality-filter cuts (i.e., DATA QUAL==1 && LAT CONFIG==1 ) were applied. Unbinned likelihood analysis was performed by the gtlike task to extract the γ -ray flux and spectrum. We used the test statistic (TS = -2ln( L 0 / L ), Mattox et al. 1996) to determine the significance in the γ -ray detection. L and L 0 correspond to the maximum likelihood values for the models with and without the target γ -ray source, respectively. During the likelihood analysis, a region of interest (ROI) of 10 degrees centered on the coordinates of 4FGL J0433.0+0522, for which the low-energy counterpart is 3C 120, was set. Parameters of the 4FGL-DR4 background source (Ballet et al. 2023) within the ROI, as well as those for the diffuse emission templates (i.e., gll iem v07.fits and iso P8R3 SOURCE V3 v01.txt ) were left free, while other parameters were frozen as the default values. In addition, the residual TS maps were produced in which potential γ -ray sources not included in the 4FGL-DR4 were checked. The model was updated by embracing these sources and the likelihood analyses were then re-performed. In the temporal analysis, weak background sources with TS < 10 were removed from the model. Meanwhile, 95% CL upper limits were obtained by the pyLikelihood UpperLimits tool, to replace the flux estimations. An analysis of the entire 16-year dataset proves that 4FGL J0433.0+0522 is a significant γ -ray source (TS ≃ 1500). Its averaged flux is estimated as (4.9 ± 0.3) × 10 -8 ph cm -2 s -1, consistent with the results listed in 4FGL-DR4 (Ballet et al. 2023). It is confirmed as a spectrally soft γ -ray source. Our localization analysis also confirms the spatial association between 3C 120 and the γ -ray source. Since detection of the neutrino event IC-180213A is highly time-sensitive, we carried out a specific analysis on half-year Fermi-LAT data, centered at the arrival time of the neutrino (i.e., from MJD 58072 to 58252). According to the 4FGL-DR4 (Ballet et al. 2023), there are three γ -ray sources likely cospatial with the neutrino. Two sources fall into the localization uncertainty box of the neutrino, 4FGL J0433.0+0522 and 4FGL J0426.5+0517, along with another one, 4FGL J0420.0+0805, lying at the edge of the box. The analysis on this specific epoch suggests that 4FGL J0426.5+0517 and 4FGL J0420.0+0805 are undetectable for Fermi-LAT then, with given TS values of < 5. On the other hand, 4FGL J0433.0+0522 is revealed as a significant γ -ray source, of which the TS value is estimated as 127, see Figure 1(a). Its photon flux in this period is obtained as (8.1 ± 1.2) × 10 -8 ph cm -2 s -1 . Furthermore, a TS map with 4FGL J0433.0+0522 included in the analysis model was extracted, to check whether sources not in 4FGL-DR4 emerge. As shown in 1(b), no such sources are found. Therefore, when the neutrino arriving, 3C 120 is the unique cospatial γ -ray source. We then investigated the temporal behavior of 4FGL J0433.0+0522. Firstly, one-year time-bin γ -ray light curve was extracted, see Figure 1(c). At the beginning of the Fermi-LAT operation, 3C 120 was at a low flux state, around MJD 56000, its flux started to rise. Two distinct γ -ray flares were followed, after the flares, it maintained at a median value flux level. The variability is proved to be significant ( > 5 σ ) by adopting the 'variability index' test (Nolan et al. 2012). Temporal behaviors of the bright nearby background sources were also investigated. No similar behaviors to those of the target are observed, and hence detections of the γ -ray flares is likely intrinsic rather than being artificial caused by the backgrounds. Moreover, the flaring epochs are identified by the Bayesian block approach (Scargle et al. 2013). As shown in Figure 1(c), incoming time of IC-180213A (i.e., MJD 58162) coincides with a γ -ray flare of 4FGL J0433.0+0522 with time range between MJD 57968 and 58333. A constant uncertainty light curve was also extracted, by employing an adaptive-binning method (Lott et al. 2012). The γ -ray variation is confirmed to be significant ( > 5 σ ) through the 'variability index' test (Nolan et al. 2012). At this case, the comparison between different flux states in long timescale is well characterized, as well as the shape of the γ -ray flares. Based on the Bayesian Block method (Scargle et al. 2013), three γ -ray flares were identified, which are marked as flare I, II and III, respectively, see Figure 2. The detections of flare I and II have been reported (Janiak et al. 2016). The most intense flare is the flare II peaking at MJD 57133. Its maximum flux, (4.3 ± 0.6) × 10 -7 ph cm -2 s -1 , is roughly 40-fold of the flux level at the initial phase of Fermi-LAT observation around MJD 55000, (1.0 ± 0.4) × 10 -8 ph cm -2 s -1 . The flare III is also distinct, ranging from MJD 58109 to 58298. The peaking flux (at MJD 58290), is (2.4 ± 0.4) × 10 -7 ph cm -2 s -1 , which suggests a roughly 20-fold γ -ray brightening. Note that arrival time of IC-180213A is at the ascent phase of flare III, consistent with the findings from the yearly bin light curve. By comparison, the flare I appears to be relatively mild, with flux level up to ≃ 10 -7 ph cm -2 s -1 . Furthermore, individual analyses on these three flares were performed. The corresponding results are summarized in Table 1. No significant differences of the spectral indexes between these flares are found. Since the spectral index of 4FGL J0433.0+0522 listed by the 4FGL-DR4 is 2.79 ± 0.03 (Ballet et al. 2023), no clear spectral hardness is indicated for the flares. Localization analyses suggest that during the flare epochs 3C 120 is cospatial with the γ -ray source. Fast γ -ray variability with timescale of several hours has been reported in year 2014 - 2015 (Janiak et al. 2016), which is atypical for MAGNs (but also see Baghmanyan et al. 2017). We also extracted a daily time bin γ -ray light curve targeting on the epoch of flare III, to search any signs of fast variability. There is a flux increase from (2.1 ± 0.7) × 10 -7 ph cm -2 s -1 at MJD 58191 to (6.5 ± 1.0) × 10 -7 ph cm -2 s -1 at MJD 58192. Meanwhile, a flux decline from (6.2 ± 0.6) × 10 -7 ph cm -2 s -1 at MJD 58297 to (2.3 ± 0.9) × 10 -7 ph cm -2 s -1 at MJD 58298 has been also detected. Therefore, intraday γ -ray variability in the epoch of flare III is revealed. The corresponding doubling timescale in the AGN frame is given as, τ doub,AGN = ∆ t × ln 2 /ln ( F 1 /F 2 ) / (1 + z ) ≲ 15 hours.", "pages": [ 2, 3, 4 ] }, { "title": "2.2. Swift-XRT Data", "content": "3C 120 is a well studied bright X-ray source (e.g., Marscher et al. 2002). In particular, there are more than 200 visits from the X-ray Telescope aboard Neil Gehrels Swift Observatory (Gehrels et al. 2004; Burrows et al. 2005). Interestingly, one Swift observation on MJD 58196 (ObsID: 00037594050, exposure time of ≃ 1 ks), about one month after the arrival of the neutrino, has been performed. The XRT photon counting mode data of this observation were analyzed with the FTOOLS software version 6.33.2. After the xrtpipeline event cleaning by the standard quality cuts, spectra from the target as well as the background were extracted by xselect task. To deal with the pile-up effect, a annular region with an inner radius of 5 pixels and outer one of 20 pixels was adopted for the source, while a circle with a radius of 60 pixels in a blank area was set for the background. Then, generation of the ancillary response files by the response matrix files taken from the calibration database was followed. The spectrum was rebinned that each bin has at least 20 photons. Channels with energy below 0.5 keV were excluded and the absorption column density was set as the Galactic value (i.e., 10 21 cm -2 ). Fitting the spectrum by xspec suggests an unabsorbed 0.5 - 10.0 keV flux of 6 . 94 +0 . 85 -0 . 81 × 10 -11 erg cm -2 s -1 ( χ 2 /d.o.f., 19.2/16), as well as a hard x-ray spectrum, Γ x = 1 . 52 ± 0 . 15. The long-term X-ray light curve data were derived from the results from analyses by swift xrtproc script (Giommi et al. 2021) 2 , in which standard data reduction procedures, corrections for pile-up effect as well as spectral fitting analyses have been employed. Our results for the XRT observation on MJD 58196 are consistent with that given by the swift xrtproc script. As shown in Figure 2, at that time, the X-ray flux level is moderate.", "pages": [ 4 ] }, { "title": "2.3. ASAS-SN Data", "content": "Optical light curve data were obtained by the All-Sky Automated Survey for Supernovae (ASASSN, Shappee et al. 2014; Kochanek et al. 2017; Hart et al. 2023) 3 . V-band and g -band magnitudes, extracted by photometry and calibrated by AAVSO Photometric All-Sky Survey (Henden et al. 2012), from different cameras were adopted. Only frames flagged with quality of 'G' were selected. The exposures were binned into weekly time bin light curves to exhibit the long-term flux temporal behavior. As shown in Figure 2, the optical emissions of 3C 120 are highly active, with variability amplitudes ≳ 0.5 mag, which is consistent with the results in literature (e.g., Le'on-Tavares et al. 2010). There are simultaneous ASAS-SN observations, V-band at MJD 58161 and g -band at 58160, during the arrival of the neutrino (i.e., MJD 58162). Although the optical flux level then is mild, a strong flare revealed by both bands with a peaking time around MJD 58335, is closely followed.", "pages": [ 5 ] }, { "title": "2.4. WISE data", "content": "Single-exposure photometric data in the W1 and W2 bands (centered at 3.4 and 4.6 µ m in the rest frame) from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and the NEOWISE Reactivation mission (Mainzer et al. 2020; WISE Team 2020) were collected. Data with poor image quality (' qi fact ' < 1), close proximity to the South Atlantic anomaly (' SAA ' < 5), as well as those flagged for moon masking (' moon mask ' = 1) were filtered out (Jiang et al. 2021). We grouped the data for each observational epoch (see Figure 2), separated by approximately half a year, using the weighted mean value. The corresponding uncertainty was determined by propagations of the measurement errors. An investigation based on a χ 2 -test statistic analysis (for more details, see Liao et al. 2019) suggests significant long-term variability ( > 5 σ ). The variability amplitude is up to ≃ 0.3 mag. One flux maximum is distinct at MJD 57264. A continuous flux increase is exhibited in recent years and the flux is at a high flux level right now. There are WISE observations on MJD 58154, about 8 days before the detection of the neutrino. The infrared fluxes at that time are at a relatively low flux state.", "pages": [ 5 ] }, { "title": "2.5. RATAN-600 data", "content": "Additional radio data were collected from observations from RATAN-600, a 600-meter circular multi-element antenna that is capable of measuring broad-band spectra across the 1-22 GHz range simultaneously (Parijskij 1993), since 3C 120 has been included in the RATAN-600 multi-frequency monitoring list for jetted AGNs 4 (Mingaliev et al. 2014; Sotnikova et al. 2022). As shown in Figure 2, the centimeter radio fluxes are variable, especially at 21.7/22.3 GHz in which the flux density at MJD 57457 (4.5 ± 0.6 Jy) is three times of that at MJD 56731 (1.3 ± 0.2 Jy). RATAN-600 observations at 21.7/22.3, 11.2 and 4.8 GHz have been performed on MJD 58156, about 6 days before the detection of IC-180213A. The measured flux densities then are mild.", "pages": [ 5 ] }, { "title": "2.6. Implications of multi-messenger observations", "content": "The broadband temporal behaviors, shown in Figure 2, strongly suggest that the activities of the relativistic jet are distinct from radio wavelengths to the high energy γ -ray regime, although as a MAGN the Doppler effect of 3C 120 is supposed to be mild. In particular, the variability in the GeV γ -ray domain is more violent than that in optical/infrared wavelengths, that in the latter case contributions from the accretion disk as well as the host galaxy are likely also significant. Since three flares have been identified in the γ -ray light curve, the temporal behaviors then at other radiation windows are examined. For the flare III, in its initial phase of the flux rise when the neutrino event IC-180213A is detected, brightening of the optical fluxes in the both g -band and V-band are followed. However, no observations in X-ray, and infrared around the peaking time of flare III are available. Although the variability amplitude of flare I is moderate, at the same time a strong rapid X-ray flare appears. The multi-wavelengths electromagnetic observations play an important role on investigation of origin of the neutrino event IC-180213A. Aforementioned evidences include that in a half-year epoch centered at the arrival time of the neutrino 3C 120 is the unique known cospatial γ -ray source. In addition, the incoming of IC-180213A temporally coincides with the second strongest γ -ray flare among the entire 16-yr observations, of which 20-fold flux increase is found. A strong accompanying optical flare is also detected. All these observational facts suggest a physical link between IC-180213A and 3C 120. Further Monte Carlo simulations are conducted to estimate the chance probability corresponding to the multi-messenger detections. The central position of IC-180213A is randomized across the sky, where the Galactic plane area is excluded, by fixing the size of localization error box (Abbasi et al. 2023). Since the detection sensitivity of IceCube is highly dependent on the sky declination, only the R.A. value is left free. Known γ -ray sources within this sky area, showing a similar variability behavior with 3C 120 are searched. Firstly, they should be statistically ( > 3 σ ) variable identified in the yearly time bin light curve 5 , and possess a significant detection (TS > 100) in the tenth time bin of the light curve (i.e., Aug. 2017 - Aug. 2018). Additional selection criterion is that there is a major brightening (i.e., with a peaking flux three times of the 14-yr averaged flux) when the neutrino arriving, revealed by adaptive-binning light curves and subsequent Bayesian Block analyses. Three sources, TXS 0506+056, GB6 J0922+0433 and TXS 1421+048, are picked out. The chance probability is calculated as p = M +1 N +1 , where N represents the number of times of simulations (i.e., 10 4 ) and M denotes the number of temporally coincident sources (but not cospatial) that fall within the simulated uncertainty box. Under this circumstance, p ≃ 0.04 is yielded. Conclusively, 3C 120 is suggested as a potential neutrino emitter. Besides the long-term variability activities, short-term variability brings crucial information on jet properties as well. Fast intraday γ -ray variations detected in 3C 120 reflect a compact radiation region. Taking the 15-hour doubling time in flare III as an example, the radius of the radiation region is constrained as, r < cδτ doub,AGN ≈ 0 . 0013 pc, where a Doppler boosting factor of δ = 2 . 4 is adopted (Jorstad et al. 2005). Under a conical jet geometry, the distance between the radiation region and the central SMBH is suggested as r b = Γ r ≲ 0 . 007 pc, where Γ is taken to be 5 (Jorstad et al. 2005). On the other hand, the time lag measured in the reverberation mapping observations between the emission lines and the continuum radiations is given as ≃ 28 light days (Pozo Nu˜nez et al. 2014), indicating that the broad line region (BLR) is at a distance of ∼ 0.02 pc away from the central SMBH. Therefore, the jet radiation region is likely within the BLR. It is also worth noting that 3C 120 is a spectrally soft γ -ray source with the most energetic photons at energies of a few GeVs, see Table 1. The proximity of the jet radiation region could lead severe γγ absorption that is consistent with the results from the γ -ray observation. The spectral energy distribution (SED) of 3C 120 is drawn in Figure 3, where a typical twobump shape is exhibited. Benefited from the multiwavelength observations in time domain, (quasi)simultaneous detections (i.e., within one week) in radio, infrared as well as optical bands are available, corresponding to the incoming of the neutrino. Complementary archival un-simultaneous data(Chang et al. 2020) are also plotted as backgrounds. As shown in the (quasi)simultaneous SED, the γ -ray emission tends to be at a high flux level compared with the 16-year averaged measurement, whereas those in other radiation windows seems to be moderate. Meanwhile, the peak frequency of synchrotron bump of 3C 120 is constrained as ∼ 10 13 Hz, and it is suggested as an low-synchrotronpeaked source (Abdo et al. 2010b). Considering its soft γ -ray spectrum, peak frequency of the high energy bump is likely at MeV γ -ray domain.", "pages": [ 5, 6, 7 ] }, { "title": "3. DISCUSSIONS AND SUMMARY", "content": "According to the detection of IC-180213A, a constraint on the neutrino luminosity of 3C 120 can be given. The number of (anti-)muon neutrinos at a declination δ detected by IceCube over a time period ∆ T is expressed as, The energy range of the neutrino population is set between ϵ ν µ , min = 80TeV and ϵ ν µ , max = 8PeV, where detections of 90% of neutrinos in the γ -ray follow-Up (GFU) channel are expected (Oikonomou et al. 2021). The neutrino spectrum is characterized by a power-law distribution ϵ -γ , with an index γ = 2. Meanwhile, ϕ ν µ represents the differential muon neutrino flux. For IC-180213A, a GFU Bronze type event, the corresponding effective area is approximately A eff ≈ 16 m 2 (Abbasi et al. 2023). As suggested by the Bayesian Blocks analysis, a time period of ∆ T = 0 . 5 years is applied. Therefore, the integrated muon neutrino energy flux is calculated as approximately 2 . 4 × 10 -10 erg cm -2 s -1 . Considering the proximity of 3C 120, the integrated muon neutrino luminosity is obtained as L ν µ ≈ 6 . 4 × 10 44 erg s -1 , or alternatively, an average muon neutrino luminosity ϵ ν µ L ϵ νµ = L νµ ln(8 PeV / 80 TeV) ≈ 1 . 4 × 10 44 erg s -1 . Considering the detected intraday γ -ray variability, the jet dissipation region is likely within the BLR. Together with the luminous accretion disk emission ( ∼ 9 λLλ [5100 ˚ A ] ∼ 1 . 3 × 10 45 erg s -1 , Peterson et al. 2004), photopion production ( p + γ → p + π ) could be responsible to the neutrino production. The energy in the cosmic rest frame for IC-180213A is ϵ ν = ϵ ν,obs (1 + z ) = 0 . 115 PeV 6 . Hence the emitting proton is at ϵ p ≈ 20 ϵ ν ≈ 2 . 3 PeV. Based on the approaches provided in Murase et al. (2018), the energy of the external photons involved in the photopion processes for 3C 120 is deduced as ϵ t ∼ 25 eV, that likely correspond to the emissions from the BLR. During the photopion interaction, protons transfer 3/8 of their energy to neutrinos, while the rest 5/8 part corresponds to the production of electrons and pionic γ rays. Subsequent electromagnetic cascades are initiated, leading a potential contribution on the detected electromagnetic radiations. Note that synchrotron cooling is likely dominant here, due to the Klein-Nishina suppression in the inverse Compton scattering. The connection between neutrino radiation and related γ -ray emission from the cascade is suggested in Murase et al. (2018), Y IC represents the Compton-Y parameter for pairs from the cascades, which is typically ≤ 1 (Murase et al. 2018). If a Doppler factor value of 2.4 as well as a magnetic filed intensity of B ' = 1 Gauss are set, the synchrotron radiations by the secondary pairs from the cascades peak at, ϵ pπ syn,obs ≈ 0 . 3 GeV ( B ' 1 Gauss ) ( ϵ p 2 . 3 PeV ) 2 ( 2 . 4 δ ) ( 1 . 033 1+ z ) . In the most optimistic scenario, the observed γ -ray emissions at such an energy could be overwhelmed by emissions from the secondary pairs. Or conservatively, emissions from the primary leptons are dominant. Since the typical relative uncertainty for FermiLAT estimations is roughly 20%, and the observed luminosity at 0.3 GeV is about 5 × 10 43 erg s -1 , ϵ γ L ϵ γ | ϵ pπ syn =10 43 erg s -1 is assumed. Therefore, a (anti-)muon neutrino luminosity of 4 × 10 42 erg s -1 is expected. Compared to the luminosity of muon neutrinos inferred from the detection of IC-180213A, the probability of observing such a neutrino due to Poisson fluctuations is approximately 0.03. Moreover, investigations on the kinetic energy of the jet based on the detection of the neutrino have been performed. As mentioned above, an all-flavor neutrino luminosity is described by ϵ ν L ϵ ν = 3 8 f pπ ϵ p L ϵ p , where the optical depth for pπ processes ( f pπ ) accounts for the efficiency of neutrino production. For 3C 120, BLR photons, which are described as a blackbody distribution with a temperature of 42000 K, are considered as target photons in the photopion processes. The optical depth for pπ processes is approximated by f pπ ≈ ˆ n BLR σ eff pπ r b ≈ 2 . 8 × 10 -4 (Murase et al. 2014), where ˆ n BLR ≈ 2 × 10 8 cm -3 denotes the photon density at ϵ t = 25 eV, σ eff pπ = 0 . 7 × 10 -28 cm 2 is the effective cross-section for pπ interactions (Murase et al. 2016), and r b represents the location of the emission region. Corresponding to the flare III, the proton luminosity is constrained as ϵ p L ϵ p ≈ 3 . 6 × 10 46 erg s -1 . Assuming a power-law spectrum for protons ( ϵ -γ p with γ = 2, ϵ p, min = δm p c 2 , and ϵ p, max = 10 18 eV, where m p c 2 is the proton rest energy), together with δ = 2 . 4, the proton luminosity L p is given by ϵ p L ϵ p ln ( 10 18 eV δm p c 2 ) ≈ 7 . 2 × 10 47 erg s -1 . In the AGN frame, the jet proton luminosity ˆ L p, jet is approximately L p / (4Γ 2 / 3) ≈ 2 . 2 × 10 46 erg s -1 . Such a value is about three times of the Eddington luminosity, 7 × 10 45 erg s -1 , consistent with the results on other neutrino emitting candidates (e.g., Cerruti et al. 2019; Gao et al. 2019; Rodrigues et al. 2021). So far there are more than 60 MAGNs detected by Fermi-LAT, that majority of them are radio galaxies (Ballet et al. 2023). As shown in Figure 4(a), 3C 120 is one of the most brightest radio galaxies. Meanwhile, it is one of the few ones in its kind possessing significant GeV γ -ray variability. Furthermore, 3C 120 is at the horizon direction of the IceCube, where the IceCube's GFU Bronze+Gold effective area is maximized. Therefore, it is not surprising that 3C 120 is the first radio galaxy proposed as a neutrino emitter. Besides 3C 120, there are a few other radio galaxies are worth noting for future neutrino detections. The first one is NGC 1275, the most powerful and active radio galaxy in the GeV domain (Ballet et al. 2023). However, since the declination angle of NGC 1275 is +41.3 · , its IceCube effective area could be significantly reduced compared with 3C 120 if the energy of the arrival neutrino reaches to ≃ 1 PeV (Abbasi et al. 2023). Other sources include M 87 and NGC 1218. Strong TeV variations have been detected in M 87 (Beilicke et al. 2008), while the GeV γ -ray behavior and the sky declination of the latter are similar with 3C 120. In the nearby universe (i.e., z < 0.05), in addition to NGC 1068 (IceCube Collaboration et al. 2022), 3C 120 is the second neutrino emitting candidate. However, origins of the neutrino production of these two sources are likely different. As a radio quiet source, neutrinos of NGC 1068 are proposed to be from outflow-ISM interaction roughly 50 pc away from the central engine (Fang et al. 2023), or at a place near the SMBH (Ajello et al. 2023). Note that the averaged γ -ray luminosity of 3C 120 is two orders of magnitude higher than that of NGC 1068 (see Figure 4(b)), and γ -ray emissions of 3C 120 are widely accepted to be from the relativistic jet (Abdo et al. 2010a; Kataoka et al. 2011; Janiak et al. 2016). The temporal coincidences between the arrival of the neutrino and the γ -ray flare further strengthen the connection between the neutrino and the jet. Results from theoretical studies support that the relativistic jets in radio galaxies are possible neutrino contributors (Becker Tjus et al. 2014; Blanco & Hooper 2017). It is intriguing to compare 3C 120 with blazars that are established as neutrino-emitting candidates. As shown in Figure 4(b), the averaged γ -ray luminosity of 3C 120 ( < 10 44 erg s -1 ) is generally two orders of magnitudes lower than that of the blazars ∼ 10 46 erg s -1 . Due to the relatively large inclination jet angle, the Doppler boost effect of 3C 120 is likely suppressed. As the first neutrino emitting radio galaxy candidate, 3C 120 serves as a unique target, which offers a different perspective than blazars for approaching the production of neutrinos. In fact, 3C 120 fills in the blank between the blazars and NGC 1068. Moreover, analyses of the IceCube data suggest that luminous sources (i.e., blazars) are unlikely the dominant population for extragalactic neutrinos (Aartsen et al. 2017). On the other hand, relatively less powerful sources, such as nearby radio galaxies, starbursts, as well as galaxy clusters and groups, are likely preferred, and a detection of the few brightest objects is enabled (Murase & Waxman 2016). Our study on 3C 120 indeed encourages such a proposition. Future more cases like 3C 120 would be crucial to a comprehensive understanding on the source of the IceCube diffuse neutrinos. In summary, based on the multi-wavelength observations of 3C 120, we suggest that it is likely associated with the neutrino event IC-180213A. 3C 120 is identified as the unique co-spatial γ -ray source at the time of the neutrino's arrival. Moreover, a prominent γ -ray flare that is second strongest one among the entire 16-year emerges then. Correlated optical flares in the V and g -bands observed by ASAS-SN have also detected. Monte Carlo simulations incorporating both spatial and temporal information yield a by-chance association probability of ∼ 0 . 04, establishing a physical connection between 3C 120 and the neutrino. It is the first radio galaxy proposed as a neutrino emitter, filling in the blank between the blazars and the NGC 1068. Theoretical constraints on the jet properties of 3C 120 have been investigated and comparisons with other neutrino emitting candidates have been performed. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center. This research has made use of data from the ASAS-SN project, which is supported by the Ohio State University and operated by the Ohio State Astronomy Department. This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the NASA and operated by the California Institute of Technology. This research uses data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research also makes use of data products from NEOWISE-R, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research has made use of the RATAN-600 data, provided by the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS). This work was supported in part by the NSFC under grants U2031120. This work was also supported in part by the Special Natural Science Fund of Guizhou University (grant No. 201911A) and the First-class Physics Promotion Programme (2019) of Guizhou University.", "pages": [ 7, 8, 9, 10 ] }, { "title": "REFERENCES", "content": "(c)", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "The gas dynamics of galaxies provide critical insights into the evolution of both baryons and dark matter (DM) across cosmic time. In this context, galaxies at cosmic noon - the period characterized by the most intense star formation and black hole activities are particularly significant. In this work, we present an analysis of the gas dynamics of PKS 0529-549: a galaxy at z ≃ 2 . 6, hosting a radio-loud active galactic nucleus (AGN). We use new ALMA observations of the [C i ] (2-1) line at a spatial resolution of 0.18 '' ( ∼ 1.5 kpc). We find that (1) the molecular gas forms a rotation-supported disk with V vrot /σ v = 6 ± 3 and displays a flat rotation curve out to 3.3 kpc; (2) there are several non-circular components including a kinematically anomalous structure near the galaxy center, a gas tail to the South-West, and possibly a second weaker tail to the East; (3) dynamical estimates of gas and stellar masses from fitting the rotation curve are inconsistent with photometric estimates using standard gas conversion factors and stellar population models, respectively; these discrepancies may be due to systematic uncertainties in the photometric masses, in the dynamical masses, or in the case a more massive radio-loud AGN-host galaxy is hidden behind the gas-rich [C i ] emitting starburst galaxy along the line of sight. Our work shows that in-depth investigations of 3D line cubes are crucial for revealing the complexity of gas dynamics in highz galaxies, in which regular rotation may coexist with non-circular motions and possibly tidal structures. Key words. dark matter - galaxies: active -galaxies: evolution - galaxies: formation - galaxies: high-redshift - galaxies: kinematics and dynamics", "pages": [ 1 ] }, { "title": "Gas dynamics in an AGN-host galaxy at z ≃ 2 . 6 : regular rotation, non-circular motions, and mass models", "content": "Lingrui Lin 1 , 2 , Federico Lelli 3 , Carlos De Breuck 4 , Allison Man 5 , Zhi-Yu Zhang 1 , 2 , Paola Santini 6 , Antonino Marasco 7 , Marco Castellano 6 , Nicole Nesvadba 8 , Thomas G. Bisbas 9 , Hao-Tse Huang 10 , 11 , 5 , and Matthew Lehnert 12 Received September 15, 1996; accepted March 16, 1997", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The study of gas dynamics provides key insights in the formation and evolution of galaxies across cosmic time. On global scales, the distributions of baryons and dark matter (DM) shape the gravitational potential of galaxies, a ff ecting their overall gas kinematics (e.g., van Albada & Sancisi 1986). In addition, the feedback e ff ects from massive stars (e.g., stellar winds and supernovae) and active galactic nuclei (AGN) inject energy into the interstellar medium (ISM), stirring the star-forming gas and possibly quenching the star formation of galaxies (e.g., Silk & Mamon 2012). In this context, galaxies at redshift z ≃ 1 -3 are particularly important because both the cosmic star formation history and the accretion history of supermassive black holes peak around this epoch, which is known as 'cosmic noon' (Madau & Dickinson 2014). Rapid developments in astronomical instruments have been boosting spatially-resolved studies of gas kinematics in highz galaxies. Near-infrared (NIR) spectroscopy with integral field units (IFUs) has enabled studies of the kinematics of warm ionized gas in galaxies at z ≃ 1 -3, using H α or [O iii ] emission lines (e.g., Förster Schreiber et al. 2009; Gnerucci et al. 2011; Wisnioski et al. 2015, 2019; Di Teodoro et al. 2016; Stott et al. 2016; Turner et al. 2017). Radio and (sub-)millimeter interferometers can resolve kinematics of cold molecular gas, which is composed mostly of molecular hydrogen (H2). However, the symmetric H2 molecule does not have a permanent dipole moment so it hardly emits any lines in the cold molecular gas. Practically, we observe H2-tracers like CO (Hodge et al. 2012; Tadaki et al. 2017; Talia et al. 2018; Übler et al. 2018; Rizzo et al. 2023; Lelli et al. 2023), [C i ] (Lelli et al. 2018; Dye et al. 2022; Gururajan et al. 2022; Rizzo et al. 2023), or [C ii ] (De Breuck et al. 2014; Jones et al. 2017; Smit et al. 2018; Rizzo et al. 2020; Lelli et al. 2021; Rizzo et al. 2021) lines. Emission lines of di ff erent species trace distinct phases of the interstellar gas. In nearby galaxies, the warm ionized gas traced by H α is often found to rotate slower and has a larger velocity dispersion than the cold molecular gas traced by lowJ CO lines (e.g., Levy et al. 2018; Su et al. 2022). In galaxies hosting starbursts and / or AGNs, emission lines of H α and [O iii ] can be dominated by galactic outflows (Arribas et al. 2014; Harrison et al. 2014; Concas et al. 2017, 2019, 2022), which further complicates analyses of galaxy rotation. For highz galaxies, the beam smearing e ff ect (Warner et al. 1973; Bosma 1978; Begeman 1989) also becomes significant as usually these galaxies can be spatially resolved only with a few independent elements with current facilities. This will lead to the observed emission lines being broadened by both the intrinsic turbulent motion of the interstellar gas and the unresolved rotation velocity ( V rot) structure within the telescope beam. In addition, the observed line-of-sight velocity is intensity-weighted so it is biased towards small galactic radii where the surface brightness is higher. To overcome the beam smearing e ff ect, various tools have been developed to fit a rotating disk model directly to the 3-dimensional (3D) emission-line cubes (e.g., Bouché et al. 2015; Di Teodoro & Fraternali 2015; Kamphuis et al. 2015). Therefore, to study the dynamics of highz galaxies, especially the extreme cases at cosmic noon, one needs multi-phase gas tracers for a panchromatic view of both circular and noncircular motions as well as careful treatment of beam smearing e ff ects (i.e., high-resolution data and reliable modeling). To date, such studies are only limited to a handful of cases (e.g., Chen et al. 2017; Übler et al. 2018; Lelli et al. 2018, 2023). PKS0529-549 is a well-studied radio galaxy at z ≃ 2 . 57 with plenty of multi-wavelength data - optical spectroscopy, NIR imaging, and radio polarimetry (Broderick et al. 2007), Spitzer Infrared Array Camera (IRAC), Infrared Spectrograph (IRS), and Multiband Imaging Photometer for Spitzer (MIPS) imaging (De Breuck et al. 2010), Herschel Photodetector Array Camera and Spectrometer (PACS) and Spectral and Photometric Imaging Receiver (SPIRE) imaging (Drouart et al. 2014), 1.1mmdata from AzTEC (Humphrey et al. 2011), Very Large Telescope (VLT) Spectrograph for INtegral Field Observations in the Near Infrared (SINFONI) imaging spectroscopy (Nesvadba et al. 2017), Atacama Large Millimeter Array (ALMA) [C i ] (2-1) line (Lelli et al. 2018) and band-6 continuum (Falkendal et al. 2019), and VLT / X-Shooter spectra from rest-frame ultra-violet (UV) to optical (Man et al. 2019). PKS0529-549 hosts a Typeii AGN and two radio lobes (Broderick et al. 2007). The Eastern lobe has the highest Faraday rotation measure ever observed to date (Broderick et al. 2007), suggesting that the galaxy is surrounded by a medium with high electron density and / or a strong magnetic field. PKS 0529-549 has an estimated stellar mass ( M ⋆ ) of 3 × 10 11 M ⊙ (De Breuck et al. 2010) derived by fitting the stellar spectral energy distribution (SED), and a star formation rate (SFR) of 1020 + 190 -170 M ⊙ yr -1 (Falkendal et al. 2019) derived by the total infrared luminosity. PKS0529-549 has experienced at least two bursts of recent star formation in the past, 6 Myr and > 20 Myr, respectively, based on an analysis of the photospheric absorption features in the restframe UV spectrum (Man et al. 2019). Using ALMA observations, Lelli et al. (2018) found that the [C i ] 3 P2 → 3 P1 emission (hereafter [C i ] (2-1)) is consistent with a rotating disk. The [O iii ] λ 5007 emission (Nesvadba et al. 2017), on the other hand, is more extended and is aligned with the radio lobes, so it is probably dominated by an AGN-driven outflow. The rotation speed of the gas disk traced by [C i ] provided a total dynamical mass consistent with the observed baryonic mass, but detailed mass models that separate the gravitational contributions of baryons and / or DM could not be constructed due to the low resolution and sensitivity of the [C i ] (21) data. For the same reasons, it was not possible to measure the gas velocity dispersion and to investigate possible non-circular motions in the molecular disk. In this work, we present new ALMA [C i ] (2-1) observations of PKS0529-549 with high spatial resolution and sensitivity. The [C i ] lines are among the most e ffi cient H2-tracers for galaxies at cosmic noon because they are accessible through ALMA band 4 and band 6. At z ∼ 2, the CO lines ( J ≥ 3) covered by ALMA are weak. The [C ii ] 158µ m line, instead, is di ffi cult to observe at z ≃ 1 -3 due to its high frequency (even though redshifted) that requires excellent weather conditions at the ALMA site, but it is cost-e ff ective for galaxies at z > 4 because it becomes observable with ALMA band 7 (e.g., De Breuck et al. 2014; Jones et al. 2017; Smit et al. 2018; Lelli et al. 2021). This paper is structured as follows. Section 2 describes the new ALMA observations and the data reduction. Section 3 describes the gas and dust distribution as well as their radial surface brightness profile. Section 4 studies the gas kinematics and measures the rotation curve of PKS 0529-549 as well as non-circular motions. Section 5 builds mass models with di ff erent combinations of baryonic and DM components, testing the consistency of our observations with the expectations from the Λ cold dark matter ( Λ CDM)cosmology. Section 6 discusses the implications of our results. Section 7 provides a summary. Throughout this paper, we assume a flat Λ CDM cosmology with H 0 = 67.4 km s -1 Mpc -1 , Ω m = 0.315, and ΩΛ = 0.685 (Planck Collaboration et al. 2020). In this cosmology, 1 arcsec corresponds to 8.22 kpc at the redshift of PKS 0529-549 ( z = 2 . 57), while the age of the Universe and the lookback time are 2.5 Gyr and 11.3 Gyr, respectively.", "pages": [ 1, 2 ] }, { "title": "2.1. ALMA observations", "content": "The ALMA band-6 observations were carried out during ALMA Cycle 6 (Project ID: 2018.1.01669.S, PI: Federico Lelli), targeting the [C i ] (2-1) line. Four spectral windows were centered at 226.200, 228.075, 240.000, and 241.875 GHz - each covers 1.875 GHz with 480 channels for a native velocity resolution of 5 km s -1 . The first spectral window was chosen to cover both the [C i ] (2-1) line (rest-frequency of 809.341970 GHz) and the CO J = 7 -6 line (rest-frequency of 806.651806 GHz); the other three spectral windows cover the continuum emission. Three execution blocks (EBs) were conducted on 9 August, 23 August, and 18 September, respectively, in 2019. The on-source times were 32.93, 32.93, and 15.20 min, respectively (1.35 hours in total). The first EB was labeled as semi-pass in the initial quality assurance (QA0) but we kept this EB because it improved the imaging quality after careful manual calibration. The latter two EBs (QA2 pass) were calibrated using the standard Common Astronomy Software Applications (CASA) pipeline (v5.6.1-8) (CASA Team et al. 2022).", "pages": [ 2 ] }, { "title": "2.2. Imaging and cleaning", "content": "Imaging of the [C i ] (2-1) cube was interactively done with the tclean task in CASA (v6.5.2.26), using Briggs' weighting with a robust parameter of 1.5 and a uv-taper of 0.05 arcsec. This gave a restored beam of 0.178 arcsec × 0.163 arcsec with a position angle PA = -20 . 9 deg. To reach an optimal compromise between resolution and sensitivity, we circularized the beam to 0.18 arcsec and rebinned the velocity resolution to 25.8 km s -1 . The root-mean-square (RMS) noise of the final [C i ] (2-1) cube is ∼ 0 . 15 mJy beam -1 . The continuum of the [C i ] (2-1) cube was subtracted by fitting a zeroth-order polynomial using the linefree channels (227.4047 to 228.9916 GHz) in the image plane. The CO J = 7 -6 line in the same spectral window was masked by visually trimming the spectrum. A band-6 continuum image was created by combining the spectral windows centered at 228.075, 240.000, and 241.875 GHz. We used tclean in interactive mode with Briggs' weighting, robust parameter of 1.5, and a uv-taper of 0.1 arcsec. This gave a restored beam of 0.199 arcsec × 0.195 arcsec with a position angle PA = 89 . 5 deg. The RMS noise of the band-6 continuum image is ∼ 0 . 012 mJy beam -1 .", "pages": [ 2, 3 ] }, { "title": "3.1. Global overview", "content": "Fig. 1 shows the ALMA band-6 (354 µ m at rest-frame of PKS0529-549) continuum map at ∼ 0 . 2 '' resolution. We confirm the two continuum components discussed in Lelli et al. (2018): the one on the East coincides with the radio lobe from the Australia Telescope Compact Array (ATCA) 18-GHz observations, while the one in the middle coincides with the [C i ] (Fig. 2) and optical emission. Falkendal et al. (2019), indeed, have shown that the Eastern compact source is consistent with the extrapolation of the power-law synchrotron emission from the radio band to the sub-mm band, while the central emission coincides with the gas disk and is contributed mostly by the cold dust in the galaxy. The new ALMA data confirm this interpretation. To create a continuum map that contains only dust emission, we used the following procedure to remove the Eastern synchrotron component. We first interactively drew a mask enclosing only the synchrotron component in the tclean task (with savemodel='modelcolumn' ). Next, we subtracted the synchrotron clean component from the original uv-data (using the uvsub task) and redid the imaging. The top-left panel of Fig. 2 shows the outcome of this procedure. The [C i ] (2-1) moment maps were constructed with 3DBased Analysis of Rotating Object via Line Observations ( 3D B arolo , Di Teodoro & Fraternali 2015), considering the [C i ] (2-1) signal within a fiducial mask created by the task Smooth & Search (see Section 4). The moment maps are shown in Fig. 2. The new ALMA data unambiguously confirm and spatially resolve the rotating disk identified in Lelli et al. (2018). In addition, the moment maps reveal a gas tail to the South-West of the rotating disk. In Section 4.4, we present a detailed 3D analysis which reveals a kinematically anomalous component in the blueshifted side of the disk and a second, weaker gas tail in the redshifted side to the East. It is possible that these three di ff erent non-circular components have a common physical origin, as we discuss in Section 6.1.", "pages": [ 3 ] }, { "title": "3.2. Radial surface brightness profiles", "content": "Radial surface brightness profiles provide an e ff ective 1-D description of the dust and gas distribution in galaxies. They are useful for measuring characteristic scale lengths, such as the effective radius ( R e) that contains half of the total flux. They are also needed to study the overall mass distribution of galaxies because one needs radial surface density profiles of di ff erent baryonic components to compute their gravitational field in the galaxy mid-plane out to infinity (see section 5). For the dust radial profile, we use the synchrotron-subtracted band-6 continuum image (top-left panel in Fig. 2). We measure the surface brightness profile averaging over a set of elliptical annuli, positioned according to the kinematic center, the inclination and position angle of the rotating disk (as derived in Section 4.1). The annulus spacing is half of the beam size of the dust continuum (0.10 arcsec). The results are shown in Fig. 3 (top panel). UV emission from both young stellar populations and the AGN can heat up the dust grains, though the latter is expected to contribute a negligible fraction to the Rayleigh-Jeans tail of the cold dust emission (Falkendal et al. 2019; Lamperti et al. 2021). To test this e ff ect, we measure the radial profile using also the ALMA band-4 continuum map at 625 µ m rest-frame (Huang et al. 2024). We find that the dust emission profile at band 4 is consistent with that at band 6, confirming that the dust continuum at band 6 is dominated by cold dust grains heated by young stars. Actually, Man et al. (2019) have shown that the UV emission from the host galaxy is dominated by the young stellar population, not the AGN. For the gas radial profile, we use a [C i ] (2-1) moment-0 map integrated within ( -550 , 550) km s -1 centered at the [C i ] (21) systematic velocity (Section 4) to account for possible faint emissions missed by the source mask. We measure the surface brightness profile using the same set of annuli as for the dust profile. The annuli spacing is half of the beam size of the [C i ] (21) cube (0.09 arcsec). The results are shown in Fig. 3 (bottom panel). Both profiles are fitted with a Sèrsic function (Sersic 1968) parameterized by the Sèrsic index ( n ) and the e ff ective radius ( R e). The fitting is done using the orthogonal-distance-regression method in scipy.odr . For the dust profile, we fit the data out 1 1 1 to radii R = 1 arcsec and obtain n = 1 . 3 ± 0 . 2. For the gas profile, we fit only the data at R < 0 . 7 arcsec because we aim to trace the inner rotating disk without contribution from the outer gas tail. We obtain n = 0 . 52 ± 0 . 01, which properly captures the inner flattening of the gas profile. The best-fit radial density profiles are shown in Fig. 3 and the best-fit parameters are given in Table 1.", "pages": [ 3, 4 ] }, { "title": "4. Gas kinematics", "content": "We study the gas kinematics of PKS 0529-549 using 3D B arolo (Di Teodoro & Fraternali 2015). In 3D B arolo , a rotating disk is modeled with a set of tilted rings, each characterized by five geometric parameters - center coordinates ( x 0, y 0), systemic velocity ( V sys), position angle ( PA ), and inclination ( i ) - and five physical parameters - rotation velocity ( V rot), radial velocity ( V rad), velocity dispersion ( σ v), surface density ( Σ gas), and vertical thickness ( z 0). The tilted-ring model is convolved with the telescope beam and then is iteratively compared with the observations to obtain the best-fit parameters. The 3D fit of 3D B arolo is performed on a masked cube which includes mostly real line emission and avoids noisy pixels. We generate the source mask by setting MASK=SMOOTH&SEARCH , FACTOR=1.8 (factor by which the cube is spatially smoothed before source search), SNRCUT = 4 (primary S / N threshold), GROWTHCUT=3 (secondary S / N threshold to growth the primary mask), and MINCHANNELS=2 (minimum number of channels for an accepted detection). A di ff erent choice of the source mask would not substantially change our general results. Within the source mask, the [C i ] (2-1) disk of PKS 0529-549 can be fitted with five rings. The width of each ring is 0.09 arcsec, which is half of the beam size of the [C i ] (2-1) cube. We set NORM=AZIM so that the observed moment-0 map is azimuthally averaged to obtain the Σ gas of each ring in the model. For the vertical density distribution, we assume an exponential profile ( LTYPE=3 ) with a fixed scale height of 300 pc ( ∼ 0.04 arcsec). The disk scale height is much smaller than the [C i ] (2-1) beam (0.18 arcsec) so it has negligible impact on the kinematic fitting. We also fix V rad = 0 because there are no indications for strong radial motions, which generally produce a non-orthogonality between the kinematic major and minor axes (e.g., Lelli et al. 2012a,b; Di Teodoro & Peek 2021). Therefore, seven free parameters need to be optimized: x 0, y 0, V sys, PA , i , V rot, and σ v. To obtain the rotation curve, we first estimate the geometric parameters and then fit the kinematic parameters ( V rot and σ v) with the disk geometry fixed.", "pages": [ 4, 5 ] }, { "title": "4.1. Disk geometry", "content": "We first run 3DFIT on the [C i ] (2-1) cube, leaving all seven parameters free. To estimate the overall geometry, all pixels are uniformly weighted ( WFUNC=0 ) and both sides of the rotating disk are considered ( SIDE=B ). We set the initial PA = 75 · and initial i = 50 · . We also set DELTAPA=15 and DELTAINC=15 such that PA and i can explore the parameter space within ± 15 · around their initial guesses. The initial value of σ v is 30 km s -1 (Lelli et al. 2018). We let 3D B arolo guess the initial values of V sys and V rot automatically. After several tests, we find that the kinematic center is di ffi -cult to measure because the best-fit value does not coincide with the kinematic minor axis (defined by the iso-velocity contour equal to V sys) as expected for a rotating disk. This is likely due to the disturbed gas kinematics on the approaching side of the disk. Therefore, we fix the kinematic center of the galaxy to (R.A., Dec.) = (5 h 30 m 25 . 447 s , -54 · 54 ' 23 . 165 '' ) so that it lies along the kinematic minor axis (see the bottom left panel of Fig. 2), and re-run the fits with five free parameters. Table 2 summarizes the disk geometric parameters fitted by 3D B arolo . The adopted values of V sys, PA , and i are measured as the median values across di ff erent rings. The uncertainties are estimated as where N = 5 is the number of rings, MAD is the median absolute deviation across the rings, and δ i are the individual errors on the given parameter at each ring. Under the radical sign, the first term considers the variation among di ff erent rings while the second term considers the uncertainty of each ring. The best-fit PA and i are perfectly consistent with the values from Lelli et al. (2018) of 75 · and 50 · , respectively. For such an inclination angle, V rot is not sensitivity to the inclination correction; for example, V rot only changes by ∼ 10% when i varies from 50 · to 60 · .", "pages": [ 5 ] }, { "title": "4.2. Rotation velocity and velocity dispersion", "content": "Fixing the geometric parameters, we run SPACEPAR in 3D B arolo to look for global minima in the parameter space of V rot -σ v. We explore V rot within [200, 450] km s -1 and σ v within [1, 200] km s -1 , both with a grid step of 1 km s -1 . The residual function to be minimized is | M -D | ( FUNC=2 ), where M and D are the intensity values at each 3D voxel of the model and the data cube, respectively. To examine the e ff ect of non-circular motions (such as the enhanced kinematic irregularities on the blueshifted side), we run SPACEPAR separately on the approaching (blueshifted, SIDE=A ) and receding sides (redshifted, SIDE=R ), as well as simultaneously on both sides ( SIDE=B ). Fig. 4 shows V rot and σ v of each ring optimized on di ff erent sides. The rotation velocities are consistent within the errors among the three di ff erent runs. When fitting only the approaching side, the velocity dispersion shows an elevated value at R ≃ 0 . 135 '' , which is likely due to complex non-circular motions rather than a real increase in the gas turbulence (see Fig. 5). The non-circular motions are examined in detail in Section 4.4. The current data are unable to properly constrain the radial profile of the gas velocity dispersion, so we calculate the median σ v from the two-sides fitting (47 ± 16 km s -1 ) and use it as our fiducial estimate of the intrinsic gas velocity dispersion. The uncertainty is calculated using Eq. 1. This measurement of σ V is consistent within the errors with the fiducial upper limit of ∼ 30 km s -1 estimated by Lelli et al. (2018). As a final step, we rerun 3DFIT , fixing σ v = 47 km s -1 and leaving only V rot free. We set WFUNC=2 to give more weights along the kinematic major axis. Fig. 5 compares the positionvelocity (PV) diagram along the kinematic major axis of the observed cube with the best-fit model cube. Overall, the disk model provides a good description of the observations. In particular, the thickness of the observed PV diagram is well reproduced by the model, indicating that the velocity dispersion is reasonable. Noncircular motions that cannot be reproduced by the rotating disk model will be described in detail in Section 4.4.", "pages": [ 5, 6 ] }, { "title": "4.3. Asymmetric drift correction", "content": "The gas disk of PKS 0529-549 is rotationally supported, having a median V rot /σ v = 6 ± 3. The uncertainty is calculated by propagating the errors on V rot and σ v, which are estimated using Eq. 1. Turbulent motions, however, may provide non-negligible pressure support, so we estimate the asymmetric drift correction (ADC) to obtain the circular velocity ( V c) that directly relates to the gravitational potential. The ADC depends on the radial gradients of σ v and Σ gas (see, e.g., Eq. 4 in Lelli 2023). In 3D B arolo , the ADC can be computed using polynomials to describe the radial profiles of σ v and Σ gas. Fig. 6 shows the resulting V c( R ) assuming a constant σ v = 47 km s -1 or a radially varying σ v( R ) (taken from the twosides fitting). We find that the values of V rot and the two versions of V c are consistent within the errors, confirming that the rotation support is dominant while pressure support is nearly negligible. Hereafter, we use V c from a radially constant σ v for simplicity.", "pages": [ 6 ] }, { "title": "4.4. Non-circular motions", "content": "The channel maps (Fig. 7) show that there are non-circular motions that cannot be reproduced by the rotating disk model: 1) a gas tail to the South-West of the rotating disk at line-of-sight (LoS) velocities from -191 to -88 km s -1 (SW-tail); 2) a second weaker gas tail to the East of the rotating disk at LoS velocities from 274 to 377 km s -1 (E-tail); and 3) an anomalous structure at R ≃ 0 . 1 -0 . 3 '' at LoS velocities from -501 to -346 km s -1 (see also Fig. 5). These non-circular components are also visible in the residual [C i ] (2-1) moment-0 map (left panel of Fig. 8), which is obtained by subtracting the best-fit model cube from the observed cube. To better visualize the tail-like structures, we construct the so-called 'Renzograms' (Sancisi 1976) from the [C i ] (2-1) cube by integrating over the velocity intervals of the two gas tails specified above, and overlay them on the dust-only continuum map (Fig. 9). While the contours around the kinematic center are influenced by emission from the rotating disk (especially for the blue contours), the outer parts trace mostly the gas tails, possibly extending beyond the main body of the galaxy disk. The SW-tail is significantly detected at S / N > 3 while the E-tail is detected at S / N ∼ 2 -3. The right panel of Fig. 8 shows a PV diagram extracted along the path in the left panel, averaging over a width of 0.225 arcsec. It is di ffi cult to tell whether the anomalous structure and the SWtail are kinematically connected because the eventual connection occurs at the same LoS velocities of the gas disk. The physical nature of these three non-circular components remains unclear and will be discussed in Section 6.1. A sensible hypothesis is that we are seeing two \"leftover\" tidal tails due to a past major merger and a gas inflow towards the galaxy center, possibly related to the SW-tail. Using the residual [C i ] (2-1) moment-0 map, we estimate the [C i ] (2-1) flux associated with the non-circular motions. We sum over pixels with S / N > 3 enclosed by the apertures shown in the left panel of Fig. 8. The fluxes and the fiducial uncertainties are given in Table 3 but we stress that these values are lower limits. In fact, given that the rotating disk model assumes axissymmetry, the flux in the observed moment-0 map is azimuthally averaged over rings, including the flux of the non-circular motions. This explains why the residuals along the minor axis of the galaxy are systematically negative. The lowS / N pixels associated with the gas tails are also discounted. The non-circular motions are responsible for at least 12% of the total flux in the gas disk (2 . 8 ± 0 . 3 Jy km s -1 ). We will further discuss the noncircular motions in Section 6.1.", "pages": [ 7 ] }, { "title": "5.1. Bayesian rotation-curve fitting", "content": "In this section, we build a set of mass models with di ff erent combinations of mass components (gas, star, dark matter halo). The model circular velocity ( V mod), therefore, is determined by several free parameters p depending on which mass components are included. To determine the parameter values and uncertainties, we use a Markov-Chain-Monte-Carlo (MCMC) method to sample the posterior probabilities of the free parameters (see Appendix A for details). In Bayesian inference, the posterior probability distribution of the free parameters is the product of the likelihood function (based on new observations) and their priors (based on previous knowledge or assumptions). We define the likelihood function as L = exp( -0 . 5 χ 2 ) with where V c is the observed circular velocity at the k -th radius R k and δ V c is the associated uncertainty. Apart from the free parameters in V mod, the disk inclination i is treated as a nuisance parameter. We impose a Gaussian prior on i centered at i 0 = 53 · and with a standard deviation of 5 · to account for the observational uncertainties (see Section 4.1 and Table 2). When sampling in the parameter space of i , V c and δ V c change by a factor of sin( i 0) / sin( i ) accordingly. In the following sections, we explore di ff erent mass models and clarify priors on the related free parameters. We start with partial mass models with a limited amount of baryonic components; these models are probably unphysical but are useful to set hard upper limits on gas and stellar masses. Next, we build complete mass models, but warn that the masses of the di ff erent components are often degenerate. The best-fit models are shown in Fig. 10 & 11 and the MCMC corner plots are shown in Fig. A.1 &A.2. Median values and associated uncertainties of parameters of each model are presented in Table 4.", "pages": [ 7 ] }, { "title": "5.2.1. Gas only", "content": "To set a hard upper limit on the gas mass, we start with a minimalist mass model where the gas disk is the only dynamically important component. This mass model is probably unphysical; as we will show, indeed, it cannot reproduce the observed rotation curve. The gas gravitational contribution ( V gas) is calculated by numerically solving the Poisson's equation for a finite-thickness disk with a density profile ρ ( R , z ) = Σ ( R ) ξ ( z ), where Σ ( R ) is the radial surface density profile and ξ ( z ) is the vertical profile. To this aim, we use the vcdisk package 1 . For Σ ( R ), we take the Sèrsic profile fitted to the [C i ] (2-1) surface brightness profile in Section 3.2. For ξ ( z ), we assume an exponential distribution with a constant scale height of 300 pc. For practical reasons, we calculate V gas for a normalization mass ( M 0) defined for R →∞ and we introduce a dimensionless scaling factor Υ gas = M gas / M 0 of the order of unity, where M gas is the actual gas mass. Therefore, we have V 2 mod = Υ gas V 2 gas . For numerical convenience, we take M 0 = 10 11 M ⊙ and apply hard - beam mJy boundaries on log( Υ gas) ∈ ( -2 , 2). Therefore, M gas has a uniform 'uninformative' prior within (10 9 , 10 13 ) M ⊙ . The best-fit model gives M gas = 8 . 6 × 10 10 M ⊙ . This value, as we will discuss in Section 6.2.1, is comparable to the molecular gas mass inferred from the CO J = 4 -3 (hereafter CO (4-3)) flux but is three times smaller than those inferred from [C i ] and dust emissions (Huang et al. 2024). The left panel of Fig. 10 shows that it is impossible to reproduce the inner parts of the rotation curve using only the gas disk component. The high rotation velocities in the innermost two rings require the existence of a central mass concentration, such as a stellar spheroid and / or a supermassive black hole.", "pages": [ 7, 8 ] }, { "title": "5.2.2. Stars only", "content": "We now consider a mass model where V mod is fully determined by the stellar component while the gas contribution is neglected. Since PKS 0529-549 is very bright in [C i ], this model corresponds to a scenario where the [C i ]-toM gas conversion factor is extremely small (see discussions in Section 6.2.1), so that the gravitational contribution from gas is much smaller than that from stars. This model is probably unrealistic but is useful for setting hard upper limits on the stellar mass. Given the lack of high-resolution optical / NIR imaging for PKS0529-549, we cannot directly compute V ⋆ using the observed stellar surface brightness profile (as for V gas). Therefore, we adopt a sensible parametric function for the stellar mass distribution by assuming a spherical stellar component described by a Sèrsic profile. The stellar gravitational contribution at radius R is then given by where the fitting parameters are the stellar mass M ⋆ , the stellar half-mass radius R e ,⋆ , and the Sèrsic index n ⋆ . The parameters p and b are functions of n ⋆ . The incomplete and complete gamma functions are denoted as γ and Γ , respectively (see Terzi'c & Graham 2005). Similarly to Section 5.2.1, we calculate V ⋆ for a nor- M 0 = 10 11 M ⊙ and introduce a dimensionless parameter Υ ⋆ = M ⋆/ M 0 so that V 2 mod = Υ ⋆ V 2 ⋆ . We apply uniform priors on n ⋆ ∈ (0 . 5 , 10), log( Υ ⋆ ) ∈ ( -2 , 2), and R e ,⋆ ∈ (0 . 1 , 5) kpc. The right panel of Fig. 10 shows that this single-component model can fit the observed rotation curve. The best-fit n ⋆ = 5 . 7 reconfirms that the stellar mass distribution should be centrally concentrated but R e ,⋆ is unconstrained (see the corner plot in Fig. A.1). Given that the model neglects gas and DM contributions, the best-fit stellar mass ( ∼ 1 . 1 × 10 11 M ⊙ ) represents a hard upper limit on the actual stellar mass of the galaxy. This value is not sensitive to R e ,⋆ , as is shown in Fig. A.1, and is about a factor of three smaller than the value estimated from SED fitting (De Breuck et al. 2010, 3 × 10 11 M ⊙ ,). We will discuss possible reasons for this discrepancy in Section 6.2.2. We have also explored a mass model (Fig. A.3) where the stellar component is given by the sum of an exponential disk and a De Vaucouleurs' bulge (with n ⋆, bul = 4). This multi-component mass model has four strongly degenerate parameters: the stellar masses and the e ff ective radii of each component. Since our main aim is to obtain an upper limit on the total stellar mass, the effective radius of the disk was fixed to be equal to that of the dust component R e , dust (Fig. 3), while that of the bulge was fixed to 0 . 1 × R e , dust. This multi-component model also gives a good fit to the rotation curve and returns a total stellar mass (bulge plus disk) of ∼ 8 × 10 10 M ⊙ . This mass is slightly smaller than the one from the single-component spherical model because of the well-known fact that a highly flattened mass distribution implies higher circular velocities than the equivalent spherical mass distribution (e.g., Lelli 2023). The bulge-to-disk ratio is ∼ 0 . 9, but this value is highly uncertain and depends on the adopted effective radii. Future high-resolution NIR images are needed to better constrain the stellar mass distribution. Notes. The values show medians (50% quantile) with 16% and 84% quantiles in sub- and super-scripts, respectively.", "pages": [ 8, 9, 10 ] }, { "title": "5.3.1. Baryons only", "content": "Compared to the single-component models, a more complete model includes the gravitational contributions of both gas and stars. In this case, V 2 mod = Υ gas V 2 gas + Υ ⋆ V 2 ⋆ . To alleviate the degeneracy among the parameters, we add the following physically-motivated priors: The left panel of Fig. 11 combines the gravitational contribution of both gas and stars. The M gas and M ⋆ values in this model are 1 . 8 × 10 10 M ⊙ and 7 . 0 × 10 10 M ⊙ , respectively. As expected, both masses decrease with respect to those in the partial models. The stellar component dominates the total gravitational contribution otherwise the circular velocities at the innermost radii cannot be recovered.", "pages": [ 10, 11 ] }, { "title": "5.3.2. Baryons plus dark matter", "content": "In Section 5.3.1, we have shown that the rotation curve of PKS0529-549 is well fitted by a baryon-only model with sensible baryonic masses. Therefore, it is immediately clear that the DM contribution is unconstrained due to the disk-halo degeneracy (van Albada et al. 1985; Lelli 2023). Nevertheless, we tentatively add a Navarro-Frenk-White (NFW) dark matter halo with constraints based on Λ CDMcosmology. In this way, we can examine whether the observed rotation curve is consistent with the expectations from the Λ CDMcosmology. The NFW-profile is parameterized by the halo concentration C 200 and the halo mass M 200 (or equivalently the halo velocity V 200). Including the contributions of both baryons and the NFW DMhalo, V mod is thus given by where V NFW is the circular velocity of the NFW halo (Eq. 10 in Li et al. 2020). In addition to the baryonic priors described in Section 5.3.1, we use two Λ CDMscaling relations as DM priors: Compared to the baryons-only model, including a DM halo decreases the gas mass and stellar mass, but these di ff erent estimates are all consistent within uncertainties. Even though the DM contribution is not constrained, the rotation curve of PKS0529-549 is consistent with the expectations from the Λ CDMcosmology.", "pages": [ 11 ] }, { "title": "6. Discussion", "content": "PKS0529-549 is about 10 -100 times more luminous in the [C i ] line than the majority of highz radio galaxies (HzRG, Kolwa et al. 2023), enabling detailed studies of its gas distribution and kinematics. On the other hand, the SFR of PKS 0529-549 (1020 + 190 -170 M ⊙ yr -1 , Falkendal et al. 2019) indicates that its ISM condition should be extreme (e.g., strong UV field, cosmic ray intensity, gas turbulence, and high gas density and temperature), which could complicate the abundances of molecular gas tracers and the excitation of molecular lines.", "pages": [ 11 ] }, { "title": "6.1. Circular and non-circular motions", "content": "In Section 4, we show that PKS 0529-549 has a regular rotating disk, with V rot /σ v = 6 ± 3. This value is larger than what is predicted by the disk-instability model from Wisnioski et al. (2015) at the redshift of PKS 0529-549, but is consistent with recent ALMA observations in a significant sample of highz starforming galaxies (Lelli et al. 2023; Rizzo et al. 2023, 2024). Given that PKS 0529-549 is an AGN-host starburst with an extreme SFR of 1020 + 190 -170 M ⊙ yr -1 (Falkendal et al. 2019), it is surprising that its gas disk is still dynamically cold. In addition to the overall regular rotation of the gas disk, there are clear signatures of non-circular motions, i.e, the SWtail, the E-tail, and the anomalous structure (Section 4.4). The gas tails may be remnants of a past major merger event, which could have triggered a gas inflow (possibly related to the anomalous kinematic structure near the center) and therefore the high star-formation rate and radio-loud AGN activity of the galaxy. Alternatively, the two gas tails may be spiral arms in a more extended gas disk, while the kinematically anomalous component may be something unrelated, such as a gas outflow. Future images from the Hubble Space Telescope (HST) or the James Webb Space Telescope (JWST) are key to elucidating their origins. Considering the fraction of non-circular motions in the total flux of PKS 0529-549 as well as the gas mass obtained using the 'gas-only' mass model, we obtain hard upper limits on the gas mass of the non-circular structures, which is about 1 . 0 × 10 10 M ⊙ . Here we assume that the flux-to-mass conversion factors are the same in the rotating disk and the non-circular structures. If we take the gas mass from the 'Baryons + DM' mass model, the mass of the non-circular motions decreases to 1 . 7 × 10 9 M ⊙ . In both cases, the molecular gas involved in the non-circular components is a minor fraction (12%) of the total gas mass that resides in the rotating disk.", "pages": [ 11 ] }, { "title": "6.2. Discrepancies in different mass estimates", "content": "Mass models fitted to the observed [C i ] rotation curve allow us to obtain dynamical upper limits on the gas and stellar masses of this galaxy. In the following, we compare our mass measurements with those from independent methods, finding some puzzling discrepancies.", "pages": [ 11 ] }, { "title": "6.2.1. Discrepancies in gas masses and conversion factors", "content": "To estimate the total molecular gas mass ( M mol) of a galaxy, one usually measure the line luminosity of an H2-tracer and adopt a luminosity-to-mass conversion factor. For example, CO lines have been widely used (Carilli & Walter 2013). The CO-to-H2 conversion factor α CO is typically defined as the ratio between M mol and the luminosity of the CO (1-0) line, L ' CO(1 -0) (Bolatto et al. 2013). This conversion factor must be calibrated with an independent measurement of M mol, such as the one derived with dynamical methods. By doing so, the underlying assumption is that molecular gas dominates the total gas mass ( M gas) in the inner galaxy regions. In the case of PKS 0529-549, using the CO (4-3) luminosity L ' CO(4 -3) = (4 . 2 ± 0 . 5) × 10 10 K km s -1 pc 2 (Huang et al. 2024), a typical CO line ratio of r 41 ≡ L ' CO(4 -3) / L ' CO(1 -0) = 0 . 5 (Carilli & Walter 2013), and the upper-limit on M gas from the gas-only mass model (Section 5.2.1), we get an upper limit on α CO < 1 . 0 ( r 41 / 0 . 5) M ⊙ (K km s -1 pc 2 ) -1 . This is similar to what is commonly used for starbursts ( ∼ 0.8, Bolatto et al. 2013; Carilli & Walter 2013), but we stress that it is a very hard upper limit because it neglects contributions from stars and DM in the mass model. If we instead consider the gas mass from the complete mass model with baryons plus DM, we find α CO = 0 . 17 ( r 41 / 0 . 5) M ⊙ (K km s -1 pc 2 ) -1 . Using the [C i ] (1-0) luminosity L ' [C I] = (3 . 12 ± 0 . 67) × 10 10 K km s -1 pc 2 (Huang et al. 2024) and the upper-limit of M gas from the gas-only model, we get an upper limit on the [C i ]-toM gas conversion factor α [C I] ≡ M gas / L ' [C I] < 2 . 8 ± 0 . 8 M ⊙ (K km s -1 pc 2 ) -1 . This value is about 1 / 7 of the mean value in highz metal-rich galaxies ( ∼ 20, Dunne et al. 2022). If we consider the gas mass from the complete mass model with baryons plus DM, the inferred value of α [C I] goes down to 0.4 M ⊙ (K km s -1 pc 2 ) -1 , which is 50 times lower than the usual value. One possibility is that PKS 0529-549 has a [C i ] / H2 abundance ratio at least seven times the value taken for local ultraluminous infrared galaxies ( ∼ 3 × 10 -5 , Papadopoulos & Greve 2004). If PKS 0529-549 is the progenitor of a local early-type galaxy (ETG), we may indeed expect that its star-forming gas is already significantly enriched (e.g., Thomas et al. 2005, 2010). Moreover, considering the intense star formation and AGN activity in PKS 0529-549, the [C i ] / H2 ratio can also be enhanced by the dissociating far-UV photons and cosmic rays (Bisbas et al. 2024). Another possibility is that the radiative transfer of [C i ] lines is complicated by the intense star formation and AGN activity, leading to enhanced [C i ] emission and the failing of the usual conversion factor. Moreover, the [C i ] (1-0) flux is very uncertain and there may be spatial variations of the [C i ] line ratio that we cannot probe with the current data.", "pages": [ 11, 12 ] }, { "title": "6.2.2. Discrepancies in stellar masses", "content": "The upper limit on M ⋆ given by the star-only mass model is about a factor of ∼ 3 smaller than the value estimated from fitting the spectral energy distribution (SED) with stellar population models (De Breuck et al. 2010). The discrepancy increases up to a factor of ∼ 6 if the gravitational contributions of gas and DM are included in the mass models. The discrepancy is quite serious, so we discuss three possibilities to explain it: (1) the SED fitting overestimates the stellar mass; (2) the rotating disk is not in full equilibrium, so the circular velocities underestimate the dynamical mass; and (3) we are observing two di ff erent galaxies along the line of sight. (1) Regarding the SED fitting, the stellar mass comes from De Breuck et al. (2010), where 70 HzRGs were studied based on Spitzer photometry. The SED fitting assumes an elliptical galaxy template for the stellar component, which may not be ideal for PKS0529-549 given its high SFR (Falkendal et al. 2019). Two or three black body functions are assumed for dust emissions. The inherent uncertainty of stellar mass derived from SED fitting (Seymour et al. 2007) is smaller than its di ff erence from the stellar mass derived using dynamic mass models. To further investigate the issue, we have constructed a new SED with 22 photometric points from the rest-frame optical to the radio, using data from Gemini Flamingos-2, VLT Infrared Spectrometer And Array Camera (ISAAC), Spitzer / MIPS and IRAC, Herschel / SPIRE and PACS, ALMA band 4 and band 6, and ATCA. Preliminary SED fittings with Code Investigating GALaxy Emission (CIGALE, Boquien et al. 2019) show that the best-fit stellar mass can range from 0 . 4 × 10 11 to 1 . 2 × 10 11 M ⊙ depending on the chosen AGN model, so it may be consistent with the dynamicallyinferred value of M ⋆ . The SED fittings, however, are not fully satisfactory, especially in the AGN-dominated far-infrared portion of the spectrum, so we will investigate this issue in more detail in a future paper, in which we will test di ff erent SED fitting codes and AGN models. (2) Regarding the dynamical equilibrium, the [C i ] (2-1) velocity field is relatively symmetric and shows regular rotation (Fig. 2), which is usually interpreted as the cold gas being in equilibrium with the gravitational potential. Our 3D kinematic modeling, however, reveals an anomalous kinematic structure in the approaching side of the disk (Fig. 5) and two extended gaseous tails. If PKS 0529-549 has indeed undergone a recent major merger, the inner disk may not have had enough time to relax with the overall gravitational potential, so that the dynamical mass is potentially underestimated (Lelli et al. 2015). Using the outermost measured point of the rotation curve, we estimate the orbital time of PKS 0529-549, which is t orb ∼ 80 Myr. This is larger than the time from the two recent bursts of star formation: 6 Myr and > 20 Myr (Man et al. 2019). If the two star formation bursts are driven by a major merger, it is therefore possible that the gas disk of PKS 0529-549 is not relaxed because it did not have enough time to complete several rotations since the time of the latest starburst. Custom-built hydrodynamical simulations are needed to investigate whether such a merger event could be strong enough to drive the rotating disk out of dynamical equilibrium, leading to a systematic underestimate of the dynamical mass. (3) Regarding the third possibility, the scenario is that we are seeing two galaxies roughly aligned along the line of sight: a [C i ]-emitting, gas-rich, star-forming galaxy on the foreground and a [O iii ]-emitting, gas-poor, AGN-dominated galaxy on the background. As strange as it may sound, there are actually several clues in this direction. First, the [O iii ] λ 5007 emission is systematically redshifted with respect to the [C i ] (2-1) emission (Fig. 12). The redshifts of [C i ] (2-1) and [O iii ] λ 5007 lines are 2 . 5706 ± 0 . 0002 and 2 . 5745 ± 0 . 0001 (Nesvadba et al. 2017), respectively. Their redshift di ff erence corresponds to either a velocity di ff erence of ∼ 350 km s -1 with respect to the [C i ] (2-1) rest frame, or a comoving distance di ff erence of ∼ 4 . 5 Mpc if we consider the [C i ] and [O iii ] redshifts as distinct reference frames. Given the cosmological scale-factor of 0.28 at z = 2 . 57, the physical distance between the [C i ] and [O iii ] emitters would be of 1.3 Mpc, so the two putative galaxies would probably be unbound. Second, the [O iii ] λ 5007 kinematic major axis is o ff set by ∼ 30 · with respect to the [C i ] disk major axis (see Fig. 12). This fact was already noticed by Lelli et al. (2018, see their Fig. 1), who interpreted the [O iii ] emission as coming from the redshifted, far-side of an ionized gas outflow, given that the [O iii ] kinematic major axis is well aligned with the AGN-driven radio lobes. Discrepant redshifts from several di ff erent lines were also found in Man et al. (2019) using rest-frame UV absorption lines from VLT / X-Shooter observations. This two-galaxies scenario is similar to the configuration of the Dragonfly Galaxy (Lebowitz et al. 2023), where two galaxies (though both gas-rich) are merging, while one of them hosts an AGN and two radio lobes. To test this scenario, we need high-resolution images from HST or JWST to possibly discern two separate stellar components.", "pages": [ 12 ] }, { "title": "6.3. The disk-halo degeneracy", "content": "In the previous sections, we discussed discrepancies between stellar and gas masses from 'photometric' and 'dynamical' methods. These discrepancies already emerge when we consider single-component mass models, which provide hard upper limits to the mass of each individual component. Clearly, the discrepancies become even more severe when we consider twocomponent models (gas and stars) or multi-component models with a DM halo. These facts highlight the severity of the diskhalo degeneracy at high z (Lelli 2023): if we cannot measure with high confidence the stellar and gas masses with 'photometric' methods, there is little hope to measure the DM content. The disk-halo degeneracy has been a long-standing issue in building mass models at z = 0 (van Albada et al. 1985). In particular, van Albada & Sancisi (1986) showed that one needs to know the baryonic mass with an accuracy of about 25% to fully break the degeneracy, even when extended rotation curves from H i observations are available (see their Fig. 5). For galaxies at cosmic noon, the stellar masses from SED fitting and the gas masses from standard methods (often based on highJ CO lines) are surely more uncertain than 25%, indicating that major observational and technical endeavours are needed to address the crucial question of the DM content of highz galaxies. In recent years, several studies reported DM fractions of galaxies at cosmic noon (Price et al. 2021; Nestor Shachar et al. 2023; Puglisi et al. 2023) and some of them even argued to find evidence for DM cores (Genzel et al. 2020; Bouché et al. 2022). These works, however, rarely discuss or investigate the disk-halo degeneracy, possibly indicating some over-confidence in knowing the true baryonic masses of highz galaxies. At z ≃ 0, one approach to break the disk-halo degeneracy has been to use NIR surface photometry in combination with dedicated stellar population models (Schombert & McGaugh 2014; Schombert et al. 2019). Even so, some systematic uncertainties remain due to the choice of the specific stellar population model and stellar initial mass function, so additional dynamical arguments are used to set the absolute calibration of the stellar mass (McGaugh & Schombert 2015; Lelli et al. 2016b,a,c). At high z , the current situation is much more uncertain, but rest-frame NIR imaging with JWST may be a promising route to measure robust stellar masses, while multi-line gas tracer observations may allow to measure robust gas masses, so that the disk-halo degeneracy could be ameliorated, using stringent, physically motivated priors when fitting the rotation curve.", "pages": [ 13 ] }, { "title": "7. Conclusions", "content": "In this work, we study the gas distribution and dynamics of a radio-loud AGN-host galaxy at z ≃ 2 . 6, PKS 0529-549, using ALMAdata of the [C i ] (2-1) line with a superb spatial resolution of 0.18 '' ( ∼ 1.5 kpc). Our results can be summarized as follows: High-resolution optical / NIR images, such as those from HST and / or JWST, are needed to probe the stellar mass distribution and break the disk-halo degeneracy, so to measure the actual DM content of highz galaxies. These images may also help to understand the discrepancies between the di ff erent methods for estimating stellar and gas masses at high z , which are key aspects to understand the formation and evolution of galaxies. Acknowledgements. L. L. and F. L. acknowledge the hospitality of ESO Garching, where most of this work was done. L.L. and Z.Y.Z acknowledge the support from the National Key R&D Program of China (2023YFA1608204). L.L. and Z.Y.Z acknowledge the support of the National Natural Science Foundation of China (NSFC) under grants 12173016 and 12041305. L.L. and Z.Y.Z acknowledge the science research grants from the China Manned Space Project, CMS-CSST-2021-A08 and CMS-CSST-2021-A07. L.L. and Z.Y.Z acknowledge the Program for Innovative Talents, Entrepreneur in Jiangsu. A. M. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through grant reference number RGPIN-2021-03046. T.G.B. acknowledges support from the Leading Innovation and Entrepreneurship Team of Zhejiang Province of China (Grant No. 2023R01008). This paper makes use of the following ALMA data: ADS / JAO.ALMA#2018.1.01669.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSTC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI / NRAO and NAOJ.", "pages": [ 13, 14 ] }, { "title": "References", "content": "Arribas, S., Colina, L., Bellocchi, E., Maiolino, R., & Villar-Martín, M. 2014, A&A, 568, A14 Broderick, J. W., De Breuck, C., Hunstead, R. W., & Seymour, N. 2007, MN- RAS, 375, 1059 Carilli, C. L. & Walter, F. 2013, ARA&A, 51, 105 Chen, C.-C., Hodge, J. A., Smail, I., et al. 2017, ApJ, 846, 108 Falkendal, T., De Breuck, C., Lehnert, M. D., et al. 2019, A&A, 621, A27 Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24 Förster Schreiber, N. M., Genzel, R., Bouché, N., et al. 2009, ApJ, 706, 1364 arXiv:2411.04290 Humphrey, A., Zeballos, M., Aretxaga, I., et al. 2011, MNRAS, 418, 74 Jones, G. C., Carilli, C. L., Shao, Y., et al. 2017, ApJ, 850, 180 Kamphuis, P., Józsa, G. I. G., Oh, S. . H., et al. 2015, MNRAS, 452, 3139 Kolwa, S., De Breuck, C., Vernet, J., et al. 2023, MNRAS, 525, 5831 Lacerna, I., Ibarra-Medel, H., Avila-Reese, V., et al. 2020, A&A, 644, A117 Lamperti, I., Harrison, C. M., Mainieri, V., et al. 2021, A&A, 654, A90 Lebowitz, S., Emonts, B., Terndrup, D. M., et al. 2023, ApJ, 951, 73 Legrand, L., McCracken, H. J., Davidzon, I., et al. 2019, MNRAS, 486, 5468 Lelli, F. 2023, arXiv e-prints, arXiv:2305.18224 Lelli, F., De Breuck, C., Falkendal, T., et al. 2018, MNRAS, 479, 5440 Article number, page 14 of 17 Lelli, F., Verheijen, M., & Fraternali, F. 2014, MNRAS, 445, 1694 Lelli, F., Verheijen, M., Fraternali, F., & Sancisi, R. 2012a, A&A, 537, A72 Lelli, F., Verheijen, M., Fraternali, F., & Sancisi, R. 2012b, A&A, 544, A145 Lelli, F., Zhang, Z.-Y., Bisbas, T. G., et al. 2023, A&A, 672, A106 Levy, R. C., Bolatto, A. D., Teuben, P., et al. 2018, ApJ, 860, 92 Li, P., Lelli, F., McGaugh, S., & Schombert, J. 2020, ApJS, 247, 31 Madau, P. & Dickinson, M. 2014, ARA&A, 52, 415 Smit, R., Bouwens, R. J., Carniani, S., et al. 2018, Nature, 553, 178 Stott, J. P., Swinbank, A. M., Johnson, H. L., et al. 2016, MNRAS, 457, 1888 Su, Y.-C., Lin, L., Pan, H.-A., et al. 2022, ApJ, 934, 173 Tadaki, K.-i., Kodama, T., Nelson, E. J., et al. 2017, ApJ, 841, L25 Talia, M., Pozzi, F., Vallini, L., et al. 2018, MNRAS, 476, 3956 Terzi'c, B. & Graham, A. W. 2005, MNRAS, 362, 197", "pages": [ 14 ] }, { "title": "Appendix A: Posterior probability distribution from Markov-Chain Monte-Carlo fits", "content": "Fig. A.1 and Fig. A.2 show 'corner plots' from MCMC fits to the rotation curves (see Section 5). The corner plots are obtained using the corner package (Foreman-Mackey 2016). The various panels of the corner plots show the posterior probability distribution of pairs of the fitting parameters (inner panels) as well as the marginalized 1D probability distribution of each parameter (outer panels). In the inner panels, individual MCMC samples outside the 2 σ confidence region are shown with black dots, while binned MCMC samples inside the 2 σ confidence region are shown by a grayscale; the black contours correspond to the 1 σ and 2 σ confidence regions. In the outer panels (histograms), red solid lines and dashed black lines correspond to the median and ± 1 σ values, respectively. The red solid lines continue in the outer panels, hitting the median value of the parameter (red square). The four corner plots correspond to the di ff erent mass models presented in Section 5, having an increasing number of mass components and free parameters. In addition, we show in Fig. A.3 a mass models where the stellar component is divided up in a thick exponential disk and spherical De Vaucouleurs' bulge. In general, the posterior probability distributions are wellbehaved and show clear peaks, indicating that the fitting quantities are well measured. The only exception is represented by the e ff ective radius of the stellar spheroid ( R e) which is poorly constrained in all models, so it should be interpreted as a fiducial upper limit.", "pages": [ 15 ] }, { "title": "A & A proofs: manuscript no. aanda", "content": "+4.82", "pages": [ 16 ] } ]

[ { "title": "Amit Kumar", "content": "Department of Physics, Royal Holloway, University of London, TW20 0EX, UK Department of Physics, University of Warwick, Coventry, CV4 7AL, UK ∗ E-mail: [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Kaushal Sharma", "content": "Forensic Science Laboratory Uttar Pradesh, Moradabad 244 001, India Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune 411 007, India E-mail: [email protected] A rapidly spinning, millisecond magnetar is widely considered one of the most plausible power sources for gamma-ray burst-associated supernovae (GRB-SNe). Recent studies have demonstrated that the magnetar model can effectively explain the bolometric light curves of most GRB-SNe. In this work, we investigate the bolometric light curves of 13 GRB-SNe, focusing on key observational parameters such as peak luminosity, rise time, and decay time, estimated using Gaussian Process (GP) regression for light curve fitting. We also apply Principal Component Analysis to all the light curve parameters to reduce the dimensionality of the dataset and visualize the distribution of SNe in lowerdimensional space. Our findings indicate that while most GRB-SNe share common physical characteristics, a few outliers, notably SNe 2010ma and 2011kl, exhibit distinct features. These events suggest potential differences in progenitor properties or explosion mechanisms, offering deeper insight into the diversity of GRB-SNe and their central engines. Keywords : Supernovae; Gamma-ray bursts; Magnetar; Principal Component Analysis.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The connection between long gamma-ray bursts (LGRBs) and broad-lined H/Hedeficient supernovae (Type Ic-BL SNe) was first established with the discovery of the nearby GRB 980425, which was accompanied by SN 1998bw 1-4 . Over the past two and a half decades, this field has evolved significantly, with the identification of more than fifty confirmed GRB-SN events 5-10 . Among these, SN 2011kl stands out as the only superluminous SN (SLSN), associated with the ultra-long (ul) GRB 111209A 11-13 ; however, only a handful of GRB-SNe have been well-observed across multiple bands. See the recent work by Ref. 14 for detailed data and a comprehensive compilation of parameters for GRBs and their associated SNe. This scarcity of well-observed GRB-SNe stems from several factors beyond the relatively low occurrence of GRB-SNe. These include the lower observed flux from SNe connected to high-redshift GRBs and dust extinction within the interstellar medium and host galaxies, particularly in the blue wavelengths, which hinders SN detection 15 . Furthermore, not all GRBs produce bright SNe, possibly due to insufficient 56 Ni production, significant fallback onto the newly formed black hole, or lower-energy explosions 16 , also see Refs. 17,18. Thus, an in-depth investigation of the limited sample of GRB-SNe is crucial for understanding their unique properties, and exploring their power sources remains one of the important keys to unlocking their nature. A widely accepted model suggests that a newborn millisecond magnetar, formed during the collapse of a massive star, could serve as the power source for both GRBs and their associated SNe 10,19-29 . Several studies have proposed the magnetar as the central engine in individual cases or small samples of GRB-SNe 11,30-33 . Additionally, millisecond magnetars have been proposed as power sources to explain the properties of various other types of SNe, including SLSNe 34-40 , classical Ic-BL SNe (e.g., SNe 1997ef and 2007ru 41 ), Type Ic (e.g., SN 2019cad 42 ), Type Ib (e.g., SN 2005bf 43 and SN 2012u 44,45 ), and fast blue optical transients (FBOTs 46-49 ). More recently, through light curve modelling of all GRB-SNe with comprehensive multi-band data, Ref. 10 (hereafter K24 ) demonstrated that the magnetar model can successfully explain the light curves of nearly all observed GRB-SNe. This study also explored magnetar parameters across a range of cosmic transients-including SLSNe, FBOTs, and both long and short GRBs -showing that variations in parameters like magnetic field strength and initial spin period can lead to the formation of different types of transients. In this study, we investigate the diversity of bolometric light curve parameters of 13 GRB-SNe in multi-dimensional space using parameters reported in K24 along with some newly derived parameters for 13 GRB-SNe.", "pages": [ 1, 2 ] }, { "title": "2. Bolometric Light Curves", "content": "The GRB-SNe used in the present work, along with the names of their associated GRBs (in parentheses), are as follows: SN 1998bw (GRB 980425), SN 2003dh (GRB 030329), SN 2003lw (GRB 031203), SN 2006aj (GRB 060218), SN 2010bh (XRF 100316D), SN 2010ma (GRB 101219B), SN 2011kl (GRB 111209A), SN 2012bz (GRB 120422A), SN 2013dx (GRB 130702A), SN 2016jca (GRB 161219B), SN 2017htp (GRB 171010A), SN 2017iuk (GRB 171205A), and SN 2019jrj (GRB 190114C). The bolometric light curves for all 13 GRB-SNe, listed in Table 1 and analyzed in this study were directly taken from K24 , which compiled these data from Refs. 6,33,50-57 and references therein. For details regarding the sample selection criteria, the method used to extract SN light curves from GRB+SN+host contributions, and the construction of bolometric light curves, refer to K24 . Additionally, light curve modelling of GRB-SNe using the MINIM code 58 , assuming a millisecond magnetar as the powering source, was performed in K24 , constraining various parameters such as initial rotational energy of the magnetar ( E p ), diffusion timescale ( t d ), spin-down timescale ( t p ), progenitor star radius ( R p ), ejecta expansion velocity ( V exp ), ejecta mass ( M ej ), initial spin period ( P i ), and magnetic field strength ( B ). As an extension of the analysis conducted in K24 , in this study, we estimate the peak luminosity ( L p ), peak time ( t r , time from explosion to peak luminosity, where the explosion time corresponds to the GRB detection time by satellite), rise time (time from half of the peak luminosity to peak in pre-peak data, t r L/ 2 ), and decay time (time from peak to half of the peak luminosity in post-peak data, t d L/ 2 ). To estimate these parameters, we used t r L/ 2 and t d L/ 2 , which effectively capture the light curve evolution around the peak. To estimate L p , t r , t r L/ 2 and t d L/ 2 , the light curve fitting was performed using Gaussian Processes (GP) regression 59,60 with a Radial Basis Function (RBF) kernel for interpolation, providing a more accurate fit compared to spline fitting. This approach replicates the bolometric light curves while naturally providing uncertainty estimates as a function of phase 61 . We performed these analyses using Python packages sklearn and scipy . The bolometric light curves of all 13 GRB-SNe in this study, along with GP interpolation for each, are shown in Figure 1. The phases where L p , L p / 2 in the pre-peak, and L p / 2 in the post-peak data are located, are marked by red, green, and blue dots on the GP fits, respectively. Due to insufficient pre-peak data, we were unable to calculate t r L/ 2 for SNe 1998bw, 2003dh, 2003lw, 2010ma, and 2019jrj. The estimated parameters L p , t r , t r L/ 2 , and t d L/ 2 for all GRB-SNe, along with their medians, are provided in Table 1, alongside other light curve fitting parameters adopted from table 1 of K24 . The median values of the parameters determined for the GRB-SNe in our sample are as follows: E p ≈ 4 . 8 × 10 49 erg, t d ≈ 17 days, t p ≈ 5 . 4 days, R p ≈ 7 . 2 × 10 13 cm, V exp ≈ 24 , 000 km s -1 , M ej ≈ 5 . 2 M ⊙ , P i ≈ 20 . 5 ms, B ≈ 20 . 1 × 10 14 G, L p ≈ 0 . 88 × 10 43 erg s -1 , t r ≈ 13 days, t r L/ 2 ≈ 8 . 5 days, and t d L/ 2 ≈ 14 days.", "pages": [ 2, 3, 4 ] }, { "title": "3. Dimensionality Reduction", "content": "To explore the estimated parameters among the GRB-SNe in this study, we apply Principal Component Analysis (PCA) to the parameters listed in Table 1, aiming to reduce the dimensionality of the dataset and visualize the distribution of SNe in lower-dimensional space. PCA transforms a multi-dimensional parameter space into a set of orthogonal components, called Principal Components (PCs). These components are linear combinations of the original parameters and are constructed to maximize the variance captured in the data while minimizing redundancy. PCA helps simplify complex datasets by identifying the most important directions of variation, allowing us to reduce the number of dimensions while preserving as much information as possible for analysis and visualization. PCA analysis reveals that the first two principal components (PC1 and PC2) capture approximately 50% of the variance in the dataset, with PC1 accounting for 25.5% and PC2 for 24.4%. The first five components collectively explain about 90% of the variance in the data, which is sufficient for visualizing and interpreting key trends. Figure 2 shows the distribution of the 13 SNe projected onto the PC1-PC2 plane. Most of the SNe form a relatively compact group near the origin, suggesting that their physical parameters share commonalities. 5 2 L/ SNe 2019jrj and 2006aj appear somewhat isolated, particularly along PC1, while SN 2017htp stands out along PC2, indicating that these objects may possess distinct parameter characteristics, e.g., SN 2017htp exhibits the highest spin-down timescale and SN 2006aj presents the lowest progenitor radius and highest magnetic field. Notably, SNe 2011kl and 2010ma, deviate significantly from the main cluster, suggesting that these SNe exhibit distinct observational and physical characteristics. As highlighted in K24 , SN 2011kl (one and only known ulGRB-associated SLSN) and SN 2010ma show the highest peak luminosity and magnetar initial rotational energy among the sample of GRB-SNe (see Table 1). Furthermore, SN 2011kl is characterized by the lowest magnetic field strength, while SN 2010ma displays the shortest spin-down timescale compared to all other events in the dataset. In addition to the diversity in the powering source parameters, this diversity in properties can likely be attributed to differences in their progenitor properties and host environment. These deviations underscore the complexity and diversity within the population of GRB-SNe, providing key insights into the diverse nature of their progenitors and explosion dynamics.", "pages": [ 4, 5, 6 ] }, { "title": "4. Discussion and Conclusion", "content": "In this study, we analyzed the bolometric light curves of 13 GRB-SNe and examined key physical and observational parameters. For each SN, we derived important light curve parameters such as peak luminosity ( L p ), rise time ( t r ), and both the prepeak ( t r L/ 2 ) and post-peak ( t d L/ 2 ) half-luminosity times using GP regression with an RBF kernel. This approach provided precise interpolations and uncertainty estimates, enabling a more robust analysis of the light curves and their associated parameters. We applied PCA to the light curve parameters estimated in this study, alongside those adopted from K24 , to explore correlations and reduce dimensionality. The PCA revealed that most SNe are clustered near the origin of the PCA plot, indicating shared physical characteristics. However, a few outliers, such as SNe 2010ma and 2011kl, diverged from the main cluster, suggesting distinct observational and physical properties. These deviations could also be attributed to the differences in progenitor properties and environments of these SNe. These findings underscore the need for more advanced modelling and theoretical studies to fully capture the diverse nature of GRB-SNe and their central enginebased powering sources. The results also emphasize the value of statistical techniques like PCA in unravelling the complex multi-dimensional parameter space of SNe. Expanding the sample size of GRB-SNe in future studies will allow for more rigorous statistical analysis, further enhancing our understanding of these extraordinary cosmic events.", "pages": [ 7 ] }, { "title": "Acknowledgement", "content": "A.K. is supported by the UK Science and Technology Facilities Council (STFC) Consolidated grant ST/V000853/1. This research has made use of data from the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory at the California Institute of Technology, under contract with NASA. We also acknowledge the invaluable support of NASA's Astrophysics Data System Bibliographic Services.", "pages": [ 7 ] }, { "title": "References", "content": "8 10", "pages": [ 8, 10 ] }, { "title": "2019).", "content": "12", "pages": [ 12 ] } ]