1001.0015.txt

arXiv:1001.0015v2  [astro-ph.CO]  10 May 2010DRAFT VERSION MAY12, 2010
Preprint typesetusingL ATEX styleemulateapjv. 11/10/09
ACOMPREHENSIVE ANALYSISOFUNCERTAINTIESAFFECTING THE
STELLARMASS –HALO MASS RELATIONFOR0 <z<4
PETERS. BEHROOZI1, CHARLIECONROY2, RISAH. WECHSLER1
Draftversion May12, 2010
ABSTRACT
We conductacomprehensiveanalysisoftherelationshipbet weencentralgalaxiesandtheirhostdarkmatter
halos, as characterized by the stellar mass – halo mass (SM–H M) relation, with rigorous consideration of
uncertainties. Our analysis focuses on results from the abu ndance matching technique, which assumes that
every dark matter halo or subhalo above a specific mass thresh old hosts one galaxy. We provide a robust
estimate of the SM–HMrelationfor 0 <z<1 anddiscussthe quantitativeeffectsof uncertaintiesin o bserved
galaxystellarmassfunctions(GSMFs)(includingstellarm assestimatesandcountinguncertainties),halomass
functions (including cosmology and uncertainties from sub structure), and the abundance matching technique
used to link galaxies to halos (including scatter in this con nection). Our analysis results in a robust estimate
of the SM–HM relation and its evolution from z=0 to z=4. The sh ape and evolution are well constrained for
z<1. The largest uncertainties at these redshifts are due to st ellar mass estimates (0.25 dex uncertainty in
normalization); however, failure to account for scatter in stellar masses at fixed halo mass can lead to errors
of similar magnitude in the SM–HM relation for central galax ies in massive halos. We also investigate the
SM–HM relation to z= 4, although the shape of the relation at higher redshifts re mains fairly unconstrained
whenuncertaintiesaretakenintoaccount. Wefindthatthein tegratedstarformationatagivenhalomasspeaks
at 10-20%ofavailable baryonsforall redshiftsfrom0 to 4. T hispeak occursat a halomass of7 ×1011M⊙at
z=0andthismassincreasesbyafactorof5to z=4. Atlowerandhighermasses,starformationissubstantia lly
lessefficient,withstellarmassscalingas M∗∼M2.3
hatlowmassesand M∗∼M0.29
hathighmasses. Thetypical
stellarmassforhaloswithmasslessthan1012M⊙hasincreasedby0 .3−0.45dexforhalossince z∼1. These
resultswill providea powerfultoolto informgalaxyevolut ionmodels.
Subject headings: dark matter — galaxies: abundances — galaxies: evolution — g alaxies: stellar content —
methods: N-bodysimulations
1.INTRODUCTION
A variety of physical processes are thought to be respon-
sible for the observed distribution of galaxy properties, a nd
distinguishing among them is one of the principal goals of
modern galaxy formation theory. Among the relevant mech-
anisms are those responsible for galaxy growth, such as star
formation and galaxy mergers, as well as those responsible
forregulatinggrowth,includingenergeticfeedbackbysup er-
novae, active galactic nuclei, cosmic ray pressure, and lon g
gascoolingtimes.
A fruitful approach to separating the influence of different
mechanisms is to constrain the redshift–dependent relatio n
between physical characteristics of galaxies, such as stel lar
mass,andthemassoftheirdarkmatterhalos. Thisispossibl e
because it is expected that many of these physical processes
depend primarily on the mass of a galaxy’sdark matter halo.
By connecting galaxies to their parent halos, one is able to
moreclearlyidentifyandconstrainthephysicalprocesses re-
sponsibleforgalaxygrowth.
The galaxystellar mass – halo massrelation hasadditional
utility because many properties of both galaxies and halos
are tightly correlated with halo mass. In addition, the stel -
lar mass – halo mass relation providesa mechanism for con-
necting predictions for the halo mass function and the mass-
1Kavli Institute for Particle Astrophysics and Cosmology; P hysics De-
partment, Stanford University, and SLAC National Accelera tor Labora-
tory, Stanford, CA 94305
2Department of Astrophysical Sciences, Princeton Universi ty, Prince-
ton, NJ 08544dependent spatial clustering of halos to the abundances and
clustering of galaxies. If a model for galaxy evolution is
able to reproduce the intrinsic galaxy mass – halo mass re-
lation in the correct cosmological model, then such a model
will match both the observed stellar mass function and the
stellar mass dependent clustering of galaxies. Simultane-
ously matching these two observational quantities and thei r
evolution has been difficult with either hydrodynamicalsim -
ulations or semi-analytic models of galaxy formation (e.g. ,
Weinbergetal. 2004; Liet al.2007).
There are several ways to constrain the galaxy mass –
halo mass relation. The first type of approach attempts
to directly measure the mass of galactic halos. Tech-
niques include weak lensing (e.g., Guzik&Seljak 2002;
Sheldonet al. 2004; Mandelbaumetal. 2006) and the use of
satellite galaxy or stellar velocities as tracers of the hal o po-
tentialwell(e.g.,Ashmanetal.1993;Zaritsky& White1994 ;
Pradaet al. 2003; vandenBoschet al. 2004; Conroyet al.
2007). While such methods are a relatively direct probe of
halo mass, they are limited in dynamic range; current obser-
vationsprobehalo massesfromroughly1012–1014M⊙at the
present epoch, and a smaller range at higher redshift. A sec-
ond approach is to identify groups and clusters of galaxies
either through optical or X-ray selected cluster catalogs, and
then directly measure their galaxy content (e.g., Lin&Mohr
2004; Hansenet al. 2009; Yanget al. 2007). This is limited
to relatively massive halos (although for optically identi fied
groupsit could extend to lower masses as new surveysprobe
dimmer galaxies in large enough volumes), and it also re-
quires accurate knowledge of the mass–observable relation2 BEHROOZI,CONROY& WECHSLER
(Yangetal. 2007; Hansenetal. 2009).
An alternative approach is to assume that the properties
of the halo population are known, for example from cos-
mological simulations, and then find a functional form re-
lating galaxies to halos which achieves agreement with a
variety of observations. This approach is less direct but
can be applied over a much larger dynamic range. Halo
occupation (e.g., Berlind&Weinberg 2002; Bullocketal.
2002; Cooray& Sheth 2002; Tinkeret al. 2005; Zhengetal.
2007) and conditional luminosity function modeling (e.g.,
Yanget al.2003; Cooray2006) fallintothiscategory.
In the past decade, a number of studies have found that
this latter approach can be greatly simplified using a tech-
nique called abundance matching. In its most basic form,
the technique assigns the most massive (or the most lumi-
nous) galaxiesto the most massive halos monotonically. The
techniquethusrequiresasinputonlytheobservedabundanc e
of galaxies as a function of mass, namely the galaxy stel-
lar mass function (alternatively the galaxy luminosity fun c-
tion) and the abundance of dark matter halos as a func-
tion of mass, namely the halo mass function. This tech-
nique has been shown to accurately reproduce a variety
of observational results including various measures of the
redshift– and scale–dependent spatial clustering of galax ies
(Colínetal. 1999; Kravtsov& Klypin 1999; Neyrincketal.
2004; Kravtsovet al. 2004; Vale&Ostriker 2004, 2006;
Tasitsiomi etal. 2004; Conroyetal. 2006; Shankaretal.
2006; Berrieret al. 2006; Marínet al. 2008; Guoet al. 2009;
Mosteretal. 2009). In the context of this technique, not
only central halos but also subhalos (halos contained withi n
the virial radii of larger halos) are included in the matchin g
process, meaning that satellite galaxies can be accounted f or
withoutanyadditionalparameters.
Applications of the abundance matching technique have
typically focused on using the default modeling assumption s
to derive statistical information about the galaxy – halo co n-
nection. Uncertaintiesinthederivedgalaxymass–halomas s
relationhavereceivedlittlesystematicattentioninthec ontext
of this technique (though see Mosteret al. 2009, for a recent
treatment of the attendant uncertainties). An accurate ass ess-
ment of the uncertainties is necessary to make strong state-
ments regarding the underlying physical processes respons i-
bleforthederivedgalaxy–halorelation. Inthepresentwor k
we undertake an exhaustive exploration of the uncertaintie s
relevantinconstructingthegalaxystellarmass–halomass re-
lationfromtheabundancematchingtechnique. Weconsidera
rangeofuncertaintiesrelatedtotheobservationalstella rmass
function,the theoretical halo mass function,and the under ly-
ing technique of abundance matching. The resulting galaxy
stellar mass – halo mass relation and associated uncertain-
tieswill provideabenchmarkagainstwhichgalaxyevolutio n
modelsmaybefruitfullytested.
This paper is divided into several sections. In §2 we detail
known sources of uncertainty which may affect our results.
Our methodologyfor modeling the effects of uncertainties i s
discussed in §3, providing simple conversions where possi-
ble to allow for different modeling choices. We present our
resulting estimates of the galaxy stellar mass – halo mass re -
lation for z<1 in §4 and describe the contribution of each
of the uncertainties to the overall error budget. Estimates for
theevolutionoftherelationoutto z∼4,forwhichtheuncer-
tainties are significantly less well-understood, are prese nted
in §5. Finally, we discuss the implications of this work in §6
andsummarizeourconclusionsin§7.Stellar masses throughout are quoted assuming a Chabrier
(2003) initial mass function (IMF), the stellar population
synthesis models of Bruzual& Charlot (2003), and the age
and dust models in Blanton& Roweis (2007). We consider
multiple cosmologies in this paper, but the main results as-
sume a WMAP5+SN+BAO concordance ΛCDM cosmology
(Komatsuet al. 2009) with ΩM=0.27,ΩΛ=1−ΩM,h=0.7,
σ8=0.8,andns=0.96.
2.UNCERTAINTIESAFFECTINGTHESTELLARMASS
TOHALO MASS RELATION
Uncertainties in the abundance matching technique for as-
signinggalaxiestodarkmatterhaloscanbeconceptuallyse p-
arated into three classes. The first is uncertainty in the abu n-
dance of galaxies as a function of stellar mass. This class
includes both uncertainties in counting galaxies due to sho t
noiseandsamplevariance,aswellasuncertaintiesinthest el-
larmassestimatesthemselves. Thesecondclassconcernsth e
darkmatter halos, andincludesuncertaintiesin cosmologi cal
parameters,theimpactofbaryoncondensation,andsubstru c-
ture. Finally, there are uncertainties in the process of mat ch-
inggalaxiesto halosarising primarilyfromthe intrinsics cat-
ter between galaxystellar mass and halo mass. Each of these
sources of uncertainty are described in detail below. The de -
tailedmodelingoftheseuncertaintiesis describedin§3.
2.1.UncertaintiesintheStellarMassFunction
Galaxystellar massesare notmeasureddirectly,but are in-
stead inferred from photometry and/or spectra. In particul ar,
as the observed stellar light is a function of many physical
processes (e.g., stellar evolution, star formation histor y and
metal–enrichmenthistory,andwavelength–dependentdust at-
tenuation),stellarmassesareestimatedviacomplicatedm od-
els to find the best fit to galaxy observations in a very large
parameter space. Assumptions and simplifications in these
models, along with the fact that the best fit may be only one
ofanumberoflikelypossibilities,meanthattherecanbesu b-
stantialuncertaintiesin calculatedstellar masses.
Different types of observations can yield different uncer-
taintiesinthesecalculations. Spectroscopicsurveysgen erally
recover more spectral features per galaxy and more accurate
redshifts than photometric surveys. However, spectroscop y
requires substantially more telescope time than photometr y.
Therefore, spectroscopic samples tend to be limited both in
area and depth, which translates into limitations in both vo l-
ume and the minimum stellar mass probed. An additional
problem for spectroscopic surveys such as the Sloan Digital
SkySurvey(SDSS,Yorket al.2000) isthatthespectraprobe
only the central regionsof galaxies (the SDSS spectra gathe r
onaverage1 /3 of the total flux fromgalaxiesat z=0.1). For
galaxies containing both a bulge and a disk, or galaxies with
radial gradients, the spectra will therefore not provide a f air
sampleoftheentiregalaxy(e.g.,Kewleyet al.2005);furth er-
more, this bias will be a function of redshift. For these rea-
sons,especiallyforrobustcomparisonsofevolution,we co n-
fine thispaperto photometric–basedstellar masses; howeve r,
samples with spectroscopic redshifts are used where avail-
able.
2.1.1.Principal Uncertainties inStellar Mass Functions
In this paper, we analyze seven main sources of uncertain-
ties in stellar mass functions applicable to photometric su r-
veys,listedbelowinroughorderofimportance.UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 3
1.Choice of stellar initial mass function (IMF): The lu-
minosity of stars scales as their mass to a large power
(dlnL/dlnM∼3.5),whiletheIMF,definedasthenum-
berofstarsformedperunitmass,scalesapproximately
asξ∝M−2.3(Salpeter 1955). Thus, the light from a
galaxy is dominated by the most massive stars, while
the total stellar mass is dominated by the lowest mass
stars. This fact, which has been known for decades
(e.g., Tinsley&Gunn 1976), implies that the assumed
form of the IMF at M/lessorsimilar0.5M⊙will have no effect on
the integratedlight from galaxies, but will have a large
effectonthetotalstellarmass. Nonetheless,constraints
onthe total dynamicalmassofspheroidalsystemspro-
vide a valuable independent check on the form of the
IMF at low masses. Cappellariet al. (2006) find, for
example, that the IMF proposed by Chabrier (2003)
for the Solar neighborhood is consistent with dynam-
ical constraintsonmassesofnearbyellipticals.
We do not marginalize over IMFs in our error calcula-
tionsforthereasonthatitisrelativelysimpletoconvert
fromourchoiceofIMF(Chabrier2003)tootherIMFs.
For our purposes the IMF becomes a serious source
of systematic uncertainty only if the IMF varies with
galaxy properties or if it evolves. It should be noted,
however,thattheIMF canalsointroducemorecompli-
catedsystematiceffectsassociatedwiththeinferredstar
formationrate,whichmayinturnimpactstellarmasses
in non–trivialways(e.g.Conroyetal. 2010).
2.Choice of Stellar Population Synthesis (SPS) model:
SPS modeling efforts have grown substantially in so-
phistication in the past decade (e.g. Leithereretal.
1999; Bruzual&Charlot 2003; LeBorgneet al. 2004;
Maraston 2005). Yet, significant uncertainties re-
main (e.g., Charlotetal. 1996; Charlot 1996; Yi 2003;
Lee etal. 2007; Conroyetal. 2009). Treatments of
convection vary, leading to different main sequence
turn off times for intermediate mass stars. Advanced
stages of stellar evolution, including blue stragglers,
thermally–pulsating AGB stars (TP–AGB), and hori-
zontal branch stars, are poorly understood, both obser-
vationally and theoretically. The theoretical spectral
libraries contain known deficiencies, especially for M
giants, where the effective temperatures are low and
where hydrodynamic effects become important. Em-
pirical stellar libraries to some extent circumventthese
issues, althoughthey are plaguedbyincompletecover-
age in the HR diagram and difficulties associated with
deriving stellar parameters. See Conroyet al. (2009)
andPercival&Salaris(2009) forrecentreviews.
Differentstellarpopulationsynthesismodelstreatthese
issues differently, which can result in large system-
atic differences in the derived stellar mass. For
instance, the model of Maraston (2005) compared
to Bruzual&Charlot (2003) has systematic differ-
ences of 0.1dex in stellar mass (Salimbeniet al. 2009;
Pérez-Gonzálezetal. 2008). However,Salimbenietal.
(2009) reports that the model in Bruzual (2007) (with
a revisedtreatmentofTP-AGBstars)yieldssystematic
differencesinstellarmassrelativetoBruzual&Charlot
(2003), which ranges from 0.05dex for 1011M⊙galax-
iesto0.3dexfor109.5M⊙galaxies. Conroyet al.(2009)
show thatuncertaintyin theluminosityofthe TP-AGBphasecanshiftstellar massesbyasmuchas ±0.2dex.
3.Parameterization of star formation histories: In or-
der to estimate stellar masses, model libraries are con-
structed with a large range in star formation histories
(SFHs),dustattenuation(seebelow),and,oftenbutnot
always,metallicity. Theadoptedfunctionalformofthe
SFHisanothersourceofsystematicuncertainty,astyp-
ically very simple functional forms are assumed. Sev-
eral authors have investigated various aspects of this
problem. When attempting to model observed pho-
tometry, Pérez-Gonzálezet al. (2008) found that a sin-
gle stellar population model (in particular, star forma-
tion proportional to e−t/τ, which is a commonly-used
parameterization) systematically underpredicts stellar
mass by 0.18 dex comparedto a double stellar popula-
tion model (exponentially decaying star formation fol-
lowed by a later starburst). The particular parameteri-
zation of SFHs may also lead to systematic differences
asafunctionofstellarmass. Leeet al.(2009)analyzed
a sample of mock Lyman–break galaxies at z∼4−5
and found that simple SFHs produced best–fit stellar
masses that were under or overestimated by ∼ ±50%
dependingontherest-framegalaxycolor. Thisbiaswas
attributedto thechaoticSFHsofthemockgalaxies.
4.Choice of dust attenuation model: Because dust red-
dens starlight, it is difficult to separate the effects of
dustfromstellarpopulationeffects,especiallywhenfit-
ting optical photometry. The effects of dust are also
knowntochangedependingongalaxyinclination(e.g.,
Driveret al. 2007). Hence, the choice of dust attenu-
ation law has a nontrivial effect on the inferred stel-
lar population ages and, consequently, star formation
histories derived from photometric and even spectro-
scopic surveys (Panteret al. 2007). In terms of the
effect on stellar masses, Pérez-Gonzálezetal. (2008)
compared the dust models of Calzetti et al. (2000) and
Charlot&Fall (2000), finding a systematic difference
of 0.10dex. Panteret al. (2007) found a similar differ-
ence between Calzetti et al. and models based on ex-
tinction curves from the Small and Large Magellanic
Clouds. The effectsof varyingdust attenuationmodels
have also been explored recently by Marchesiniet al.
(2009) and Muzzinet al. (2009), with similar results.
We have used the stellar population fitting proce-
dure described in Conroyet al. (2009) to compare the
Calzetti et al. dust attenuation law to the dust model
used in the kcorrect package (Blanton&Roweis
2007). We find a median offset of 0.02dex but also a
systematic trend such that two galaxies whose stellar
massesareestimatedwithCalzettidustattenuationand
are separated by 1.0dex will have kcorrect masses
separatedby(onaverage)only0.92dex.
5.Statistical errors in individual stellar mass estimates:
Stellar mass estimates for each galaxy are subject to
statistical errors due to uncertainty in photometry as
well as uncertainty in the SPS parameters for a given
set of model assumptions. We treat this here as a ran-
dom statistical error. While it may seem that random
scatter in individual stellar masses should on average
have no systematic effect, it in fact introduces a sys-
tematic error analogous to Eddington bias (Eddington4 BEHROOZI,CONROY& WECHSLER
1940) observed in luminosity functions. As the stellar
mass function drops off steeply beyond a certain char-
acteristic stellar mass, there are many more low stellar
mass galaxies that can be up-scattered than there are
high stellar mass galaxies that can be down-scattered
by errors in stellar mass estimates. This asymmetric
scatter implies that the drop-off in number density at
highmassesbecomesshallowerinthepresenceofscat-
ter. We discuss this effect in detail in §3.1.2 (see also
the AppendixinCattaneoet al.2008).
6.Sample variance: Surveysof finite regionsof the Uni-
verse are susceptible to large–scale fluctuations in the
number density of galaxies. This is no longer a dom-
inant source of uncertainty for the volumes probed at
lowredshiftbytheSDSS,butitisanimportantconsid-
eration for higher–redshift surveys, which cover much
smaller comoving volumes. Most authors who con-
sidersamplevarianceattempttominimizeitbyaverag-
ingoverseveralfields(e.g.,Pérez-Gonzálezetal.2008;
Marchesinietal. 2009). Very few surveys at z>0 at-
tempt to estimate the magnitude of the error except by
computing the field–to–field variance, which is often
an underestimate when insufficient volume is probed
(Crocceet al.2009). Wedetailamoreaccuratemethod
based on simulations to model the error arising from
samplevariancein §3.2.4.
7.Redshift errors: Photometric redshift errors blur the
distinction between GSMFs at different redshifts.
While a galaxy may be scattered either up or down in
redshift space, volume-limited survey lightcones will
contain larger numbers of galaxies at higher redshifts,
meaning that the GSMF as reported at lower redshifts
willbeartificiallyinflated. Moreover,asgalaxiesatear-
liertimeshavelowerstellarmasses,surveyswilltendto
report artificially larger faint-end slopes in the GSMF.
However, as these errors are well known, it is easy to
correct for their effects on the stellar mass function,
as has been done for the data in Pérez-Gonzálezetal.
(2008) (seetheappendixofPérez-Gonzálezet al.2005
fordetailsonthisprocess).
For completeness, we remark that galaxy-galaxy lensing
will also result in systematic errorsin the GSMF at high red-
shifts because galaxy magnification will result in higher ob -
served luminosities. However, from ray-tracing studies of
the Millennium simulation (Hilbertet al. 2007), the expect ed
scatter in galaxystellar masses fromlensingis minimal (e. g.,
0.04 dex at z= 1) compared to the other sources of scatter
above (e.g., 0.25 dex from different model choices). For tha t
reason,we donot modelgalaxy-galaxylensing effectsin thi s
paper.
2.1.2.Additional Systematics atz >1
Recently, it has become clear that current estimates of
the evolution in the cosmic SFR density are not consistent
with estimates of the evolution of the stellar mass density
atz>1 (Nagamineet al. 2006; Hopkins&Beacom 2006;
Pérez-Gonzálezet al. 2008; Wilkinset al. 2008a). The ori-
gin of this discrepancy is currently a matter of debate. One
solution involves allowing for an evolving IMF with red-
shift (Davé 2008; Wilkinsetal. 2008a). While such a so-
lution is controversial, a number of independent lines ofevidence suggest that the IMF was different at high red-
shift(Lucatelloetal. 2005;Tumlinson2007a,b; vanDokkum
2008). Reddy&Steidel (2009) offer a more mundane ex-
planation for the discrepancy. They appeal to luminosity–
dependentreddeningcorrectionsin the ultraviolet lumino sity
functionsat highredshift,anddemonstratethat the purpor ted
discrepancythenlargelyvanishes.
In contrast to results at z>1, there does seem to be
an accord that for z<1 both the integrated SFR and the
total stellar mass are in good agreement if one assumes
(as we have) a Chabrier (2003) IMF (see Wilkinset al.
2008b; Pérez-Gonzálezetal. 2008; Hopkins& Beacom
2006;Nagamineet al.2006; Conroy&Wechsler 2009).
Because of the discrepancy between reported SFRs and
stellar massesin the literature,it is clearthat estimates ofun-
certaintiesin galaxystellar mass functionsandSFRs at z>1
tend to underestimate the true uncertainties; for this reas on,
we separately analyze results for z<1 in §4 and z>1 in §5
ofthispaper.
2.2.Uncertaintiesin theHaloMassFunction
Darkmatterhalopropertiesoverthemassrange1010−1015
M⊙have been extensively analyzed in simulations (e.g.,
Jenkinset al. 2001; Warrenet al. 2006; Tinkeret al. 2008),
and the overall cosmology has been constrained by probes
such as WMAP (Spergeletal. 2003; Komatsuetal. 2009).
As such, uncertainties in the halo mass function have on the
wholemuch less impact thanuncertaintiesin the stellar mas s
function. We present our primary results for a fixed cosmol-
ogy (WMAP5), but we also calculate the impact of uncer-
tain cosmological parameters on our error bars. We do not
marginalize over the mass function uncertainties for a give n
cosmology,astherelevantuncertaintiesareconstraineda tthe
5% level (when baryonic effects are neglected, see below;
Tinkeret al. 2008). Additionally, in Appendix A, a simple
method is described to convert our results to a different cos -
mology using an arbitrary mass function. For completeness,
wementionthethreemostsignificantuncertaintieshere:
1.Cosmologicalmodel: Thestellarmass–halomassrela-
tionhasdependenceoncosmologicalparametersdueto
the resulting differences in halo number densities. We
investigate this both by calculating the relation for two
specific cosmological modes (WMAP1 and WMAP5
parameters)andthenbycalculatingtheuncertaintiesin
the relation over the full range of cosmologiesallowed
by WMAP5 data. We findthat in all casesthese uncer-
tainties are small compared to the uncertainties inher-
entinstellarmassmodeling(§2.1.1),althoughtheyare
larger than the statistical errors for typical halo masses
at lowredshift.
2.Uncertainties in substructure identification: Different
simulations have different methods of identifying and
assigning masses to substructure. Our matching meth-
ods make use only of the subhalo mass at the epoch
of accretion ( Macc) as this results in a better match to
clustering and pair–count results (Conroyetal. 2006;
Berrieret al. 2006), so we are largely immune to the
problem of different methods for calculating subhalo
masses. Ofgreaterconcernistheabilitytoreliablyfol-
lowsubhalosinsimulationsastheyaretidallystripped.
Two related issues apply here. The first is that it is not
clear how to account for subhaloswhich fall below theUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 5
resolution limit of the simulation. The second is that
theformationofgalaxieswilldramaticallyincreasethe
binding energy of the central regions of subhalos, po-
tentiallymakingthemmoreresilienttotidaldisruption.
Hydrodynamicsimulationssuggestthatthislattereffect
issmallexceptforsubhalosthatorbitnearthecentersof
themostmassiveclusters(Weinbergetal.2008). How-
ever, while these details are important for accurately
predicting the clustering strength on small scales ( /lessorsimilar1
Mpc), they are not a substantial source of uncertainty
fortheglobalhalomass—stellarmassrelationbecause
satellites are always sub-dominant ( /lessorsimilar20%) by num-
ber. We discuss the analytic method we use to model
the satellite contribution to the halo mass function in
§3.2.2.
3.Baryonic physics: Recent work by Staneket al. (2009)
suggests that gas physics can affect halo masses rela-
tive to dark matter-onlysimulations by -16% to +17%,
leading to number density shifts of up to 30% in the
halo massfunctionat 1014M⊙. Withoutevidencefora
clear bias in one direction or the other—the models of
gasphysicsstillremaintoouncertain—wedonotapply
a correction for this effect in our mass functions. Un-
certainties of this magnitude are larger than the statis-
tical errors in individual stellar masses at low redshift,
but are still small in comparisonto systematic errorsin
calculatingstellar masses.
For completeness, we note that the effects of sample vari-
ance on halo mass functions estimated from simulations are
small. Current simulations readily probe volumes of 1000
(h−1Mpc)3(Tinkeretal. 2008), and so the effects of sample
varianceonthe halomassfunctionaredwarfedbythe effects
of sample variance on the stellar mass function; we therefor e
donotanalyzethemseparatelyinthispaper.
We also remark on the issue of mass definitions. Al-
though abundance matching implies matching the most mas-
sive galaxiesto the most massivehalos, thereis little cons en-
susonwhichhalomassdefinitiontouse,withpopularchoices
beingMvir(mass within the virial radius), M200(mass within
a sphere with mean density 200 ρcrit), andMfof(mass deter-
minedby a friends-of-friendsparticle linkingalgorithm) . We
chooseMvirfor this paper and note that the largest effect of
choosinganothermassalgorithmwill beapurelydefinitiona l
shift in halo masses. We expect that scatter between any two
of these mass definitions is degenerate with and smaller than
the amountofscatter in stellar massesat fixedhalo mass(the
lattereffectisdiscussedin§2.3).
2.3.Uncertaintiesin AbundanceMatching
Finally, there are two primary uncertainties concerningth e
abundancematchingtechniqueitself:
1.Nonzero scatter in assigning galaxies to halos: While
host halo mass is strongly correlated with stellar mass,
the correlation is not perfect. At a given halo mass,
the halomergerhistory,angularmomentumproperties,
and cooling and feedback processes can induce scatter
between halo mass and galaxystellar mass. This is ex-
pectedtoresultinscatterinstellarof ∼0.1–0.2dexata
given halo mass, see §3.3.1 for discussion. The scatter
between halo mass and stellar mass will have system-
atic effects on the mean relation for reasons analogousto those mentioned for statistical error in stellar mass
measurements. At the high mass end where both the
halo and stellar mass functions are exponential, scat-
ter in stellar mass at fixed halo mass (or vice versa)
will alter the average relation because there are more
low mass galaxies that are upscattered than high mass
galaxiesthataredownscattered.
2.Uncertainty in Assigning Galaxies to Satellite Halos:
It is not clear that the halo mass — stellar mass rela-
tion should be the same for satellite and central galax-
ies. Once a halo is accreted onto a larger halo, it starts
to lose halo mass because of dynamicaleffects such as
tidal stripping. While stripping of the halo appears to
be a relatively dramatic process (e.g., Kravtsovet al.
2004), the stripping of the stellar component proba-
bly does not occur unless the satellite passes very near
to the central object because the stellar component is
muchmoretightlyboundthanthehalo. Itisclearfrom
the observed color–density relation (Dressler 1980;
Postman&Geller 1984; Hansenet al. 2009) that star
formation in satellite galaxies must eventually cease
with respect to galaxiesin the field. It is less clear how
quicklystar formationceases, andwhetherornot there
is a burst ofstar formationuponaccretion. All ofthese
issues can potentially alter the relation between halo
andstellarmassforsatellites(althoughthemodelingre-
sults ofWang etal. 2006suggestthat the halo–satellite
relation is indistinguishable from the overall galaxy–
halorelation).
3.METHODOLOGY
Ourprimarygoalistoprovidearobustestimateofthestel-
lar mass – halo mass relation over a significant fraction of
cosmic time via the abundance matching technique. We aim
to constructthis relation by taking into account all of the r el-
evant sources of uncertainty. This section describes in de-
tail a number of aspects of our methodology, including our
approach for incorporating uncertainties in the stellar ma ss
function ( §3.1), a summary of the adopted halo mass func-
tionsand associateduncertainties( §3.2), the uncertaintiesas-
sociatedwithabundancematching(§3.3),ourchoiceoffunc -
tionalformforthestellarmass–halomassrelation,includ ing
adiscussionofwhycertainfunctionsshouldbepreferredov er
others (§3.4), and the Markov Chain Monte Carlo parameter
estimationtechnique( §3.5). Forreadersinterestedinthegen-
eral outline of our process but not the details, we conclude
witha briefsummaryofourmethodology(§3.6).
3.1.ModelingStellarMassFunctionUncertainties
Asdiscussedin§2,thereareseveralclassesofuncertainti es
affectingthewaythestellarmassfunctionisusedintheabu n-
dance matching process. In this section, we discuss system-
aticshiftsinstellarmassestimatesandtheeffectsofstat istical
errorsonthestellar massfunction.
3.1.1.Modeling Systematic ShiftsinStellar Mass Estimates
Most studies on the GSMF report Schechter function fits
as well as individual data points; many also provide statist i-
calerrors. However,evenwhensystematicerrorsarereport ed
(either in Schechter parameters or at individual data point s),
the systematic error estimates are of limited value unless o ne
is also able to model shifts in the GSMF caused by such er-
rors.6 BEHROOZI,CONROY& WECHSLER
Fortunately, based on the discussion in §2.1.1, there seem
to be two main classes of systematic errors causing shifts in
theGSMF:
1. Over/underestimationofallstellarmassesbyaconstant
factorµ. This appears to cover the majority of errors,
includingmostdifferencesinSPSmodeling,dustatten-
uationassumptions,andstellar populationagemodels.
2. Over/underestimation of stellar masses by a factor
which depends linearly on the logarithm of the stel-
lar mass (i.e., depends on a power of the stellar mass).
Thiscoversthemajorityoftheremainingdiscrepancies
between different SPS models and different stellar age
models.
Bothformsoferrorare modeledwith theequation
log10/parenleftbiggM∗,meas
M∗,true/parenrightbigg
=µ+κlog10/parenleftbiggM∗,true
M0/parenrightbigg
.(1)
Without loss of generality, we may take M0= 1011.3M⊙(the
fixed point of the variation between the Bruzual 2007 and
Bruzual& Charlot 2003 models found by Salimbenietal.
2009), allowing the prior on M0to be absorbed into the prior
onµ.
For the prior on µ, we consider four contributing sources
of uncertainty. We adopt estimates of the uncertainty from
the SPS model( ≈0.1dex),the dust model( ≈0.1dex),and as-
sumptions about the star formation history ( ≈0.2dex) from
Pérez-Gonzálezet al. (2008) as detailed in §2.1.1. Additio n-
ally, we have the variation in κlog10(M0) (at most 0.1dex, as
|κ|/lessorsimilar0.15 — see below). Assuming that these are statisti-
cally independent, they combine to give a total uncertainty
of 0.25dex, which is consistent with the accepted range for
systematicuncertaintiesinstellarmass(Pérez-González etal.
2008; Kannappan& Gawiser 2007; vanderWel et al. 2006;
Marchesiniet al. 2009). For lack of adequate information
(i.e., different models) to infer a more complicated distri bu-
tion, we assume that µhas a Gaussian prior. As more stud-
ies ofthe overallsystematic shift µbecomeavailable,ouras-
sumptions for the prior on µand the probability distribution
will likely need corrections. We remark, however, that our
results can easily be converted to a different assumption fo r
µ, asµsimply imparts a uniform shift in the intrinsic stellar
massesrelativeto theobservedstellar masses.
For the prior on κ, the result of Salimbeniet al. (2009)
would suggest |κ|/lessorsimilar0.15. As mentionedin §2.1.1, we found
that|κ| ≈0.08 between the Blanton& Roweis (2007) and
Calzetti et al.(2000)modelsfordustattenuation. Li &Whit e
(2009) finds |κ|/lessorsimilar0.10 between Blanton& Roweis (2007)
and Bell etal. (2003) stellar masses. Without a large num-
ber of other comparisons, it is difficult to robustly determi ne
the priordistributionfor κ; however,motivatedby the results
just mentioned, we assume that the prior on κis a Gaussian
ofwidth0.10centeredat0.0.
We remark that some authors have considered much more
complicated parameterizations of the systematic error. Fo r
example, Li &White (2009) considers a four-parameter hy-
perbolic tangent fit to differences in the GSMF caused by
different SPS models, as well as a five-parameter quartic fit.
However,wedonotconsiderhigher-ordermodelsforsystem-
atic errors for several reasons. First, given that second- a nd
higher-ordercorrectionswill resultonlyinverysmall cor rec-
tions to the stellar masses in comparison to the zeroth-orde rcorrection ( µ≈0.25dex), the corrections will not substan-
tiallyeffectthesystematicerrorbars. Second,wedonotkn ow
ofanystudieswhichwouldallowustoconstructpriorsonthe
higher-order corrections. Finally, with higher-order mod els,
there is the serious danger of over-fitting—that is, with ver y
loose priors on systematic errors, the best-fit parameters f or
the systematic errors will be influenced by bumps and wig-
gles in the stellar mass function due to statistical and samp le
variance errors. Hence, the interpretive value of the syste m-
aticerrorsbecomesincreasinglydubiouswitheachadditio nal
parameter.
3.1.2.Modeling Statistical ErrorsinIndividual Stellar Mass
Measurements
In addition to the systematic effectsdiscussed in the previ -
oussection,measurementofstellarmassesissubjecttosta tis-
ticalerrors. Evenforafixedsetofassumptionsaboutthedus t
model, SPS model, and the parameterization of star forma-
tion histories, stellar masses will carry uncertainties be cause
the mapping between observables and stellar masses is not
one-to-one. This additional source of uncertainty has uniq ue
effects on the GSMF. Observers will see an GSMF ( φmeas)
which is the true or “intrinsic” GSMF ( φtrue) convolved with
theprobabilitydistributionfunctionofthemeasurements cat-
ter. Forinstance,ifthescatterisuniformacrossstellarm asses
and has the shape of a certain probability distribution P, we
have:
φmeas(M)=/integraldisplay∞
−∞φtrue(10y)P/parenleftbig
y−log10(M)/parenrightbig
dy,(2)
whereyis the integrationvariable,in units of log10mass. As
derived in Appendix B, the approximate effect of the convo-
lutionis
log10/parenleftbiggφmeas(M)
φtrue(M)/parenrightbigg
≈σ2
2ln(10)/parenleftbiggdlogφtrue(M)
dlogM/parenrightbigg2
,(3)
whereσis the standard deviation of P. That is to say, the
effectof the convolutiondependsstronglyon the logarithm ic
slope ofφtrue. Where the slope is small (i.e., for low-mass
galaxies), there is almost no effect. Above 1011M⊙, where
the GSMF becomes exponential, there can be a dramatic ef-
fect, with the result that φtrueis more than an order of mag-
nitudelessthan φmeasbecauseit becomesfar morelikelythat
stellar mass calculation errors produce a galaxy of very hig h
perceived stellar mass than it is for there to be such a galaxy
inreality(seeforexampleCattaneoet al. 2008).
For the observed z∼0 GSMF, we take the probabil-
ity distribution Pto be log-normal with 1 σwidth 0.07dex
fromtheanalysisofthephotometryoflow–redshiftluminou s
red galaxies (LRGs) (Conroyet al. 2009). Kauffmannet al.
(2003) found similar results regarding the width of P. This
function only accounts for the statistical uncertainties m en-
tioned above and does not include additional systematic un-
certainties. In light of Equation 3, we use LRGs to esti-
matePbecause LRGs occupy the high stellar mass regime
where measurementerrors are most likely to affect the shape
of the observed GSMF. However, the single most important
attribute of the distribution Pis its width; the main results do
not change substantially if an alternate distribution with non-
Gaussiantailsbeyondthe1 σlimitsofPisused.
For higher redshifts, we scale the width of the probabil-
ity distributionto accountfor the fact that mass estimates be-
come less certain at higher redshift (e.g., Conroyet al. 200 9;UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 7
Kajisawaet al. 2009):
P(∆log10M∗,z)=σ0
σ(z)P0/parenleftbiggσ0
σ(z)∆log10M∗/parenrightbigg
,(4)
whereP0is the probability distribution at z=0 (as discussed
above),σ0is the standard deviation of P0, andσ(z) gives the
evolutionofthe standarddeviationasa functionofredshif t.
Conroyet al. (2009) did not give a functional form for
σ(z), but they calculate fora handfulof massive galaxiesthat
σ(z= 2) is≈0.18dex, as compared to σ(z= 0)≈0.07dex.
Kajisawaet al. (2009) performeda similar calculation (alb eit
with a differentSPS model)ofthe distributioninseveralre d-
shift bins; their resultsshow gradualevolutionfor σ(z) out to
z=3.5 for high stellar mass galaxies consistent with a linear
fit:
σ(z)=σ0+σzz. (5)
The results of Kajisawaet al. (2009) suggest that σz=0.03-
0.06dexforLRGs. Asthisisconsistentwiththevalueof σz=
0.05dexwhichwouldcorrespondtoConroyet al.(2009),we
adopt the linear scaling of Equation 5 with a Gaussian prior
ofσz=0.05±0.015dex.
Note that the effect of this statistical error on the stellar
mass functionis minimalbelow 1011M⊙, andthereforedoes
notaffectthestellarmass–halomassrelationforhalosbel ow
∼1013M⊙,asdiscussedin §4.2. Whilethisscatterdoeshave
an effect on the shape of the stellar mass function for high-
mass galaxies, the qualitative predictions we make from thi s
analysisaregenericto alltypesofrandomscatter.
3.2.HaloMassFunctions
The halo mass function specifies the abundance of halos
as a function of mass and redshift. A number of analytic
modelsandsimulation–basedfittingfunctionshavebeenpre -
sented for computing mass functions given an input cos-
mology (e.g., Press& Schechter 1974; Jenkinset al. 2001;
Warrenet al. 2006; Tinkeret al. 2008). For most of our re-
sultswewilladopttheuniversalmassfunctionofTinkereta l.
(2008), as described below. Analytic mass functions are
preferableasthey1)allowmassfunctionstobecomputedfor
arangeofcosmologiesand2)donotsuffersignificantlyfrom
sample variance uncertainties, because the analytic relat ions
are typically calibrated with very large or multiple N−body
simulations.
For some purposes it will be useful to also consider full
halo merger trees derived directly from N−body simulations
that have sufficient resolution to follow halo substructure s.
The simulations used herein will be described below, in ad-
ditiontoourmethodsformodelinguncertaintiesintheunde r-
lyingmassfunction,includingcosmologyuncertainties,s am-
plevarianceinthegalaxysurveys,andourmodelsforsatell ite
treatment.
3.2.1.Simulations
For the principal simulation in this study (“L80G”), we
used a pure dark matter N-body simulation based on Adap-
tive Refinement Tree (ART) code (Kravtsovet al. 1997;
Kravtsov&Klypin 1999). The simulation assumed flat, con-
cordance ΛCDM (ΩM=0.3,ΩΛ=0.7,h=0.7, andσ8=0.9)
and included 5123particles in a cubic box with periodic
boundary conditions and comoving side length 80 h−1Mpc.
These parameters correspondto a particle mass resolution o f≈3.2×108h−1M⊙. For this simulation, the ART code be-
gins with a spatial grid size of 5123; it refines the grid up to
eight times in locally dense regions, leading to an adaptive
distance resolution of ≈1.2h−1kpc (comoving units) in the
densest parts and ≈0.31h−1Mpc in the sparsest parts of the
simulation.
In this simulation, halos and subhalos were identified
using a variant of the Bound Density Maxima algorithm
(Klypinetal. 1999). Halo centers are located at peaks in the
density field smoothed over a 24-particle SPH kernel (for a
minimumresolvable halomass of 7 .7×109h−1M⊙). Nearby
particles are classified as bound or unbound in an iterative
process;onceall thelocallyboundparticleshavebeenfoun d,
halo parameters such as the virial mass Mvirand maximum
circularvelocity Vmaxmaybe calculated. (See Kravtsovet al.
2004 for complete details on the algorithm). The simulation
is complete down to Vmax≈100 km s−1, corresponding to a
galaxystellar massof108.75M⊙atz=0.
The ability of L80G to track satellites with high mass and
forceresolutiongivesitseveraluses. MergertreesfromL8 0G
informourprescriptionforconvertinganalytical central -only
halo mass functions to mass functions which include satel-
lite halos (see §3.2.2). Additionally, the merger trees all ow
forevaluationofdifferentmodelsofsatellitestellar evo lution
with full consistency (see §3.3.2). Finally, the knowledge of
which satellite halos are associated with which central hal os
allowsforestimatesofthetotalstellarmass(inthecentra land
allsatellite galaxies)— halomassrelation(see §4.3.6).
We also make use of a secondary simulation from the
Large Suite of Dark Matter Simulations (LasDamas Project,
http://lss.phy.vanderbilt.edu/lasdamas/) in our sample vari-
ancecalculations. TheL80Gsimulationistoosmallforusei n
calculatingthesamplevariancebetweenmultipleindepend ent
mocksurveys,butthelargersizeoftheLasDamassimulation
(420h−1Mpc,14003particles)makesitidealforthispurpose.
However, the LasDamas simulation has poorer mass resolu-
tion (a minimum particle size of 1 .9×109M⊙) and force
resolution (8 h−1kpc), making it unable to resolve subhalos
(particularlyafteraccretion)aswell asL80G.TheLasDama s
simulation assumes a flat, ΛCDM cosmology ( ΩM= 0.25,
ΩΛ= 0.75,h= 0.7, andσ8= 0.8) which is very close to the
WMAP5best-fitcosmology(Komatsuetal.2009). Collision-
less gravitational evolution was provided by the GADGET-2
code (Springel 2005). Halos are identified using friends of
friendswith a linkinglengthof 0.164. The subfind algorithm
Springel(2005) isusedtoidentifysubstructure.
Asmentioned,theprimaryuseoftheLasDamassimulation
is in sampling the halo mass functions in mock surveys to
model the effects of sample variance on high-redshiftpenci l-
beam galaxy surveys. The mock surveys are constructed so
as to mimic the observationsin Pérez-Gonzálezet al. (2008) .
Ineachmocksurvey,threepencil-beamlightcones(matchin g
the angular sizes of the three fields in Pérez-Gonzálezet al.
2008) with random orientations are sampled from a random
originin the simulationvolumeoutto z=1.3. Thus,bycom-
paring the halo mass functionsin individualmock surveysto
themassfunctionoftheensemble,theeffectsofsamplevari -
ancemaybecalculatedwithfullconsiderationofthe correl a-
tionsbetweenhalocountsat differentmasses.
3.2.2.AnalyticMass Functions
TheanalyticmassfunctionsofTinkeret al.(2008)areused
to calculate the abundance of halos in several cosmological8 BEHROOZI,CONROY& WECHSLER
0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
Scale Factor-1-0.9-0.8-0.7-0.6Δlog10φ0 L80G
 Fit
0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
Scale Factor-0.16-0.12-0.08-0.0400.04Δlog10M0 L80G
 Fit
0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
Scale Factor-0.16-0.0800.080.16Δlog10α
 L80G
 Fit
Figure1. Differences between the fitted Schechter function paramete rs for the satellite halo mass (at accretion) function and th e central halo mass function, as
a function of scale factor; e.g., ∆log10φ0corresponds to log10(φ0,sats/φ0,centrals). The black lines are calculated from a simulation using WMA P1 cosmology
(L80G),and the red lines represent the fits to the simulation results in Equation 7.
models. We calculate mass functions defined by Mvir, using
the overdensity specified by Bryan&Norman (1998)3This
results in an overdensity (compared to the mean background
density)∆virwhichrangesfrom337at z=0to203at z=1and
smoothly approaches 180 at very high redshifts. Following
Tinkeret al. (2008), we use spline interpolation to calcula te
mass functions for overdensities between the discrete inte r-
valspresentedintheirpaper.
ThemassfunctionsinTinkeret al.(2008)onlyincludecen-
tral halos. We model the small ( ≈20% atz=0) correctionto
the mass function introduced by subhalos to first order only,
as the overall uncertaintyin the central halo mass function is
alreadyoforder5%(Tinkeret al.2008). Inparticular,weca l-
culate satellite (massat accretion)and centralmass funct ions
in our simulation (L80G) and fit Schechter functionsto both,
excluding halos below our completeness limit (1010.3M⊙).
Then, we plot the difference between the Schechter param-
eters (the difference in characteristic mass, ∆log10M∗; the
difference in characteristic density, ∆log10φ0; and the dif-
ference in faint-end slopes, ∆α) as a function of scale factor
(a). This gives the satellite mass function ( φs) as a function
of the central mass function ( φc), which allows us to use this
(first-order)correction for central mass functionsof diff erent
cosmologies:
φs(M)=10∆log10φ0/parenleftbiggM
M0·10∆log10M0/parenrightbigg−∆α
φc(M/10∆log10M0).
(6)
Fromoursimulation,we findfitsasshownin Figure1:4
∆log10φ0(a)=−0.736−0.213a,
∆log10M0(a)=0.134−0.306a, (7)
∆α(a)=−0.306+1.08a−0.570a2.
Themassfunctionused heremaybe beeasily replacedby an
arbitrarymassfunction,asdetailedinAppendixA.
3.2.3.Modeling Uncertainties inCosmological Parameters
Our fiducial results are calculated assuming WMAP5 cos-
mologicalparameters. In orderto modeluncertaintiesin co s-
mological parameters, we have sampled an additional 100
setsofcosmologicalparametersfromtheWMAP5+BAO+SN
3∆vir=(18π2+82x−39x2)/(1+x);x=(1+ρΛ(z)/ρM(z))−1−1
4Comparing these fits to satellite mass functions from a more r ecent sim-
ulation (Klypin etal. 2010, the “Bolshoi” simulation), we h ave verified that
applying these fits to mass functions for the WMAP5 cosmology introduces
errorsonly onthelevel of5%inoverall number density, simi lar totheuncer-
tainty with which the mass function isknown.MCMC chains (from the models in Komatsuet al. 2009)
and generated mass functions for each one according to the
methodinthe previoussection. Hence,todeterminethevari -
anceinthederivedstellarmass–halomassrelationcausedb y
cosmology uncertainties, we recalculate the relation for e ach
sampled mass function according to the method described in
AppendixA.
3.2.4.EstimatingSample Variance Effectsforthe Stellar Mass
Function
Large–scalemodesinthematterpowerspectrumimplythat
finitesurveyswillobtainabiasedestimateofthenumberden -
sities of galaxies and halos as compared to the full universe .
That is to say, matching observed GSMFs measured from a
finitesurveytothehalomassfunctionestimatedfromamuch
largervolumewill introducesystematic errorsintothe res ult-
ingSM–HMrelation. Theseerrorscannotbecorrectedunless
one has knowledge of the halo mass function for the specific
surveyin question,whichisin generalnotpossible.
However,wecanstillcalculatetheuncertaintiesintroduc ed
by the limited sample size. While we cannot determine the
true halo mass function for the survey, we can calculate the
probabilitydistribution of halo mass functionsfor identi cally
shaped surveys via sampling lightcones from simulations. I f
we rematch galaxy abundances from the observed GSMF to
the abundances of halos in each of the sampled lightcones,
thenthe uncertaintyintroducedbysample varianceis exact ly
capturedin thevarianceoftheresultingSM–HMrelations.
In detail, we create our distribution of halo mass func-
tions by sampling one thousand mock surveys from the Las-
Damassimulation(see§3.2.1)correspondingtotheexactsu r-
vey parameters used in Pérez-Gonzálezet al. (2008). We fit
Schechter functions to the halo mass functionsof each mock
survey (over all redshifts), and we calculate the change in
Schechter parameters ( ∆log10φ0,∆log10M0, and∆α) as
compared to a Schechter fit to the ensemble average of the
mass functions. Using the distribution of the changes in
Schechter parameters, we may mimic to first order the ex-
pecteddistributionofhalomassfunctionsforanycosmolog y.
In particular, we use an equation exactly analogous to Equa-
tion6to convertthe massfunctionforthe fulluniverse( φfull)
and the distribution of ∆log10φ0,∆log10M0, and∆αinto a
distributionofpossiblesurveymassfunctions( φobs):
φobs(M)=10∆log10φ0/parenleftbiggM
M0·10∆log10M0/parenrightbigg−∆α
φfull(M/10∆log10M0).
(8)
Hence,toobtainthevarianceinthestellarmass–halomassUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 9
relation caused by finite survey size, we recalculate the rel a-
tionforeachoneofthesurveymassfunctionsthuscomputed
accordingtothemethoddescribedin AppendixA.
3.3.Uncertaintiesin AbundanceMatching
3.3.1.Scatter inStellar Mass at FixedHalo Mass
An important uncertainty in the abundance matching pro-
cedure is introduced by intrinsic scatter in stellar mass at a
given halo mass. Suppose that M∗(Mh) is the average (true)
galaxy stellar mass as a function of host halo mass. For a
perfect monotonic correlation between stellar mass and hal o
mass, i.e., without scatter between stellar and halo mass, i t
is straightforwardto relate the true or “intrinsic” stella r mass
function( φtrue)to thehalomassfunction( φh) via
dN
dlog10M∗=dN
dlog10MhdlogMh
dlogM∗, (9)
whereNisthenumberdensityofgalaxies,sothat
φtrue(M∗(Mh))=φh(Mh)/parenleftbiggdlogM∗(Mh)
dlogMh/parenrightbigg−1
.(10)
Intuitively,asthehalosofmass Mhgetassignedstellarmasses
ofM∗(Mh),thenumberdensityofgalaxieswithmass M∗(Mh)
willbeproportionaltothenumberdensityofhaloswithmass
Mh. Theaboveequationsaresimplyamathematicalrepresen-
tationofthetraditionalabundancematchingtechnique.
Equation10remainsusefulinthepresenceofscatter. Ifwe
knowthe expectedscatter aboutthe meanstellar mass, sayin
the formof a probabilitydensity function Ps(∆log10M∗|Mh),
then we may still relate φtruetoφhvia an integral similar to a
convolution:
φtrue(x)=/integraltext∞
0φh(Mh(M∗))dlogMh(M∗)
dlogM∗×
×Ps(log10x
M∗|Mh(M∗))dlog10M∗,(11)
whereMh(M∗)istheinversefunctionof M∗(Mh).
This similarity to a convolution is no coincidence—
mathematically,it isanalogoustohowwemodelrandomsta-
tisticalerrorsinstellarmassmeasurementsin§3.1.2. Nam ely,
ifwedefine φdirecttoequaltheright-handsideofEquation10,
φdirect(M∗)≡φh(Mh(M∗))dlogMh
dlogM∗, (12)
and if we assume a probability density distribution indepen -
dent of halo mass (i.e., scatter in stellar mass at fixed halo
mass is independent of halo mass), then φtrueis exactly re-
latedtoφdirectbyaconvolution:
φtrue(M∗)=/integraldisplay∞
−∞φdirect(10y)Ps(y−log10M∗)dy,(13)
whichis mathematicallyidenticaltoEquation2in§3.1.2.
Then, if one calculates φdirectfromφtrue, one may find
Mh(M∗) via direct abundance matching. Namely, integrating
equation12,we have:
/integraldisplay∞
Mh(M∗)φh(M)dlog10M=/integraldisplay∞
M∗φdirect(M∗)dlog10M∗.(14)
Equivalently, letting Φh(Mh)≡/integraltext∞
Mhφh(M)dlog10Mbe the
cumulative halo mass function, and letting Φdirect(M∗)≡/integraltext∞
M∗φdirect(M∗)dlog10M∗be the cumulative “direct” stellar
massfunction,wehave
Mh(M∗)=Φ−1
h(Φdirect(M∗)), (15)
andonemaysimilarlyfind M∗(Mh)byinvertingthisrelation.
Our approach in all equations except for Equation 13 al-
lows a halo mass-dependentscatter in the stellar mass, but t o
date the data appears to be consistent with a constant scatte r
value. For example, using the kinematics of satellite galax -
ies, Moreet al. (2009) finds that the scatter in galaxy lumi-
nosity at a given halo mass is 0 .16±0.04 dex, independent
of halo mass. Using a catalog of galaxy groups, Yanget al.
(2009b) find a value of 0 .17 dex for the scatter in the stel-
lar massat a givenhalomass, also independentof halomass.
Here, we thus assume a fixed value for the scatter in stellar
mass at fixed halo mass, ξ, to specify the standard deviation
ofPs(∆log10M∗). As the Yangetal. (2009b) value is consis-
tent with the Moreet al. (2009) value, we set the prior using
the Moreetal. (2009) value and error bounds on ξ, We as-
sume a Gaussian prior on the probability distribution for ξ,
andwe assumethatthescatter itself islog-normal.
3.3.2.The Treatment of Satellites
Whenagalaxyisaccretedintoalargersystem,itwilllikely
bestrippedofdarkmattermuchmorerapidlythanstellarmas s
because the stars are much more tightly bound than the halo.
It has been demonstratedthat variousgalaxyclusteringpro p-
erties compare favorably to samples of halos where satellit e
halos— i.e., subhalos— are selected accordingto their halo
mass at the epoch of accretion, Macc, rather than their cur-
rent mass (e.g., Nagai&Kravtsov 2005; Conroyet al. 2006;
Vale&Ostriker 2006; Berrieretal. 2006). Theseresultssup -
port the idea that satellite systems lose dark matter more
rapidlythanstellar mass.
As commonly implemented (e.g. Conroyetal. 2006), the
abundancematchingtechniquematchesthestellarmassfunc -
tionataparticularepochtothehalomassfunctionatthesam e
epoch, using Maccrather than the present mass for subhalos.
AsMaccremainsfixedaslongasthesatelliteisresolvable,the
standard technique implies that the satellite galaxy’s ste llar
mass will continue to evolve in the same way as for centrals
ofthat halomass. Therefore,a subtle implicationof thesta n-
dardtechniqueisthatsatellitesmaycontinuetogrowinste llar
mass, even though Maccremainsthe same. A differentmodel
forsatellitestellarevolution(e.g.,inwhichstellarmas swhich
does not evolve after accretion) would therefore involve di f-
ferentchoicesinthesatellite matchingprocess.
The fiducial results presented here use the standard model
where satellites are assigned stellar masses based on the cu r-
rent stellar mass function and their accretion–epoch masse s.
However, we also present results for comparison in which
satellite masses are assigned utilizing the stellar mass fu nc-
tion at the epoch of accretion, correspondingto a situation in
which satellite stellar masses do not change after the epoch
ofaccretion. In orderto maintainself-consistencyforthe lat-
ter method, we use full merger trees (from L80G, the simu-
lation described in §3.2.1) to keep track of satellites and t o
assure that, e.g.,mergersbetween satellites beforethey r each
thecentralhalopreservestellar mass.
Finally, we note that any specific halo–finding algorithm
may introduce artifacts in the halo mass function in terms
of when a satellite halo is considered absorbed/destroyed.
This can have a small effect on satellite clustering as well a s10 BEHROOZI,CONROY & WECHSLER
number density counts. Wetzel & White (2009) suggest an
approach that avoids some of the problems associated with
resolving satellites after accretion. Namely, they sugges t a
model where satellites remain in orbit for a duration that is a
function of the satellite mass, the host mass, and the Hubble
time, after which time they dissolve or merge with the cen-
tralobject. Althoughwehavenotmodeledthisexplicitly,o ur
satellite counts are consistent with their recommendedcut off
—theysuggestconsideringasatellitehaloabsorbedwhenit s
presentmassislessthan0.03timesitsinfallmass;inoursi m-
ulation,only0.1%ofall satellitesfall belowthisthresho ld.
3.4.FunctionalFormsfortheStellarMass–HaloMass
Relation
Inordertodeterminetheprobabilitydistributionofourun -
derlying model parameters, we must first define an allowed
parameterspaceforthestellarmass–halomassrelation. Id e-
ally, one would like a simple, accurate, physically intuiti ve,
andorthogonalparameterization;inpractice,weseektheb est
compromise with these four goals in mind. We consider one
of the most popular methods for choosing a functional form
(indirect parameterization via the stellar mass function) be-
fore discussing the method we use in this paper (parameteri-
zationvia deconvolutionofthe stellarmassfunction).
3.4.1.Parameterizing the Stellar Mass Function
In abundance matching, knowledge of the halo mass func-
tion and the stellar mass function uniquely determines the
stellar mass – halo mass relation. Hence, parameterizing
the stellar mass function yields an indirect parameterizat ion
for the stellar mass – halo mass relation as well. Numer-
ouspapers(e.g.Cole et al.2001;Bell etal.2003;Pantereta l.
2004; Pérez-Gonzálezet al.2008) havefoundthat theGSMF
iswell-approximatedbyaSchechterfunction:
φ(M∗,z)=φ⋆(z)/parenleftbiggM∗
M(z)/parenrightbigg−α(z)
exp/parenleftbigg
−M∗
M(z)/parenrightbigg
,(16)
where the Schechter parameters φ⋆(z),M(z), andα(z) evolve
as functions of the redshift z. In many previous works
on abundance matching (e.g. Conroyetal. 2009), it is the
Schechter function for the stellar mass function that sets t he
formoftheSM–HMrelation.
More recently, however, several authors have noted that
the GSMF cannot be matched by a single Schechter function
forz<0.2 to within statistical errors (e.g. Li &White 2009;
Baldryetal. 2008), in part because of an upturn in the slope
of the GSMF for galaxies below 109M⊙in stellar mass. It
is possible that a conspiracy of systematic errors causes th e
observeddeviations,butthereisnofundamentalreasontoe x-
pecttheintrinsicGSMF tobefitexactlybyaSchechterfunc-
tion (see discussion in AppendixC). In any case, our full pa-
rameterization —either the stellar mass function or the err or
parameterization— mustbe able to capture all the subtleties
of the observedstellar massfunction. Hence, we are incline d
toadopta moreflexiblemodelthanthe Schechterfunctionof
equation16. Otherauthors,wrestlingwiththesameproblem ,
have chosen to adopt multiple Schechter functions, includ-
ing the eleven-parameter triple piecewise Schechter-func tion
fit used by Li& White (2009). While accurate, these models
oftenaddcomplicationwithoutincreasingintuition.
3.4.2.Deconvolving the Observed Stellar Mass Function11 12 13 14 15
log10(Mh) [MO•]8.89.29.61010.410.811.211.6log10(M*) [MO•]
Direct Deconvolution
Functional Fit
Figure2. Relation between halo massandstellar massinthelocalUniv erse,
obtained via direct deconvolution of the stellar mass funct ion in Li&White
(2009) matched to halos in a WMAP5 cosmology. The deconvolut ion in-
cludes the most likely value of scatter in stellar mass at a gi ven halo mass as
wellasstatisticalerrorsinindividualstellarmasses. Th edirectdeconvolution
(solid line) is compared to thebest fitto Eq. 21 ( red dashed line ).
Rather than attempting to parameterize the stellar mass
function, we could use abundance matching directly to de-
rive the stellar mass – halo mass relation for the maximal-
likelihoodstellar mass function,and thenfind a fit which can
parameterize the uncertainties in the shape of the relation .
This process is complicated by the various errors which we
musttakeintoaccount. Recall fromEquations2and13that
φmeas(M∗)=φdirect(M∗)◦Ps(∆log10M∗)◦P(∆log10M∗),
(17)
(where “◦” denotes the convolutionoperation, Psis the prob-
ability distribution for the scatter in stellar mass at fixed halo
mass, and Pis the probability distribution for errors in ob-
served stellar mass at fixed true stellar mass). However,
if we obtain φdirectby deconvolution of the observed stellar
mass function φmeas, we may use direct abundance matching
(Equation 15) to determine the maximum likelihood form of
Mh(M∗).
Figure 2 shows the result of calculating Mh(M∗) atz∼0.1
via deconvolution and direct matching of the stellar mass
function as described in the previous section. We choose
themaximum-likelihoodvalueforthedistributionfunctio nPs
(namely, 0.16 dex log-normal scatter), and we use WMAP5
cosmologyforthehalomassfunction φhinthederivation.
While deconvolutionplusdirectabundancematchinggives
anunbiasedcalculationoftherelation,thereareseveralp rob-
lems which prevent it from being used directly to calculate
uncertainties:
1. Deconvolutionwilltendtoamplifystatisticalvariatio ns
in the stellar mass function—that is, shallow bumps
in the GSMF will be interpreted as convolutions of a
sharperfeature.
2. Deconvolutionwill give different results depending on
the boundary conditions imposed on the stellar mass
function (i.e., how the GSMF is extrapolated beyond
the reporteddata points)—the effects of which may be
seenat theedgesofthedeconvolutioninFigure2.
3. Deconvolution becomes substantially more problem-UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 11
atic when the convolutionfunctionvaries over the red-
shift range, as it does for our higher-redshift data ( z>
0.2).
4. Deconvolution cannot extract the relation at a single
redshift—instead, it will only return the relation aver-
aged over the redshift range of galaxiesin the reported
GSMF.
For these reasons, we choose to find a fitting formula in-
stead. In the discussion that follows, we fit Mh(M∗) (the halo
massforwhichtheaveragestellarmassis M∗)ratherthanthe
moreintuitive M∗(Mh)(theaveragestellarmassatahalomass
Mh) primarily for reasons of computational efficiency. From
Equations12and17,thecalculationofwhatobserverswould
see(φmeas)foratrialstellarmass–halomassrelationrequires
many evaluations of Mh(M∗) and no evaluations of M∗(Mh).
Ifwehadinsteadparameterized M∗(Mh),andtheninvertedas
necessary in the calculation of φmeas, our calculations would
havetakenanorderofmagnitudemorecomputertime.
3.4.3.Fittingthe Deconvolved Relation
It is well-known from comparing the GSMF (or the lu-
minosity function) to the halo mass function that high-mass
(M∗/greaterorsimilar1010.5M⊙) galaxieshave a significantly differentstel-
lar mass-halo mass scaling than low-mass galaxies, which
is usually attributed to different feedback mechanisms dom -
inating in high-mass vs. low-mass galaxies. The transition
point between low-mass and high-mass galaxies—seen as a
turnoverintheplotof Mh(M∗)aroundM∗=1010.6M⊙inFig-
ure 2—defines a characteristic stellar mass ( M∗,0) and an as-
sociated characteristic halo mass ( M1). Hence, we consider
functionalformswhichrespectthisgeneralstructureofa l ow
stellarmassregimeandahighstellarmassregimewithachar -
acteristictransitionpoint:
log10(Mh(M∗))= log10(M1) [CharacteristicHaloMass]
+flow(M∗/M∗,0) [Low-massfunctionalform]
+fhigh(M∗/M∗,0) [High-massfunctionalform]
whereflowandfhighare dimensionless functions dominating
belowandabove M∗,0, respectively.
Forlow-massgalaxies( M∗<1010.5M⊙),wefindthestellar
mass–halomassrelationtobeconsistentwithapower–law:
Mh(M∗)
M1≈/parenleftbiggM∗
M∗,0/parenrightbiggβ
,or
log(Mh(M∗))≈log(M1)+βlog/parenleftbiggM∗
M∗,0/parenrightbigg
.(18)
Forhigh-massgalaxies,we findthestellar mass–halomass
relation to be inconsistent with a power–law. In particu-
lar, the logarithmic slope of Mh(M∗) changes with M∗, with
dlogMh/dlogM∗always increasing as M∗increases. This
may seem like a small detail; after all, by eye, it appears tha t
a power law could be a reasonable fit for high-mass galax-
ies in Figure 2. In addition, because previous authors (e.g. ,
Mosteretal. 2009; Yanget al. 2009a) have used power laws,
it maynotseem necessarytouse a differentfunctionalform.
In order to explore this issue, we tried a general dou-
ble power–law functional form for Mh(M∗) which parame-
terized a superset of the fits used in Mosteretal. (2009) and
Yanget al. (2009a) (in particular, the same form as in Equa-
tionC2inAppendixC). Wefoundthatthisapproachhadtwo
majorproblemscommonto anysuchpower–lawform:1. As the logarithmic slope of Mh(M∗) increases with
increasing M∗, the best-fit power–law for high-mass
galaxies will depend on the upper limit of M∗in the
available data for the GSMF. Thus, the best-fit power–
law will depend on the number density limit of the
observational survey used—rather than on any funda-
mental physics. Moreover, for studies such as this one
whichconsiderredshiftevolution,thedifferentnumber
densities probed at different redshifts result in a com-
pletelyartificial“evolution”ofthebest-fitpower–law.
2. The best–fit power–law will not depend on the high-
est mass galaxies alone; instead, it will be something
of an average overall the high-massgalaxies. Because
thelogarithmicslopeisincreasingwith M∗,thismeans
thatthebest-fitpowerlawfor Mh(M∗)willincreasingly
underestimate the true Mh(M∗) at high M∗. Namely,
the fit will underestimate the halo mass correspond-
ing to a given stellar mass, and therefore (as lower-
masshaloshavehighernumberdensities)resultinstel-
larmassfunctions systematically biasedaboveobserva-
tional values. However, a systematic bias in our func-
tional form will influencethe best-fit valuesof the sys-
tematic error parameters. The systematic bias caused
byassumingapower–lawformturnsouttobemostde-
generate with the scatter in stellar mass at fixed halo
mass (ξ). As a result, for the MCMC chains which as-
sumed a double power–law form for Mh(M∗), the pos-
terior distribution of ξwas 0.09±0.02 dex, which just
barelylieswithin2 σoftheconstraintsfromMoreet al.
(2009).
These problemsare not as significant if one only considers
thestellarmassfunctionatasingleredshift,orifonedoes not
allowforthesystematicerrorswhichchangetheoverallsha pe
ofthestellarmassfunction( κ,ξ,andσ(z)). However,wefind
that the issues listed above exclude the use of a power–law
for our purposes. Instead, we find that Mh(M∗) asymptotes
toasub-exponential functionforhigh M∗, namely,afunction
which climbsmore rapidly than any power–lawfunction,but
lessrapidlythananyexponentialfunction. Wefindthathigh –
massgalaxies( M∗>1010.5M⊙)arewell fit bytherelation
Mh(M∗)∼∝10/parenleftBig
M∗
M∗,0/parenrightBigδ
,or
log10(Mh(M∗))→log10(M1)+/parenleftbiggM∗
M∗,0/parenrightbiggδ
(19)
whereδsets how rapidly the function climbs; δ→0 would
correspond to a power–law, and δ= 1 would correspond to a
pureexponential. Typicalvaluesof δatz=0rangefrom0 .5−
0.6. It is not obvious what physical meaning can be directly
inferredfromthechoiceofa sub-exponentialfunction—aft er
all, the stellar mass of a galaxyis a complicatedintegralov er
the merger and evolution history of the galaxy—but it could
suggest that the physics drivingthe Mh(M∗) relation at high–
massis notscale–free.
Although this form now matches the asymptotic behavior
for the highest and lowest stellar mass galaxies, one addi-
tional parameteris necessary to match the functionalform o f
the deconvolution. That is to say, galaxiesin between the ex -
tremes in stellar mass will lie in a transition region, as the y
may have been substantially affected by multiple feedback
mechanisms. The width of this transition region will depend
on many things—e.g., how long galaxies take to gain stellar12 BEHROOZI,CONROY & WECHSLER
mass,howmuchofthestellar masspresentcamefromquies-
cent star formation as opposed to mergers, and the degree of
interaction between multiple feedback mechanisms. Hence,
instead of having Mh(M∗) become suddenly sub-exponential
forgalaxieslargerthan M∗,0,weallowforaslow“turn-on”of
the morerapid growth. The behaviorof Mh(M∗) is best fit by
modifyingthepreviousequationto
log10(Mh(M∗))→log10(M1)+/parenleftBig
M∗
M∗,0/parenrightBigδ
1+/parenleftBig
M∗
M∗,0/parenrightBig−γ(20)
The denominator,1 +(M∗/M∗,0)−γ, is largefor M∗<M∗,0,
anditfallstounityfor M∗>M∗,0ataratecontrolledby γ. A
larger value of γimplies a more rapid transition between the
power–law and sub-exponential behavior (typical values fo r
(γ)atz=0are1.3-1.7). Asthenon-constantpieceof Mh(M∗)
inEquation20is1
2forM∗=M∗,0, weadda finalfactorof −1
2tocompensatesothat Mh(M∗,0)=M1.
To summarize, our resulting best–fit functional form has
fiveparameters:
log10(Mh(M∗))=
log10(M1)+βlog10/parenleftbiggM∗
M∗,0/parenrightbigg
+/parenleftBig
M∗
M∗,0/parenrightBigδ
1+/parenleftBig
M∗
M∗,0/parenrightBig−γ−1
2.(21)
WhereM1isacharacteristichalomass, M∗,0isacharacteristic
stellar mass, βis the faint-end slope, and δandγcontrol the
massive-end slope. The best fit using this functional form is
shown in Figure 2, and it achieves excellent agreement over
theentirerangeofstellar masses.
Deconvolving the GSMF at higher redshifts does not sug-
gest that anything more than linear evolution in the parame-
tersisnecessary,at least outto z=1. While the characteristic
mass of the GSMF and the characteristic mass of the halo
mass function certainly evolve, the change in the shapesof
thetwofunctionsisrelativelyslight. Aswewishforthefun c-
tionalformtohaveanaturalextensiontohigherredshifts, we
parameterizethe evolutionin termsofthescale factor( a):
log10(M1(a))=M1,0+M1,a(a−1),
log10(M∗,0(a))=M∗,0,0+M∗,0,a(a−1),
β(a)=β0+βa(a−1), (22)
δ(a)=δ0+δa(a−1),
γ(a)=γ0+γa(a−1),
wherea=1isthescale factortoday.
3.5.CalculatingModelLikelihoods
We make use of a Markov Chain Monte Carlo (MCMC)
method to generate a probability distribution in our com-
plete parameter space of stellar mass function parame-
ters (M1,0,M1,a,M∗,0,0,M∗,0,a,β0,βa,δ0,δa,γ0,γa), systematic
modeling errors ( κ,µ,σz), and the scatter in stellar mass at
fixedhalomass( ξ). Abriefsummaryofeachoftheseparam-
eters appears in Table 1 along with a reference to the section
inwhichitwasfirstdescribed. Usingthisfullmodel,wemay
calculate the stellar mass functions expected to be seen by
observers ( φexpect) for a large number of points in parameter
space, and compare them to observed GSMFs (Li&White2009; Pérez-Gonzálezet al. 2008). Note that, as the observa -
tionaldataalwayscoversarangeofredshifts,wemustmimic
thisin ourcalculationof φexpect:
φexpect=/integraltextz2
z1φfit(z)dVC(z)/integraltextz2
z1dVC(z), (23)
wheredVC(z) is the comoving volume element per unit solid
angle as a functionof redshift. Then, we can write the likeli -
hoodasL=exp/parenleftbig
−χ2/2/parenrightbig
, where
χ2=/integraldisplay/bracketleftbigglog10[φexpect(M∗)/φmeas(M∗)]
σobs(M∗)/bracketrightbigg2
dlog10(M∗),(24)
andwhere σobs(M∗)isthereportedstatistical errorin φmeasas
afunctionofstellarmass.
Note that, as defined above, the equation for χ2contains
the assumption that there is only one independent observa-
tion point for the GSMF per decade in stellar mass (from the
weightof dlog10(M)). Wemaytunethisassumptionintroduc-
inganotherparameter n—thenumberofnon-correlatedobser-
vations per decade in stellar mass—which would change the
likelihood function to L= exp/parenleftbig
−nχ2/2/parenrightbig
. Here, we assume
that each of the data points reported by Li &White (2009)
and Pérez-Gonzálezet al. (2008) are independent—suchthat
n=10fortheformerpaperand n=5forthelatterpaper.
The MCMC chains each contain 222≈4×106points.
We verify convergence according to the algorithm in
Dunkleyet al. (2005); in all cases, the ratio of the sample
mean variance to the distribution variance (the “convergen ce
ratio”)isbelow0.005.
3.6.MethodologySummary
Our procedureto calculate the stellar mass – halo mass re-
lation, taking into account all mentioned uncertainties, m ay
besummarizedin sevensteps:
1. We select a trial point in the parameter space of SM–
HM relations as well as a trial point in our parameter
space of systematics ( µ,κ,σz,ξ). A complete list of
parametersanddescriptionsisgiveninTable1.
2. The trial SM–HM relation gives a one-to-onemapping
between halo masses and stellar masses, giving a di-
rect conversion from the halo mass function to a trial
galaxystellarmassfunction(correspondingto φdirectin
§3.3.1).
3. This trial GSMF is convolvedwith the probability dis-
tributions for scatter in stellar mass at fixed halo mass
(controlledby ξ,see§3.3.1)andforscatterinobserver-
determined stellar mass at fixed true stellar mass (par-
tially controlledby σz,see §3.1.2).
4. The resulting GSMF is shifted by a uniform offset in
stellar masses (controlled by µ) to account for uni-
formsystematicdifferencesbetweenouradoptedstellar
masses and the true underlyingmasses. Also, its shape
is stretched or compressed to account for stellar mass–
dependentoffsets between our masses and the true un-
derlyingmasses(controlledby κ, see §3.1.1).
5. We repeat steps 2-4 for all redshifts in the range cov-
ered by the observed data set. We may then calculateUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 13
Table 1
Summaryof Model Parameters
Symbol Description PrioraSection
Mh(M∗) Thehalo massfor which the average stellar massis M∗ N/A 3.4.3
M1 Characteristic Halo Mass Flat (Log) 3.4.3
M∗,0 Characteristic Stellar Mass Flat (Log) 3.4.3
β Faint-end power law ( Mh∼Mβ
∗) Flat (Linear) 3.4.3
δ Massive-end sub-exponential (log10(Mh)∼Mδ
∗) Flat (Linear) 3.4.3
γ Transition width between faint- and massive-end relations Flat (Linear) 3.4.3
(x)0Value of thevariable ( x) atthe present epoch, where ( x) is oneof ( M1,M∗,0,β,δ,γ) (see above) 3.4.3
(x)a Evolution of the variable ( x) with scale factor (same as for ( x)0) 3.4.3
µ Systematic offset in M∗calculations G(0,0.25) (Log) 3.1.1
κ Systematic mass-dependent offset in M∗calculations G(0,0.10) (Linear) 3.1.1
σz Redshift scaling of statistical errors in M∗calculations G(0.05,0.015) (Log) 3.1.2
ξ Scatter in M∗at fixedMh G(0.16,0.04) (Log) 3.3.1
aSee Equations 1, 5, 21-23. G(x,s) denotes a Gaussian prior centered at xwith standard deviation s, in either linear or logarithmic
units. ‘Flat’ denotes auniform prior in either linear or log arithmic units.
the expected GSMF in each redshift bin for which ob-
servers have reported data. The likelihood of the ex-
pectedGSMFsgiventhemeasuredGSMFsisthenused
to determinethe nextstep intheMCMCchain.
6. To account for sample variance in the observed stel-
lar mass functions above z∼0.2, we recalculate each
SM–HMrelationinthechainforanalternatehalomass
function taken from a randomly sampled mock survey
(see §3.2.4) and re-fit our functional form to the red-
shift evolutionof the relation. Similarly, for the results
which include cosmology uncertainty, we recalculate
each SM–HM relation for an alternate halo mass func-
tion randomly selected from the MCMC chain used to
determinetheWMAP5 cosmologyuncertainties.
7. We repeat steps 1-6 to build a joint probability distri-
butionfortheSM–HMrelationandthesystematicspa-
rameter space. The steps are repeated until the joint
probabilitydistributionhasconvergedtotheunderlying
posteriordistribution.
4.RESULTS FOR0 <z<1
We now present the results of this approach to determine
theSM–HMrelationandrelatedquantities. In§4.1, wecom-
pareGSMFsgeneratedfromourbestfitstoobserveddataand
comment on the effects of systematic observational biases.
We present our best-fitting results for the SM–HM relation
with full error bars in §4.2. We evaluate the relative impor-
tance of each of the contributing types of error in §4.3 and
summarize the most relevant contributionsin §4.3.7. Final ly,
our derived SM–HM relation is compared to other published
resultsin §4.4.
4.1.GalaxyStellarMassFunctions
To demonstrate that our functional form for Mh(M∗) is ca-
pable of reproducingobserved galaxy stellar mass function s,
we show a comparison between our best–fit models and the
observed data in Figs. 3 and 4 at several redshifts. For our
best-fit models, both φtrue(the true or “intrinsic” stellar mass
function)and φmeas(theGSMFthatobserverswouldmeasure)
are shown. Recall that φmeasincorporates the effects of the
systematic observational biases; namely, the overall shif t in
stellarmasscalculations, µ,thelinearlymass-dependentshift,
κ, and the statistical errorsin stellar mass calculationsfo r in-
dividual galaxies, σ(z). The fact that the best-values of thesystematic parameters ( µ,κ,ξ,σz) are very close to the cen-
ters of their prior distributionsprovidesconfirmation tha t the
functional form for the SM–HM relation outlined in §3.4.3
doesnotbiasourbest-fit results.
As our best-fit values for µandκare close to zero (see
Table 2), the differencebetween φtrueandφmeasis almost ex-
clusively due to the scatter σ(z) in calculated stellar masses.
The differencebetween φtrueandφmeasonly becomesevident
for galaxies above 1011M⊙, where the falling slope of the
GSMF becomes severe enough for the scatter σ(z) to signifi-
cantly raise number counts in the observed GSMF. At z∼0,
thesystematiceffectof σ(z)putstheintrinsicGSMF wellbe-
lowthesmall statistical errorbars.
At higher redshifts, although the effect of σ(z) is larger,
current surveys at z>0.2 do not yet cover sufficient volume
to constrain the shape of the GSMF well at the massive end.
Nonetheless, for future wide-field surveys at z>0.2, correc-
tion to the GSMF for scatter in calculated stellar masses wil l
beanimportantconsideration.
4.2.TheBest-Fit StellarMass–HaloMassRelations
We plotthe averagestellar massas a functionofhalo mass
forz= 0−1 in Figure 5 to show the evolution of the stellar
mass – halo mass relation. Note that as the stellar mass at a
givenhalomasshasalog-normalscatter(see §2.3),weusege-
ometricaveragesforstellarmassesratherthanlinearones . To
highlighttheeffectsofhalomassonstarformationefficien cy,
we also present the SM–HM relation in terms of the average
stellar mass fraction (stellar mass / halo mass) for z= 0−1
as a function of halo mass in the same figure. We focus on
this quantity for the remainder of the paper. The best-fit pa-
rameters for the function Mh(M∗) are given in Table 2, and
thenumericalvaluesforthestellarmassfractionsarelist edin
AppendixD.
The stellar mass fractions for central galaxies consistent ly
show a maximum for halo masses near 1012M⊙. While the
location of this maximum evolves with time, it clearly il-
lustrates that star-formation efficiency must fall off for b oth
higher and lower-mass halos. The slopes of the SM–HM re-
lation above and below this characteristic halo mass are in-
dicative of at least two processes limiting star-formation effi-
ciency,althoughmergerscomplicate direct analysis for hi gh-
masshalos. Atthelow-massend,theSM–HMrelationscales
asM∗∼M2.3
hatz= 0 and as M∗∼M2.9
hatz= 1. However,
giventhe lack of informationabout low stellar-mass galaxi es
atz>0.5,thestatisticalsignificanceofthisevolutionisweak;14 BEHROOZI,CONROY & WECHSLER
-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
z = 0.1, φtrue 
z = 0.1, φmeas 
z = 0.1, Li & White (2009)
9 10 11 12
log10(M*) [MO•]-0.300.3log10(φ/φmeas)
Figure3. Comparison of the best fit φtrue(the true or “intrinsic” GSMF)
in our model to the resulting φmeas(what an observer would report for the
GSMF, which includes the effects of the systematic biases µ,κ, andσ) at
z=0. Sincethebest–fitvaluesof µandκareveryclosetozero,thedifference
betweenφmeasandφtruealmost exclusively comes from the uncertainty in
measuring stellar masses ( σ).
Table 2
Bestfits for the redshift evolution of Mh(M∗)
Parameter Free ( µ,κ)µ=κ=0 Free ( µ,κ)
0<z<1 0<z<1 0.8<z<4
M∗,0,010.72+0.22
−0.2910.72+0.02
−0.1211.09+0.54
−0.31
M∗,0,a0.55+0.18
−0.790.59+0.15
−0.850.56+0.89
−0.44
M∗,0,a2 N/A N /A6.99+2.69
−3.51
M1,012.35+0.07
−0.1612.35+0.02
−0.1512.27+0.59
−0.27
M1,a0.28+0.19
−0.970.30+0.14
−1.02−0.84+0.87
−0.58
β00.44+0.04
−0.060.43+0.01
−0.050.65+0.26
−0.20
βa0.18+0.08
−0.340.18+0.06
−0.340.31+0.38
−0.47
δ00.57+0.15
−0.060.56+0.14
−0.050.56+1.33
−0.29
δa0.17+0.42
−0.410.18+0.41
−0.42−0.12+0.76
−0.50
γ01.56+0.12
−0.381.54+0.03
−0.401.12+7.47
−0.36
γa2.51+0.15
−1.832.52+0.03
−1.89−0.53+7.87
−2.50
µ0.00+0.24
−0.25N/A0.00+0.25
−0.25
κ0.02+0.11
−0.07N/A0.00+0.14
−0.04
ξ0.15+0.04
−0.020.15+0.04
−0.010.16+0.07
−0.01
σz0.05+0.02
−0.010.05+0.02
−0.010.05+0.02
−0.01
Note. —See Table1 and Equations 1,5, 21-23,25.
noevolutioninthelow-massslopeoftherelationisconsist ent
within our one-sigma errors. Several studies (most recentl y,
Baldryetal. 2008; Droryetal. 2009) have reported that the
GSMF has an upturn in slope for very low stellar masses,
particularly below 108.5M⊙; this would imply that our best
fits may overestimate the scaling relation for galaxies belo w
108.5M⊙. At the high mass end, our best fitting function re-
sults in a progressively shallower relation for the growth o f
stellar mass with halo mass, so that no single power law can
describe the scaling. However, for halos close to 1014M⊙,
the best-fit relation scales locally as M∗∼M0.28
hatz=0 and9 10 11 12
log10(M*) [MO•]-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
z = 0.5, φtrue 
z = 0.5, φmeas 
z = 0.5, Pérez-González et. al. (2008)
9 10 11 12
log10(M*) [MO•]-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
z = 1.15, φtrue 
z = 1.15, φmeas 
z = 1.15, Pérez-González et. al. (2008)
Figure4. Comparison of the best fit φtrue(the true or “intrinsic” GSMF) to
theresulting φmeas(as in Figure 3),for z=0.5 andz=1.15. Statistical errors
inindividual stellar masseshavealarger effect athigher r edshift, resulting in
asteeper intrinsic bright end than measured.
M∗∼M0.34
hatz=1,inaccordwithpreviousstudies(see§4.4).
The results for high mass halos are also consistent with no
evolutionin theslopeoftheSM–HMrelation.
Figure 6 shows the stellar mass fraction for 0 <z<1 ex-
cluding the effects of systematic shifts in stellar mass cal cu-
lations (i.e., assuming µ=κ=0). Under the assumption that
systematic errorsin stellar mass calculations result in si milar
biasesin stellar masses at z=0 as they do at higherredshifts,
thisallowsustoconsidertheevolutioninnormalizationof the
SM–HM relation. Low-mass halos (below 1012M⊙) display
clearlyhigherstellar massfractionsat lateredshiftstha nthey
doatearlyredshifts. Bycontrast,theevolutioninstellar mass
fractionsfor high mass halos (above 1013.5M⊙) is not statis-
ticallysignificant,anditisconstrainedtobesubstantial lyless
than for low-mass halos. In the time since z= 1, this means
thatthe star formationrates forhigh-masshalostypically fall
relative to their dark matter accretion rates, whereas the o p-
posite is true for low-mass halos (Conroy&Wechsler 2009).
The best-fitting parameters for the SM–HM relation assum-
ingµ=κ=0 appear in Table 2, and the data pointsin Figure
6appearinAppendixD.
4.3.ImpactofUncertaintiesUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 15
11 12 13 14 15
log10(Mh) [MO•]89101112log10(M*) [MO•]
z = 0.1
z = 0.5
z = 1.0
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1
z = 0.5
z = 1.0
Figure5. Top panel : Stellar mass – halo mass relation as a function of red-
shift for our preferred model. Bottom panel : Evolution of the derived stellar
mass fractions ( M∗/Mh). In each case, the lines show the mean values for
central galaxies. These relations also characterize the sa tellite galaxy pop-
ulation if the horizontal axis is interpreted as the halo mas s at the time of
accretion. Errors bars include both systematic and statist ical uncertainties,
calculated for afixed cosmological model (with WMAP5parame ters).
4.3.1.Systematic ShiftsinStellar Mass Calculations
ByfarthelargestcontributortotheerrorbudgetoftheSM–
HM relation is the systematic error parameter µ. As the ef-
fect ofµis to multiply all stellar masses by a constant factor,
and as the width of the error bars in Figure 5 correspondsal-
mostexactlytotheprioron µ,wemayconcludethatreducing
the error on the systematic shifts in stellar mass calculati ons
wouldrepresent the single largest improvementin ourunder -
standingoftheshapeoftheSM–HMrelation. Figure6shows
thesubstantiallysmallererrorbarsthatresultifsystema ticer-
rors(µandκ)inthe stellarmasscalculationsareneglected.
4.3.2.Scatter inStellar Mass at FixedHalo Mass
The effect of ignoring scatter in stellar mass at fixed halo
mass(i.e.,setting ξ=0)isshownat tworedshiftsinFigure7.
We find that the changeis insignificant below halo masses of
1012M⊙, and is within statistical error bars below 1013M⊙
forz=1. Thisisaresultofthefactthattheslopeofthestellar
mass function below 1010.5M⊙in stellar mass (correspond-
ing to 1012M⊙in halo mass) is not steep enough for scat-11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1
z = 0.5
z = 1.0
Figure6. Evolution of the derived stellar mass fractions ( M∗/Mh) in the
absence of systematic errors. This result is analogous to Fi gure 5, bottom
panel, calculated under theassumption thatthetrue values ofthesystematics
µandκin thestellar mass function are zero at all redshifts.
tertohavesignificantimpact(seealsoTasitsiomi et al.200 4).
Becauseξ >0 results in high stellar–mass galaxies being as-
signedtolower-masshalosthantheywouldbeotherwise(due
to the higher numberdensity of lower-mass halos), the effec t
is that higher-masshalos contain fewer stars on average tha n
they would for ξ=0. The effect of setting ξ=0 exceedssys-
tematicerrorbarsonlyfortheveryhighestmasshalos,abov e
1014.5M⊙.
We note that our posterior distribution constrains ξto be
less than 0.22 dex at the 98% confidence level. Higher val-
ues forξwould result in GSMFs inconsistent with the steep
falloff of the Li &White (2009) GSMF (see also discussion
inGuoetal. 2009).
4.3.3.Statistical ErrorsinStellar Mass Calculations
The significance of includingor excludingrandomstatisti-
calerrorsinstellarmasscalculations, σ(z),isalsoshownFig-
ure7. TheeffectofthistypeofscatterontheSM–HMrelation
is mathematically identical to the effect of scatter in stel lar
mass at fixed halo mass. As σ(z= 0) (∼0.07 dex) is much
smaller than the expected value of ξ(∼0.16 dex), the con-
volution of the two effects is only marginally different fro m
including ξaloneatz=0;thisresultsinonlyaminoreffecton
the SM–HM relation. The effect becomes more pronounced
atz=1forthereasonthat σ(z=1)(∼0.12dex)becomesmore
comparableto ξ—andsoincludingtheeffectsofstatisticaler-
rorsin stellar massbecomesas importantasmodelingscatte r
instellar massat fixedhalomass.
4.3.4.Cosmology Uncertainties
InFigure8,weshowacomparisonofbestfitsforthestellar
mass fraction using abundance matching with three differen t
halo mass functions: analytic prescriptions for WMAP5 and
WMAP1 (see §3.2.2) as well as the mass function taken di-
rectly from the L80G simulation(see §3.2.1). The differenc e
betweentheL80GsimulationandtheanalyticWMAP1mass
functionisslight,astheL80GsimulationusesWMAP1initia l
conditions( h=0.7,Ωm=0.3,ΩΛ=0.7,σ8=0.9,ns=1); the
differenceisconsistentwithsamplevariancefortherelat ively
small (80 h−1Mpc)size ofthe simulation. Thedifferencebe-
tween SM–HM relations using WMAP1 and WMAP5 cos-16 BEHROOZI,CONROY & WECHSLER
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1 (incl. σ(z), ξ=0.16dex)
z = 0.1 (excl. σ(z), ξ=0.16dex)
z = 0.1 (incl. σ(z), ξ=0.0dex)
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 1.0 (incl. σ(z), ξ=0.16dex)
z = 1.0 (excl. σ(z), ξ=0.16dex)
z = 1.0 (incl. σ(z), ξ=0.0dex)
Figure7. Comparison between SM–HM relations derived in the preferre d model (including the effects of the statistical errors σ(z) and taking the scatter in
stellar mass at a given halo mass to be ξ= 0.16dex) to those excluding the effects of σ(z) or taking ξ= 0, atz= 0 (left panel ) andz= 1 (right panel ). Light
shaded regions denote 1- σerrors including both systematic and statistical errors; d ark shaded regions denote the 1- σerrors if the systematic offsets in stellar
masscalculations ( µandκ)are fixed to 0.
mologies is within the systematic errors at all masses. When
systematic errors are neglected, the two cosmologies yield
SM–HM relations that are noticeably different only at low
halomasses( M<1012M⊙).
Figure 9 show the results of including uncertainties in the
WMAP5cosmologicalparameters. Asdescribedin§3.1,this
is doneusinghalo mass functionscalculated with parameter s
resampled from the cosmological parameter chains provided
by the WMAP team. Only at z∼0 are the changes in error
bars significant enough to justify mention. Here, the uncer-
tainty in cosmology begins to exceed other sources of statis -
tical error for halos below 1012M⊙due to the small errors
on the GSMF at the stellar masses associated with such ha-
los(Li &White2009). However,thecosmologyuncertainties
arestill well withinthesystematicerrorbars.
4.3.5.Sample Variance
Because of the large volumeof the SDSS, sample variance
contributesinsignificantlytotheerrorbudgetfortheSM–H M
relationbelow z=0.2. Abovethatredshift,thecomparatively
limitedsurveyvolumeofPérez-Gonzálezetal.(2008)resul ts
in sample variance becoming an important contributor to the
statistical error for halos below 1012M⊙(Poisson noise dom-
inatesforlargerhalos). Iftheeffectsofsamplevariancew ere
ignored, the statistical error spreads for our derived SM–H M
relations at z=1 would shrink from 0.12 dex to 0.09 dex for
1011M⊙halos, and from 0.05 dex to 0.04 dex for 1012.25M⊙
halos. As with other types of errors, these considerationsa re
well belowthelimitsofthesystematic errorbars.
We caution that our error bars including sample variance
atz>0 have a very specific meaning. Namely, they include
the standard deviation in our fitting form which might be ex-
pected if the surveyin Pérez-Gonzálezet al. (2008) had been
conductedonalternatepatchesofthesky. Samplevariancea t
redshiftsz>0 impacts only the linear evolution of the SM–
HM relations we derive, as the large volume probed by the
SDSSconstrainstheSM–HMrelationverywell at z∼0. Be-
cause our fit is matched to the ensemble of reported data be-
tween 0<z<1, it is less vulnerableto the effects of sample
varianceinindividualredshiftbins. Instead,itisaffect edmost
by overall shifts in the number densities reported for the en -tire high-redshift survey. While this means that our fit give s
a more robust SM–HM relation at all redshifts, some caution
must be used when comparing our relation to results derived
from the GSMF in a single redshift bin (e.g., 0 .2<z<0.4).
Thesewillhavemuchlargeruncertaintiesduetosamplevari -
ancethan SM–HM relationsderived(like ours)fromGSMFs
alongalightconeprobinga largeredshiftrange.
Todemonstratethecredibilityofourapproachforcalculat -
ing the appropriate error bars including sample variance, w e
repeated our analysis of the SM–HM relation using GSMFs
from Droryetal. (2009) (which appeared as we were com-
pleting this work) instead of Pérez-Gonzálezetal. (2008)
for 0.2<z<1 and retaining the GSMF from Li &White
(2009) for z<0.2. Although the COSMOS survey in
Droryetal. (2009) covers a much larger area ( ∼9x the area
in Pérez-Gonzálezet al. 2008), the fact that it is a single
fieldmeansthatthe expectedsamplevarianceis onlyslightl y
smaller than for the combined fields in Pérez-Gonzálezet al.
(2008). Asmightbeexpected,theSM–HMrelationat z=0.1
using the Droryet al. (2009) GSMF is identical to the result
in Figure 6 because of the strong constraining power of the
SDSS data sample. At z=1, the SM–HM relation generated
by using the Droryet al. (2009) GSMF is within our quoted
statistical and sample varianceerrors, as shown in Figure 1 2.
This may not be surprising unless one considers that clus-
tering results suggest an overdensity at the 2-3 σlevel in the
COSMOS redshift bin z=1 (Meneuxetal. 2009). However,
becauseourmethodfitstheDroryetal.(2009)GSMFsacross
the entire redshift range, the excess at z=1 is partially offset
by an underdensity at z= 0.5. This demonstrates the robust-
ness of our fitting method to the effects of sample variance
exceptonthescaleofthe entiresurvey.
4.3.6.Satellite Treatment
Finally, we consider the changes in both the stellar mass
fraction and total stellar mass fraction (total stellar mas s in
central galaxy and all satellites / total halo mass) induced by
different satellite evolution models (see §3.3.2 for detai ls on
thetwomodels). AsshowninFigure10,fixingsatellitestell ar
mass at the redshift of accretion (lines labeled as “SMF acc”)
hasvirtuallynoeffectoneitherfractionascomparedtoall ow-UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 17
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1 (WMAP5)
z = 0.1 (L80G)
z = 0.1 (WMAP1)
Figure8. Comparison between stellar mass fractions in different cos molo-
gies. Lightshaded regionsdenotesystematicerrorspreads whiledarkshaded
regions denote error spreads assuming µ=κ= 0, both about the WMAP5
model. The dot-dashed blue line shows the fiducial relation f or a WMAP1
cosmological model (using our analytic model) The dashed re d line shows
therelation forasimulation oftheWMAP1cosmology. Differ ences between
this and the analytic modelare within the expected sampleva riance errors.
ing satellite stellar mass to evolve the same way as centrals
with the same mass (labeled as SMF now). Because compar-
ison of different satellite evolution models requires trac king
satellites through merger trees, Figure 10 shows results on ly
forsatellites intheL80Gsimulation.
Thetreatmentofsatellitesmayhaveasomewhatlargerim-
pact on the total stellar mass fraction, including the stell ar
massofbothcentralandsatellite galaxieswithin a halo. Th is
is shown for both models in Figure 10. Because of the steep
fall-off in stellar mass for low mass galaxies, the total ste llar
mass fraction has only minimal contribution from satellite s
forlowmasshalos,anddeviatessignificantlyfromthestell ar
massfractionforcentralsonlyathalomasses Mh>1012.5M⊙.
At cluster-scale masses ( Mh∼1014M⊙), accreted satellites
haveonaveragea higherratio ofstarsto darkmatter thanthe
centralgalaxy,andthetotalstellarmassfractioncanbema ny
times the central stellar mass fraction. However, the impac t
ofthetwomodelsforsatellitetreatmentonthisratioissma ll.
Profilesofsatellitegalaxiesinclustersshouldbeabletob etter
distinguishbetweensuchmodels.
4.3.7.Summary of Most Important Uncertainties
Systematic stellar mass offsets resulting from modeling
choices result in the single largest source of uncertaintie s
(∼0.25 dex at all redshifts). The contribution from all other
sourcesof error is much smaller, rangingfrom 0.02-0.12dex
atz= 0 and from 0.07-0.16 dex at z= 1. On the other hand,
this statement is only true when all contributing sources of
scatter in stellar masses are considered. Models that do not
accountforscatterinstellarmassatfixedhalomasswillove r-
predict stellar masses in 1014.25M⊙halos by 0.13-0.19 dex,
depending on the redshift. Models that do not account for
scatterincalculatedstellarmassatfixedtruestellarmass will
overpredictstellar masses in 1014.25M⊙halos by 0.12 dex at
z= 1. Hence, it is important to take both these effects into
account when considering the SM–HM connection either at
highmassesorat highredshifts.11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1
Figure9. Effect of cosmological uncertainties on the stellar mass fr action
atz= 0.1. The error bars show the spread in stellar mass fractions in clud-
ing both statistical errors and cosmology uncertainties (f rom WMAP5 con-
straints, Komatsu etal. 2009). For comparison, the light sh aded region in-
cludesstatistical andsystematicerrors,whilethedarksh adedregionincludes
only statistical errors.
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.0 (L80G) [SMFnow]
z = 0.0 (L80G) [SMFacc]
z = 0.0 (L80G) [SMFnow] (Total M*/Mh)
z = 0.0 (L80G) [SMFacc] (Total M*/Mh)
Figure10. Comparison between stellar massfractions and total stella r mass
fractions(labeled as“TotalM ∗/Mh”)derived byassumingdifferentmatching
epochs for satellite galaxies. The L80G simulation was used here in order
to follow the accretion histories of the subhalos. The relat ions terminate at
highmasseswherethehalo statistics becomeunreliable due tofinite–volume
effects.
4.4.Comparisonwith otherwork
Acomparisonofourresultswithseveralresultsintheliter -
atureatz∼0.1isshowninFigure11. Suchcomparisonisnot
always straightforward, as other papers have often made dif -
ferentassumptionsforthecosmologicalmodel,thedefiniti on
of halo mass, or the measurement of stellar mass. In addi-
tion, some papers report the average stellar mass at a given
halomass(aswedo),andothersreporttheaveragehalomass
at a given stellar mass. Given the scatter in stellar mass at
fixedhalomass,theaveragingmethodcanaffecttheresultin g
stellarmassfractions,particularlyforgroup-andcluste r-scale
halo masses. To facilitate comparison with both approaches ,
we plot our main results (labeled as “ ∝angbracketleftM∗/Mh|Mh∝angbracketright”) along
withresultsforwhichthestellarmassfractionshavebeena v-
eraged at a given stellar mass (labeled as “ ∝angbracketleftM∗/Mh|M∗∝angbracketright”).18 BEHROOZI,CONROY & WECHSLER
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)This work, < M*/Mh | Mh >
This work, M* / < Mh | M* >
Moster et al. 2009 (AM)
Guo et al. 2009 (AM)
Wang & Jing 2009 (AM+CC)
Zheng et al. 2007 (HOD)
Mandelbaum et al. 2006 (WL)
Klypin et al. in prep. (SD)
Gavazzi et al. 2007 (SL)
Yang et al. 2009a (CL)
Hansen et al. 2009 (CL)
Lin & Mohr 2004 (CL)
Figure11. Comparison of our best-fit model at z= 0.1 to previously published results. Results shown include ot her results from abundance matching
(Moster etal. 2009 and Guo et al. 2009); abundance matching p lus clustering constraints (Wang &Jing 2009); HOD modeling (Zheng etal. 2007); direct mea-
surements from weak lensing (Mandelbaum etal. 2006), state llite dynamics (Klypin et al. 2009) and strong lensing (Gava zzi etal. 2007); and clusters selected
from SDSS spectroscopic data (Yang etal. 2009a), SDSS photo metric data (the maxBCG sample Hansen et al. 2009), and X-ray selected clusters (Lin &Mohr
2004). Dark grey shading indicates statistical and sample v ariance errors; light grey shading includes systematic err ors. Thered line shows our results averaged
over stellar mass instead of halo mass;scatter affects thes e relations differently athigh masses. Theresults of Mande lbaum et al. (2006)and Klypin etal. (2009)
are determined by stacking galaxies in bins of stellar mass, and so aremoreappropriately compared to this red line.
In the comparisons below, we have not adjusted the assump-
tions used to derive stellar masses, because such adjustmen ts
can be complex and difficult to apply using simple conver-
sions. Additionally,we haveonlycorrectedfordifference sin
the underlyingcosmology for those papers using a variant of
abundance matching method (Mosteret al. 2009; Guoetal.
2009; Wang &Jing 2009; Conroyetal. 2009) using the pro-
cess described in Appendix A, as alternate methods require
corrections which are much more complicated. We have,
however,adjustedtheIMFofall quotedstellarmassesto tha t
of Chabrier (2003), and we have converted all quoted halo
massestovirialmassesasdefinedin §3.2.2.
Theclosestcomparisonwithourwork,usingaverysimilar
method, is the result from Mosteretal. (2009). This result i s
in excellent agreementwith oursat the high mass end, and is
within our systematic errorsfor all masses considered. How -
ever, their less flexible choice of functional form, and thei r
use of a different stellar mass function(estimated from spe c-
troscopy using the results of Panteret al. 2007) results in a
differentvalueforthehalomass Mpeakwithpeakstellarmass
fractionandashallowerscalingofstellarmasswithhaloma ssat the low mass end. Their error estimates only account for
statisticalvariationsingalaxynumbercounts,andtheydo not
include sample variance or variations in modeling assump-
tions. Guoet al.(2009)useasimilarapproachtoMosteret al .
(2009),usingstellarmassesfromLi& White(2009),butthey
do not account for scatter in stellar mass at fixed halo mass.
Consequently, their results match ours for 1012M⊙and less
massive halos, but overpredict the stellar mass for larger h a-
los.
Wang&Jing (2009) use a parameterization for the SM–
HM relation for both satellites and centrals, and they attem pt
to simultaneously fit both the stellar mass function and clus -
tering constraints, including the effects of scatter in ste llar
massatfixedhalomass. At z∼0.1,theirdatasourcematches
ours (Li&White 2009), but their approach finds a best-fit
scatter in stellar mass at fixed halo mass of ξ= 0.2 dex, es-
sentially the highest value allowed by the stellar mass func -
tion(Guoet al.2009). Asthisishigherthanourbest-fitvalu e
forξ, their SM–HM relation falls below ours for high-mass
galaxies. Possiblybecauseofthelimitedflexibilityofthe irfit-
tingform(theyuseonlyafour-parameterdoublepower-law) ,UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 19
their SM–HM relation is in excess of ours for halo masses
near1012M⊙.
Zhengetal. (2007) used the galaxy clustering for
luminosity-selectedsamplesintheSDSStoconstraintheha lo
occupation distribution. This gives a direct constraint on the
r−band luminosity of central galaxies as a function of halo
mass. Stellar masses for this sample were determined us-
ing theg−rcolor and the r-band luminosity as given by
theBell etal. (2003) relation,anda WMAP1 cosmologywas
assumed. This method allows for scatter in the luminosity
at fixed halo mass to be constrained as a parameter in the
model; results for this scatter are consistent with Moreeta l.
(2009), although they are less well constrained. According
toLi &White (2009), stellar massesfortheBell et al. (2003)
relation are systematically larger than those calculated u sing
Blanton&Roweis(2007) by0.1–0.3dex. However,as Ωmin
WMAP1 is larger than in WMAP5, halo masses in WMAP1
will be higher at a given number density than in WMAP5,
somewhatcompensatingforthehigherstellarmasses.
We next compare to constraints from direct measure-
mentsofhalomassesfromdynamicsorgravitationallensing .
Mandelbaumetal. (2006) have used weak lensing to mea-
sure the galaxy–mass correlation function for SDSS galax-
ies and derive a mean halo mass as a function of stellar
mass. Mandelbaumet al. (2006) assume a WMAP1 cos-
mology and uses spectroscopic stellar masses, calculated p er
Kauffmannetal. (2003). Klypin et al (in preparation) have
derived the mean halo mass as a function of stellar mass us-
ing satellite dynamicsof SDSS galaxies(see also Pradaetal .
2003; vandenBoschet al. 2004; Conroyet al. 2007). Their
results are generally within our systematic errors but lowe r
than others at the lowest masses and with a somewhat dif-
ferent shape. This may be due to selection effects, as their
work uses only isolated galaxies, which may have somewhat
loweraveragestellarmasses. Gavazziet al.(2007)useaset of
stronglensesfromthe SLACS surveyalong with a modelfor
simultaneouslyfitting the stellar anddarkmatter componen ts
ofthestackedlensprofiles. Thisresult,atonemassscale,i sa
bithigherthanourerrorrangebutwithin1.5 σ. Theselection
effects relevant to strong lenses are beyond the scope of thi s
paper; however, within the effective radius, the stellar ma ss
can easily contribute more to the lensing effect than the dar k
matter. Thus,atanygivenhalomass,thehaloswithlessmas-
sivegalaxiesaremuchlesslikelytobestronglenses,resul ting
inabiastowardshigherstellarmassfractionsinstronglen ses
ascomparedtohalosselectedat random.
Atthehighmassend,onecandirectlyidentifyclustersand
groups corresponding to dark matter halos, and measure the
stellar masses of their central galaxies. Yanget al. (2009a )
useagroupcatalogmatchedtohalostodeterminehalomasses
(viaaniteratively-computedgroupluminosity–massrelat ion).
StellarmassesinthisworkaredeterminedusingtheBell eta l.
(2003)relationbetween g−rcolorand M/L; a WMAP3 cos-
mologywasassumed. Theirresultsagreeverywell withours
for low-masshalos, but they beginto differ at highermasses .
This may be partially due to scatter between their calculate d
halo masses (based on total stellar mass in the groups) and
the true halo masses, resulting in additional scatter in the ir
stellar masses at fixed halo mass. It could also be due to dif-
ferences in stellar modeling; their results remain at all ti mes
within oursystematic errors. We also compareto directmea-
surements of massive clusters by Hansenet al. (2009) and
Lin&Mohr (2004). In order to convert luminosities to stel-
lar masses, we assume M/Li0.25= 3.3M⊙/L⊙,i0.25andM/LK= 0.83M⊙/L⊙,Kbased on the population synthesis code of
Conroyetal.(2009). Thesemeasurementsarebothsomewhat
higherthanourresultsformassiveclusters,theone-sigma er-
ror estimates overlap. The discrepancies may be due to is-
sues with cluster selection and with modeling scatter in the
mass-observable relation; in each case the cluster mass is a n
average mass for the given observable (X-ray luminosity or
cluster richness), and can result in a bias if central galaxi es
are correlated with this observable. More detailed modelin g
of the scatter and correlations will be required to determin e
whetherthisis canaccountfortheoffsets.
A comparison of our results to others at z∼1 is shown in
Figure 12. As may be expected, it is much harder to directly
measurethe SM–HM relationat higherredshifts, resultingi n
relatively fewer published results with which we may com-
pare. We first note that we have compared the impact of
two independent measurements of the GSMF from different
surveys. As discussed in 4.3.5, because we simultaneously
fit our model with linear evolution to the GSMF at redshifts
0<z<1, our results are less sensitive to sample variance.
In contrast to the conclusion of Droryet al. (2009) which fit
theirresultstospecificredshiftbins,Figure12showsthat the
Droryetal. (2009) and Pérez-Gonzálezetal. (2008) results
are in agreement within statistical errors at z∼1 when fit-
tingthefullredshiftrange. TheresultsinMosteret al.(20 09)
are also very similar to ours at z∼1. However, Mosteret al.
(2009)donotgivefitsfortheSM–HMrelationwhichinclude
the effects of scatter in stellar mass at fixed halo mass excep t
atz∼0anddonotingeneralincludescatterinmeasuredstel-
lar mass with respect to the true stellar mass. Hence, we in-
clude for comparison an SM–HM relation derived using our
analysis but excluding both of these effects. The remaining
deviance most likely stems from sample variance due to the
muchsmallersurveyvolumeonwhichtheirfit isbased.
Atz∼1, Wang& Jing (2009) make use of clustering
data and stellar mass functions from the VVDS survey
(Meneuxet al. 2008; Pozzettiet al. 2007) at z∼0.8.At the
sametime,thehigh-massandlow-massslopesoftheirpower-
law relation are not re-fit for the higher redshift, leading t o
similar deviations from our results as at z∼0.1. The results
in Zhenget al. (2007) lie slightly below our results at z∼1.
This is partially due to their use of a WMAP1 cosmology.
However,it isdifficultto say exactlyhowmuch,as theirstel -
lar masses at z∼1 are a hybrid of Ks-derived masses from
Bundyetal.(2006)andcolor–basedmassesderivedinaman-
ner analogousto Bell et al. (2003). Nonetheless, they remai n
wellwithinoursystematicerrorbars.
We also include a comparison to the z= 1 SM–HM rela-
tion presented in Conroy& Wechsler (2009), who also use
an abundance matching technique to assign galaxies to ha-
los. Unique to the work of Conroy& Wechsler (2009) is
their attempt to jointly fit both the redshift–dependent ste l-
lar mass functions and the redshift–dependentstar formati on
rate – stellar mass relations. In their model, halo growth is
tracked through time using results derived from halo merger
trees, allowing galaxies to be identified across epochs. The
evolution of halos in conjunction with standard abundance
matchingprovidesmodelpredictionsforstarformationrat es.
The SM–HM relation from Conroy&Wechsler (2009) lies
slightly above our best–fit relation, and it is within statis tical
errorbars exceptat the veryhighest halo masses. Thisis due
tothedifferentassumptionsmadeabouttheGSMFinthetwo
worksandalso to the absenceofcorrectionsfor thescatter i n
stellarmassat fixedhalomassinConroy&Wechsler (2009).20 BEHROOZI,CONROY & WECHSLER
Finally, the strong lensing survey of Gavazzietal. (2007)
has been extended by (Lagattutaetal. 2009) out to z∼0.9
using lenses observed in the CASTLES program, as well as
in COSMOS and in the EGS. While the same caveats about
selection effects apply as for lower redshifts, Lagattutae tal.
(2009) find that the evolution in the stellar mass fraction fo r
Mh∼1013.5M⊙halosiswithin0-0.3dexgreaterat z∼0.9as
compared to z∼0, consistent with our limits of 0-0.15 dex
forthe allowedevolutionoverthatredshiftinterval.
5.RESULTS BEYOND z=1
5.1.MethodologyandDataLimitations
As discussed in §2.1.2, published results for the galaxy
stellar mass function beyond z= 1 suffer from the important
caveat that integrated SFRs are inconsistent with galaxy st el-
lar mass functions when both sets of observations are taken
at face value with a constant IMF. Nonetheless, one may use
similar methodology as in §3 to derive stellar mass – halo
mass relations at higher redshifts under the assumption tha t
the observed stellar mass functions are correct. Here we as-
sume the GSMFs of Marchesiniet al. (2009), which cover a
redshiftrangeof1 .3<z<4.
AsthereisnoguaranteethattheevolutionoftheSM–HM
relation at high redshifts will have the same form as its evo-
lutionatlowredshifts,were-examinetheassumptionsaffe ct-
ing our evolution parameterization in Equation 23. As with
all current high-redshift data, the results in Marchesinie tal.
(2009) are limited to luminous (massive) galaxies, so littl e
information about the value of β(the faint-end slope of the
galaxy-halo mass relation) is available. Hence, we continu e
to assume a linear functional form for its evolution; as the
valueofβevolveslinearlywith scalefactor( a)inourfit, this
means that it is largely constrained to be consistent with th e
evolutionat lowerredshifts(1 <z<2). Naturally,if theevo-
lution ofβat high redshifts is significantly different than for
1<z<2,thenourerrorbarsfor βmayunderestimatethefull
uncertaintiesintheparameter.
Additionally, the systematics affecting high-mass galaxi es
at high redshifts are much more severe than for z<1. Not
onlyaretheerrorsinstellarmasscalculationssignificant (due
to larger photometry errors, limited templates, etc.), but any
miscalibration in correcting photometric redshift errors will
result in low-redshift galaxies masquerading as very brigh t
high-redshift galaxies. These combined uncertainties res ult
in poor constraints on high-mass galaxies. For that reason,
we do not attempt to assume a more complicated functional
formforthe evolutionof δandγ, whichmeansasbeforethat
theirratesofevolutionarelargelyconstrainedtobeconsi stent
withlowerredshifts.
However, we find that individual fits at each redshift do
suggestapossibleevolutionforthecharacteristicstella rmass
which is nonlinear in the scale factor. Hence, we expand the
form of the evolution of M∗,0to include a quadratic depen-
denceonscale factor:
M∗,0(a)=M∗,0,0+M∗,0,a(a−1)+M∗,0,a2(a−0.5)2,(25)
where (a−0.5)2is used instead of ( a−1)2to minimize the
degeneracy between M∗,0,aandM∗,0,a2. We parameterize the
evolutionof all other parametersas in Equation23. All othe r
methodologyremainsthe same as for lower redshifts, as out-
linedin§3.6.
5.2.Results11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
This work
This work, Drory et al. 2009 GSMF
This work, ( σ(z) = 0, ξ=0)
Moster et al. 2009 (AM)
Wang & Jing 2009 (AM+CC)
Zheng et al. 2007 (HOD)
Conroy & Wechsler 2009 (AM)
Figure12. Comparison of our best-fit model at z= 1.0 for different model
assumptionsandtopreviously published results. Darkgrey shading indicates
statistical and sample variance errors; light grey shading includes system-
atic errors. The error bars for the red line, calculated usin g the Drory etal.
(2009)GSMFincludestatistical errorsonly—i.e.,theydon otincludesample
variance. Theresults ofMoster etal.(2009)(green line) do notinclude mod-
eling of scatter or statistical errors in stellar masses, so for comparison, we
present ourresults excluding the effects of σ(z) andξ(blue line). Theresults
of Conroy &Wechsler (2009) made slightly different assumpt ions about the
stellar massfunction evolution.
To maintain some overlapwith the z<1 results, we evalu-
atethelikelihoodfunctionforeachSM–HMrelationagainst
GSMFsfor0 .8<z<4. Thisdatarangeincludestworedshift
bins from Pérez-Gonzálezet al. (2008) (0.8-1.0, 1.0-1.3)a nd
three redshift bins from Marchesiniet al. (2009) (1.3-2, 2- 3,
3-4). IncludingmoreredshiftbinsfromPérez-Gonzálezet a l.
(2008) would improve the continuity of the fits to the low-
redshift results; however, doing so would require a more
complicated redshift parameterization than what we have as -
sumed. The evolution of the best-fit stellar mass fraction fo r
0<z<4isshowninFigure13. AlldatapointsforFigure13
arelistedin AppendixD.
Asmaybeexpected,uncertaintiesathighredshiftsaresub-
stantially larger than at lower redshifts. The contributio n of
systematic errors in stellar masses to the error budget (0.2 5
dex) is still important, but it is no longer the only dominant
factor. Statistical errorsdue to the comparativelysmall n um-
ber of galaxy observationsat highredshifts can contribute an
equaluncertainty(up to 0.25dex)to the derivedSM–HM re-
lation. The statistical errors are large not only for massiv e
galaxies with low number counts, but also for halos below
1012M⊙, where magnitude limits on surveys make observa-
tionsofthecorrespondinggalaxiesdifficult.
The contribution from other sources of uncertainties (e.g. ,
sample variance, cosmology uncertainties) is substantial ly
smaller than the current statistical errors. The effectsof sam-
ple variance on uncertainties at high redshift is less than f or
low redshifts because the volume probed in the high redshift
sample is five times larger than for the low redshift sample
(≈5×106Mpc3vs. 106Mpc3, respectively). While cos-
mology uncertainties are somewhat larger at high redshifts ,
theircontributiontotheoveralllevelsofuncertaintyare again
much smaller than the statistical errors, as was true even by
z=1forthe low-redshiftsample.
The statistical uncertainties at high redshifts mean that i t
is difficult to draw strong conclusions about the evolutionUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 21
11 12 13 14 15
log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
z = 0.1
z = 1.0
z = 2.0
z = 3.0
z = 4.0
Figure13. Evolution of the derived stellar mass fractions for central galax-
ies, from z=4 to the present. The best–fit relations are shown only over t he
mass range where constraining data are available. At higher redshifts, cer-
tainty about the shape of the curves drops precipitously owi ng to a lack of
constraining data beyond the knee of the stellar mass functi on. Combined
systematic and statistical error bars areshown for three re dshift bins only.
of the SM–HM relation. However, the indication is that the
mass corresponding to the peak efficiency for star formation
evolves slowly, and is roughly a factor of five larger at z=4.
At all redshifts, the integrated star formation peaks at ∼10-
20 per cent of the universal baryon fraction; the current dat a
indicatesthat this value may start high at veryhigh redshif ts,
shrinkashalosgrowfasterthantheyformstars,andthensta rt
growing again after z= 2. However, with current uncertain-
ties these results are tenuous. The single most effective wa y
to reduce current uncertainties on both the SM–HM relation
at individualredshiftsandonitsevolutionis to conductmo re
high-redshiftgalaxysurveys,bothtoprobefaintergalaxi esto
determinthe shapeofthe GSMF andto get betterstatistics at
thehighmassend.
6.DISCUSSION AND IMPLICATIONS
Atz∼0, the majority of published results are in accord
within our full systematic error bars, regardless of the tec h-
nique used. All reported results appear to be consistent wit h
the principlesnecessary for abundancematching over a wide
rangeofhalomasses(1011−1015M⊙)—thateachdarkmatter
halo and subhaloabovethe masses we have consideredhosts
a galaxy with a reasonably tight relationship between their
masses, and that average stellar mass — halo mass relation
increasesmonotonicallywith halomass.
Because of the available statistics of halo and galaxy stel-
lar mass functions,especially at z=0, the techniqueof abun-
dancematchingoffersthetightestconstraintsontheSM–HM
relationcurrentlyavailable,anditisinagreementwithre sults
from a broad variety of additional techniques. Under the as-
sumptionthatsystematicerrorsinstellarmasscalculatio nsdo
not change substantially with redshift, abundance matchin g
offers tight constraints on the evolution of the SM–HM rela-
tion from z=1 to the present. These in turn will serve as im-
portant new tests for star formation prescriptionsand reci pes
in both hydrodynamical simulations and semi-analytic mod-
els, as they will applyon the level of individualhalosinste ad
ofonthesimulatedvolumeasa whole.
At the same time, abundance matching offers these con-straints with a minimal number of parameters. The Halo
OccupationDistribution (HOD) technique requiresmodelin g
P(N|Mh), the probabilitydistributionof the numberof galax-
iesperhaloasafunctionofhalomass,inseveraldifferentl u-
minosity bins. In the model proposed in Zhenget al. (2007),
this results in 45 fitted parametersjust to models the occupa -
tionatz=0(fiveparametersfornineluminositybins). Condi-
tional Luminosity Function (CLF) modeling requires param-
eterizing a form for φ(L,Mh), the number density of galaxies
as a functionof luminosityand host halo mass, which results
in approximately a dozen parameters to model occupation at
z=0(Cooray2006). Becauseoftheadditionalconstraintsim-
posedbyassumingthateachhalohostsagalaxy,ourapproach
uses fewer parameters. Abundancematching, as discussed in
this paper, results in a model with only six independent pa-
rameters(fiveto empiricallyfit the derivedSM–HMrelation,
andonetomodelthescatterinobservedstellarmassesatfixe d
halomass) todescribethe populationofgalaxiesinhalos.
Theabundancematchingapproachto the SM–HM relation
requires so few parameters in comparison to other methods
becauseofthefairlysmallscatter( ≈0.16dex)betweenstellar
mass and halo mass at high masses (the scatter has a negligi-
bleimpactonthe averageSM–HMrelationat lowermasses),
and the requirement that satellite galaxies live in satelli te ha-
los(subhalos). Itmaywellbe thatamorecomplicatedmodel
must be adopted for satellites to quantitatively match the
small-scale clustering observations (e.g. Wang etal. 2006 ).
However, such changes will affect the clustering much more
than the derived SM–HM relation, as suggested by the min-
imal changes in Figures 7 and 10 for mass scales ( /lessorsimilar1012.5
M⊙) wheresatellites are a non-negligiblefractionofthe tota l
halopopulation.
ThelargestuncertaintiesintheSM—HMrelationat z<1
comefromassumptionsinconvertinggalaxyluminositiesin to
stellar masses, which amount to uncertaintieson the order o f
0.25 dex in the normalization of the relation. However, the
systematicbiasesintroducedbythecombinedsourcesofsca t-
terbetweencalculatedstellarmassesandhalomassescanri se
toequivalentsignificanceforhalosabove1014.5M⊙. Because
the GSMF is monotonicallydecreasing, results which do not
account for all sources of scatter in stellar mass will over-
predictthe average stellar mass in halos by 0.17-0.25dex fo r
thesemassivehalos.
Using abundance matching to find confidence intervals for
the SM–HM relation is an even more involved process, as
each of the ways in which the systematics might vary must
also be taken into account. While future work on constrain-
ing stellar masses will be the most valuable in terms of re-
ducing uncertainties for the lowest redshift data, wider an d
deeper surveys and some resolution to the discrepancy be-
tween high-redshift cosmic star formationdensity and stel lar
massfunctionsmust occurin orderto improveconstraintson
therelationat highredshifts.
Asmentionedintheintroduction,abundancematchingmay
be used equallywell to assign galaxyluminositiesand color s
to halos. In this case, the galaxy luminosity — halo mass
relation may be derived using identical methodology to that
presentedin§3,withtheexceptionthatthesystematics µand
κmaybeneglected,leavingonly σ(z)(effectively,theredshift
scalingof photometryerrors)and ξ(effectively,the scatter in
luminosityatfixedhalomass). Asthesesystematicsaremuch
betterconstrainedthantheirstellarmasscounterparts(s imply
as luminosities may be measured directly),this approachca n22 BEHROOZI,CONROY & WECHSLER
yield very powerful constraints on the normalization as wel l
as the evolution of the luminosity–mass relation. The highe r
accuracy possible compared to the stellar mass–mass rela-
tion will generally notremove uncertainties in comparing to
galaxyformationmodels. Galaxyformationcodeswhichcal-
culate luminosities must include modeling for all the effec ts
in §2.1.1, meaning that constraintson the underlyingphysi cs
are subject to the same uncertainties. However, tighter con -
straints on the luminosity–mass relation will be nonethele ss
helpful for applicationswhich are concernedwith cosmolog -
icalconstraintsfromlargeluminosity-selectedsurveys.
7.CONCLUSIONS
We have performed an extensive exploration of the uncer-
taintiesrelevantto determiningthe relationshipbetween dark
matter halos and galaxy stellar masses from the halo abun-
dancematchingtechnique. Errorsrelatedtotheobservedst el-
lar mass function, the theoretical halo mass function, and t he
underlying technique of abundance matching are all consid-
ered. We focus on the mean stellar mass to halo mass ratio
forcentralgalaxiesasafunctionofhalomass,andpresentr e-
sultsforthisrelationshipatthepresentepochandextendi ngto
z∼4. Weaccountseparatelyforstatistical errorsandforsys-
tematic errors resulting from uncertainties in stellar mas s es-
timation, and also investigatethe relative contributiono f var-
ious sources of error including cosmological uncertainty a nd
the scatter between stellar mass and halo mass. An analytic
modelhasbeendevelopedwhichcanbeusedtoconstrainthis
connectionintheabsenceofhighresolutionsimulations.
Ourprimaryconclusionsareasfollows:
1. The peak integrated star formation efficiency occurs at
a halo mass near 1012M⊙, with a relatively low frac-
tion — 20% at z= 0 — of baryons currently locked
up in stars. This peak value declines to z∼2 but re-
mains between10–20%for all redshiftsbetween z=0–
4. Thisimpliesthat30 −40%ofbaryonswereconverted
intostarsoverthelifetimeofagalaxywithcurrenthalo
massof1012M⊙.
2. At low masses, the stellar mass – halo mass relation at
z=0scalesas M∗∼M2.3
h. Athighmasses,around1014
M⊙, stellar mass scales as M∗∼M0.29
h. However, the
highmassscalingmaynotbeapowerlaw,asourmodel
indicates that this slope decreases with increasing halo
mass.
3. Within statistical uncertainties,the stellar mass cont ent
of halos has increased by 0 .3−0.45dex for halos with
mass less than 1012M⊙sincez∼1. Systematic biases
in stellar mass calculations between different redshifts
could broaden the uncertaintiesin this number, but the
conclusion that significant evolution has occurred for
low-mass halos would remain robust. For halos with
mass greater than 1013M⊙, our best-fit results indicate
more growth in halo masses than stellar masses since
z∼1,butareconsistentwithnoevolutioninthestellar
massfractionsoverthistime.
4. Systematic, uniform offsets in the galaxy stellar mass
functionand its evolutionare the dominantuncertainty
in the stellar mass – halo mass relation at low redshift.
Statistical errors in the estimation of individual stellar
masses impact the high mass end of the GSMF, and athigher redshifts may result in an observed GSMF that
deviatesfroma Schechterfunction.
5. Current uncertainties in the underlying cosmological
model are sub-dominant to the systematic errors, but
are larger than other sources of statistical error for ha-
losbelow Mh∼1012M⊙forlowredshifts( z<0.2).
6. Given current constraints from other methods, uncer-
tainty in the value of scatter between stellar mass and
halo mass is important in the mean relation for masses
aboveMh∼1012.5M⊙, although it is subdominant to
systematicerrorsforallmassesbelow Mh∼1014.5M⊙.
7. Other uncertainties in the galaxy–halo assignment, in-
cluding different assumptions about the treatment of
satellite galaxies, are subdominant when considering
themeanrelationforcentralgalaxies.
8. At higher redshifts (1 <z<4), systematic uncertain-
tiesremainimportant,butstatisticaluncertaintiesreac h
equal significance. The shape of the relation is fairly
unconstrained at z>2, where improved statistics and
constraintsonthe GSMFbelow M∗areneeded.
Wehavepresentedabest–fitgalaxystellarmass–halomass
relation including an estimate of the total statistical and sys-
tematic errors using available data from z= 0−4, although
caution should be used at redshifts higher than z∼1. We
also presentan algorithmto generalizethis relationforan ar-
bitrary cosmological model or halo mass function. The fact
that assignment errors are sub-dominant and scatter can be
well–constrained by other means gives increased confidence
inusingthesimpleabundancematchingapproachtoconstrai n
this relation. These results provide a powerful constraint on
modelsofgalaxyformationandevolution.
PSB andRHW receivedsupportfromthe U.S. Department
of Energy under contract number DE-AC02-76SF00515 and
froma TermanFellowship at StanfordUniversity. CC is sup-
ported by the Porter Ogden Jacobus Fellowship at Princeton
University. We thank Michael Blanton, Niv Drory, Raphael
Gavazzi, Qi Guo, Sarah Hansen, Anatoly Klypin, Cheng
Li, Yen-TingLin, Pablo Pérez-González, Danilo Marchesini ,
Benjamin Moster, Lan Wang, Xiaohu Yang, Zheng Zheng,
as well as their co-authors for the use of electronic ver-
sions of their data. We appreciate many helpful discus-
sionsandcommentsfromIvanBaldry,MichaelBusha,Simon
Driver,NivDrory,AnatolyKlypin,AriMaller,DaniloMarch -
esini, Phil Marshall, Pablo Pérez-González, Paolo Salucci ,
Jeremy Tinker, Frank van den Bosch, the Santa Cruz Galaxy
Workshop, and the anonymous referee for this paper. The
ART simulation (L80G) used here was run by Anatoly
Klypin, and we thank him for allowing us access to these
data. We are grateful to Michael Busha for providing the
halo catalogs we used to estimate sample variance errors.
These simulations were produced by the LasDamas project (
http://lss.phy.vanderbilt.edu/lasdamas/ );
we thank the LasDamas collaboration for providing us with
thisdata.
REFERENCES
Ashman,K.M.,Salucci, P.,& Persic, M.1993, MNRAS, 260, 610UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 23
Baldry, I.K.,Glazebrook, K.,&Driver, S.P.2008, MNRAS,38 8, 945
Bell, E.F.,McIntosh, D.H.,Katz, N.,&Weinberg, M.D.2003, ApJ, 585,
L117
Berlind, A.A.,&Weinberg, D.H.2002, ApJ, 575,587
Berrier, J.C.,Bullock, J.S.,Barton, E.J.,Guenther, H. D. ,Zentner, A.R.,
& Wechsler, R. H.2006, ApJ,652, 56
Blanton, M. R.,&Roweis, S.2007, AJ,133, 734
Bruzual, G.2007, arXiv:astro-ph/0703052
Bruzual, G.,&Charlot, S.2003, MNRAS,344, 1000
Bryan, G. L.,&Norman, M.L.1998, ApJ,495, 80
Bullock, J.S.,Wechsler,R.H.,&Somerville, R.S.2002,MNR AS,329,246
Bundy, K.,et al. 2006, ApJ,651, 120
Calzetti, D.,Armus,L.,Bohlin, R.C., Kinney, A.L.,Koornn eef, J.,&
Storchi-Bergmann, T.2000, ApJ,533, 682
Cappellari, M.,etal. 2006, MNRAS, 366, 1126
Cattaneo, A.,Dekel, A.,Faber, S.M.,& Guiderdoni, B. 2008, MNRAS,
389, 567
Chabrier, G.2003, Publications of the Astronomical Societ y of thePacific,
115, 763
Charlot, S.1996, in Astronomical Society of thePacific Conf erence Series,
Vol. 98,From Stars to Galaxies: the Impact of Stellar Physic s on Galaxy
Evolution, 275
Charlot, S.,&Fall, S.M.2000, ApJ, 539,718
Charlot, S.,Worthey, G.,&Bressan, A.1996, ApJ,457, 625
Cole, S.,et al. 2001, MNRAS,326, 255
Colín, P.,Klypin, A.A.,Kravtsov, A.V.,& Khokhlov, A.M.19 99, ApJ,
523, 32
Conroy, C.,Gunn, J.E.,&White, M.2009, ApJ,699, 486
Conroy, C.,etal. 2007, ApJ,654, 153
Conroy, C.,&Wechsler, R.H.2009, ApJ, 696,620
Conroy, C.,Wechsler, R. H.,&Kravtsov, A.V. 2006, ApJ,647, 201
Conroy, C.,White, M.,&Gunn, J.E.2010, ApJ,708, 58
Cooray, A.2006, MNRAS,365, 842
Cooray, A.,& Sheth, R. 2002, Phys.Rep., 372,1
Crocce, M.,Fosalba, P.,Castander, F.J.,& Gaztanaga, E.20 09,
arXiv:0907.0019 [astro-ph]
Davé, R.2008, MNRAS, 385, 147
Dressler, A. 1980, ApJ,236, 351
Driver, S.P.,Popescu, C.C.,Tuffs, R.J.,Liske, J.,Graham , A.W.,Allen,
P.D.,&dePropris, R. 2007, MNRAS,379, 1022
Drory, N.,et al. 2009, arXiv:0910.5720 [astro-ph]
Dunkley, J.,Bucher, M.,Ferreira, P.G.,Moodley, K.,&Skor dis, C.2005,
MNRAS, 356,925
Eddington, Sir, A.S.1940, MNRAS, 100,354
Gavazzi, R.,Treu,T.,Rhodes, J.D.,Koopmans,L.V. E.,Bolt on, A.S.,
Burles, S.,Massey, R.J.,&Moustakas, L.A.2007, ApJ,667, 1 76
Guo,Q.,White, S.,Li, C.,&Boylan-Kolchin, M. 2009, arXiv: 0909.4305
[astro-ph]
Guzik, J.,&Seljak, U.2002, MNRAS,335, 311
Hansen, S.M.,Sheldon, E.S.,Wechsler, R. H.,&Koester, B. P .2009, ApJ,
699, 1333
Hilbert, S.,White, S. D.M.,Hartlap, J.,&Schneider, P.200 7, MNRAS,
382, 121
Hopkins, A.M.,&Beacom, J.F.2006, ApJ,651, 142
Hopkins, A.M.,&Beacom, J.F.2006, ApJ,651, 142
Jenkins, A.,et al. 2001, MNRAS,321, 372
Kajisawa, M.,et al. 2009, ApJ,702, 1393
Kannappan, S.J.,& Gawiser, E.2007, ApJ,657, L5
Kauffmann, G.,etal. 2003, MNRAS, 341, 33
Kewley, L.J.,Jansen, R. A.,& Geller, M.J.2005, PASP,117, 2 27
Klypin, A.,Gottlöber, S.,Kravtsov, A.V., &Khokhlov, A.M. 1999, ApJ,
516, 530
Klypin, A.,Trujillo-Gomez, S.,&Primack, J.2010, ArXiv e- prints
Klypin, A.,et al. 2009, ApJ,in preparation
Komatsu, E.,et al. 2009, ApJS,180, 330
Kravtsov, A.,&Klypin, A.1999, ApJ,520, 437
Kravtsov, A.V.,Berlind, A.A.,Wechsler, R.H.,Klypin, A.A .,Gottloeber,
S.,Allgood, B.,&Primack, J.R.2004, ApJ, 609, 35
Kravtsov, A.V., Gnedin, O.Y., &Klypin, A.A.2004, ApJ,609, 482
Kravtsov, A.V.,Klypin, A.A.,&Khokhlov, A.M.1997, ApJ,11 1, 73
Lagattuta, D.J.,et al. 2009, arXiv:0911.2236 [astro-ph]
LeBorgne, D.,Rocca-Volmerange, B.,Prugniel, P.,Lançon, A.,Fioc, M.,&
Soubiran, C. 2004, A&A,425,881
Lee, H.-c.,Worthey, G.,Trager, S.C.,&Faber, S. M.2007, Ap J,664, 215
Lee, S.,Idzi, R.,Ferguson, H.C.,Somerville, R. S.,Wiklin d, T.,&
Giavalisco, M.2009, ApJS,184, 100
Leitherer, C., etal. 1999, ApJS,123, 3
Li,C.,Jing, Y. P.,Kauffmann, G.,Börner, G.,Kang, X.,&Wan g, L.2007,
MNRAS, 376,984Li,C.,&White, S.D.M.2009, MNRAS, 398, 2177
Lin,Y.-T.,&Mohr, J.J.2004, ApJ,617, 879
Lucatello, S.,Gratton, R.G.,Beers, T.C.,&Carretta, E.20 05, ApJ, 625,
833
Mandelbaum, R.,Seljak, U.,Kauffmann, G.,Hirata, C. M.,&B rinkmann, J.
2006, MNRAS,368, 715
Maraston, C.2005, MNRAS, 362,799
Marchesini, D.,van Dokkum, P.G.,Förster Schreiber, N.M., Franx, M.,
Labbé, I.,&Wuyts, S.2009, ApJ,701, 1765
Marín, F.A.,Wechsler, R. H.,Frieman, J.A.,&Nichol, R. C.2 008, ApJ,
672, 849
Meneux, B., etal. 2008, A&A,478, 299
—.2009, A&A,505, 463
More, S.,van den Bosch, F.C.,Cacciato, M.,Mo, H.J.,Yang, X .,&Li,R.
2009, MNRAS,392, 801
Moster, B.P.,Somerville, R.S.,Maulbetsch, C.,van den Bos ch, F.C.,
Maccio’, A.V.,Naab, T.,&Oser, L.2009, arXiv:0903.4682 [a stro-ph]
Muzzin, A.,Marchesini, D.,van Dokkum, P.G.,Labbé, I.,Kri ek, M.,&
Franx, M.2009, ApJ, 701, 1839
Nagai, D.,&Kravtsov, A. V.2005, ApJ,618, 557
Nagamine, K.,Ostriker, J.P.,Fukugita, M.,& Cen, R.2006, T he
Astrophysical Journal, 653, 881
Nagamine, K.,Ostriker, J.P.,Fukugita, M.,&Cen, R.2006, A pJ,653, 881
Neyrinck, M.C.,Hamilton, A. J.S.,& Gnedin, N. Y.2004, MNRA S, 348,1
Panter, B.,Heavens, A.,& Jimenez, R. 2004, MNRAS,355, 764
Panter, B.,Jimenez, R.,Heavens, A. F.,&Charlot, S.2007, M NRAS,488
Percival, S.M.,&Salaris, M. 2009, ApJ,703, 1123
Pérez-González, P.G.,etal. 2005, ApJ,630, 82
—.2008, ApJ,675, 234
Postman, M.,& Geller, M.J.1984, ApJ,281, 95
Pozzetti, L.,et al. 2007, A&A,474, 443
Prada, F.,et al. 2003, ApJ,598, 260
Press,W.H.,&Schechter, P.1974, ApJ,187, 425
Reddy, N.A.,&Steidel, C. C.2009, ApJ,692, 778
Salimbeni, S.,Fontana, A.,Giallongo, E.,Grazian, A.,Men ci, N.,
Pentericci, L.,&Santini, P.2009, in American Institute of Physics
Conference Series, Vol. 1111, American Institute of Physic s Conference
Series, ed.G. Giobbi, A.Tornambe, G. Raimondo, M. Limongi,
L.A.Antonelli, N.Menci, &E.Brocato, 207–211
Salpeter, E.E.1955, ApJ,121, 161
Shankar, F.,Lapi, A.,Salucci, P.,DeZotti, G.,&Danese, L. 2006, ApJ,643,
14
Sheldon, E.S.,et al. 2004, AJ,127, 2544
Spergel, D.N.,et al. 2003, ApJS,148, 175
Springel, V.2005, MNRAS,364, 1105
Stanek, R.,Rudd, D.,&Evrard, A.E.2009, MNRAS,394, L11
Tasitsiomi, A.,Kravtsov, A.V.,Wechsler, R. H.,& Primack, J.R. 2004,
ApJ,614, 533
Tinker, J.,Kravtsov, A.V.,Klypin, A.,Abazajian, K.,Warr en, M.,Yepes,
G.,Gottlöber, S.,& Holz, D.E.2008, ApJ,688, 709
Tinker, J.L.,Weinberg, D.H.,Zheng, Z.,&Zehavi, I.2005, A pJ,631, 41
Tinsley, B.M.,&Gunn, J.E.1976, ApJ,203, 52
Tumlinson, J.2007a, ApJ,665, 1361
—.2007b, ApJ, 664, L63
Vale, A.,&Ostriker, J.P.2004, MNRAS,353, 189
—.2006, MNRAS, 371,1173
van den Bosch, F.C.,Norberg, P.,Mo,H.J.,&Yang, X.2004, MN RAS,
352, 1302
van der Wel,A.,Franx, M.,Wuyts,S.,van Dokkum, P.G.,Huang , J.,Rix,
H.-W.,&Illingworth, G.D.2006, ApJ,652, 97
van Dokkum, P.G.2008, ApJ,674, 29
Wang,L.,& Jing, Y.P.2009, arXiv:0911.1864 [astro-ph]
Wang,L.,Li, C.,Kauffmann, G.,&deLucia, G.2006, MNRAS,37 1, 537
Warren, M.S.,Abazajian, K.,Holz, D.E.,&Teodoro, L.2006, ApJ, 646,
881
Weinberg, D.H.,Colombi, S.,Davé, R.,&Katz, N.2008, ApJ,6 78, 6
Weinberg, D.H.,Davé, R.,Katz, N.,&Hernquist, L.2004, ApJ ,601, 1
Wetzel, A.R.,&White, M.2009, arXiv:0907.0702 [astro-ph]
Wilkins, S. M.,Trentham, N.,& Hopkins, A.M.2008a, MNRAS,3 85, 687
—.2008b, MNRAS, 385, 687
Yang, X.,Mo,H.J.,&van den Bosch, F.C.2009a, ApJ,695, 900
—.2009b, ApJ, 693, 830
Yang, X.,Mo,H.J.,van den Bosch, F.C.,Pasquali, A.,Li,C., &Barden, M.
2007, ApJ,671, 153
Yang, X.,etal. 2003, MNRAS,339, 1057
Yi, S.K.2003, ApJ,582, 202
York,D.G.,etal. 2000, AJ,120, 1579
Zaritsky, D.,&White, S.D.M.1994, ApJ, 435,599
Zheng,Z.,Coil, A.L.,&Zehavi, I.2007, ApJ,667, 76024 BEHROOZI,CONROY & WECHSLER
APPENDIX
CONVERTING RESULTS TO OTHER HALO MASS FUNCTIONS
FromEquation14,itispossibletosimplyconvertfromourha lomassfunction φhtoanyhalomassfunctionofchoice( φh,r). In
particular,the function Mh(M∗) is defined by the fact that the numberdensity of halos with ma ss aboveMh(M∗) must match the
numberdensityofgalaxieswithstellarmassabove M∗(withtheappropriatedeconvolutionstepsapplied). Recal lfromEquation
14that
/integraldisplay∞
Mh(M∗)φh(M)dlog10M=/integraldisplay∞
M∗φdirect(M∗)dlog10M∗. (A1)
Naturally,thecorrectmass-stellarmassrelationforthea lternatehalomassfunction φh,r(whichwewilllabelas Mh,r(M∗))must
satisfythissameequation,withtheresult that:
/integraldisplay∞
Mh(M∗)φh(M)dlog10M=/integraldisplay∞
Mh,r(M∗)φh,r(M)dlog10M. (A2)
Tomakethecalculationevenmoreexplicit,let Φh(M)=/integraltext∞
Mφhdlog10Mbeourcumulativehalomassfunction,andlet Φh,r(M)
bethecorrespondingcumulativehalomassfunctionfor φh,r. Then,we find:
Mh,r(M∗)=Φ−1
h,r(Φh(Mh(M∗))). (A3)
Massfunctionsfromdifferentcosmologiesthanthose assum edin thispaperwill alsorequireconvertingstellar masses if their
choicesof hdifferfromtheWMAP5 best-fitvalue.
EFFECTS OF SCATTER ON THE STELLAR MASS FUNCTION
Thissectionisintendedtoprovidebasicintuitionforthee ffectsofboth ξandσ(z),whichmaybemodeledasconvolutions. The
classic examplein this case is convolutionof the GSMF with a log-normaldistributionof some width ω. While the convolution
(evenofa Schechterfunction)with a Gaussian hasno knownan alyticalsolution, we may approximatethe result byconside ring
the case where the logarithmic slope of the GSMF changes very little over the width of the Gaussian. Then, locally, the ste llar
mass function is proportional to a power function, say φ(M∗)∝(M∗)α. Then, if we let x= log10M∗(so thatφ(10x)∝10αx),
findingtheconvolutionisequivalenttocalculatingthefol lowingintegral:
φconv(10x)∝/integraldisplay∞
−∞10αb
√
2πω2exp/parenleftbigg
−(x−b)2
2ω2/parenrightbigg
db
= 10αx101
2α2ω2ln(10). (B1)
That is to say, the stellar mass function is shifted upward by approximately1 .15(αω)2dex. Hence, for parts of the stellar mass
functionwith shallow slopes, the shift is completely insig nificant, as it is proportionalto α2. However,it matters much more in
thesteeperpartofthestellarmassfunction,tothepointth atforgalaxiesofmass1012M⊙,theobservedstellarmassfunctioncan
beseveralordersofmagnitudeabovethe intrinsicGSMF.
A SAMPLE CALCULATION OF THE FUNCTIONAL FORM OF THE STELLAR MA SS FUNCTION
Galaxy formation models typically assume at least two feedb ack mechanisms to limit star formation for low-mass galaxie s
and for high-mass galaxies. Thus, one of the simplest fiducia l star formation rate (SFR) as a function of halo mass ( Mh) would
assumea doublepower-lawform:
SFR(Mh)∝/parenleftbiggMh
M0/parenrightbigga
+/parenleftbiggMh
M0/parenrightbiggb
. (C1)
We mightexpectthe total stellar massas a functionofhaloma ssto take a similar form,exceptperhapswith a wider regiono f
transitionbetween galaxieswhose historiesare predomina ntlylow mass, and those with historieswhich are predominan tlyhigh
mass, for the reason that some galaxies’ accretion historie s may have caused them to be affected comparably by both feedb ack
mechanisms.
Hence, assuming that the relation between halo mass and stel lar mass follows a double power-law form, we adopt a simple
functionalformto convertfromthestellar massofagalaxyt othehalomass:
Mh(M∗)=M1/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBiggγ
. (C2)
Here,βmaybethoughtofasthefaint-endslope, δasthemassive-endslope(although βandδarefunctionallyinterchangeable),
andγasthetransitionwidth(larger γmeansa slowertransitionbetweenthe massiveandfaint-end slopes).
We first calculatedlog(Mh)
dlog(M∗):UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 25
log(Mh)=log(M1)+γlog/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBigg
, (C3)
dlog(Mh)
dlog(M∗)=dlog(Mh)
dM∗dM∗
dlog(M∗)(C4)
=M∗ln(10)dlog(Mh)
dM∗(C5)
=β/parenleftBig
M∗
M∗,0/parenrightBigβ/γ
+δ/parenleftBig
M∗
M∗,0/parenrightBigδ/γ
/parenleftBig
M∗
M∗,0/parenrightBigβ/γ
+/parenleftBig
M∗
M∗,0/parenrightBigδ/γ(C6)
=β+(δ−β)/parenleftBigg
1+/parenleftbiggM∗
M∗,0/parenrightbiggβ−δ
γ/parenrightBigg−1
. (C7)
This justifies the earlier intuition that the functional for m forMh(M∗) transitions between slopes of βandδwith a width that
increases with γ. Note thatdlog(Mh)
dlog(M∗)is always of order one, as the stellar mass is always assumed t o increase with the halo mass
andvice versa(namely, β >0andδ >0).
Next,we approachdN
dlogMh. Fromanalyticalresults, weexpectaSchechterfunctionfo rthehalomassfunction,namely:
dN
dlog(Mh)=φ0ln(10)/parenleftbiggMh
M0/parenrightbigg1−α
exp/parenleftbigg
−Mh
M0/parenrightbigg
. (C8)
Substitutinginthe equationfor Mh(M∗),we have
dN
dlog(Mh)=φ0ln(10)/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
×/parenleftbiggMh
M0/parenrightbigg1−α
exp/parenleftBigg
−M1
M0/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBiggγ/parenrightBigg
. (C9)
Already evident is the generic result that there will be sepa rate faint-end and massive-end slopes in the stellar mass fu nction,
and that the falloff is not generically specified by an expone ntial. We may make one simplification in this model—namely, t o
note that Mh(M∗,0) correspondsto the halo mass at which the slope of Mh(M∗) begins to transition from βtoδ. We expect this
transition to correspondto the transition between superno vafeedbackand AGN feedbackin semi-analyticmodels—namel y,for
a halo mass which is too large to be affectedmuch by supernova feedback,but which is yet too small to host a large AGN. This
implies that Mh(M∗,0) is expected to be around 1012M⊙or less, meaning that Mh/M0is small until stellar masses well beyond
M∗,0, meaning that we may neglect the faint-end slope of the Mh(M∗) relation in the exponential portion of the stellar mass
function:
dN
dlog(Mh)=φ0ln(10)/parenleftbiggM1
M0/parenrightbigg1−α/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
×exp/parenleftBigg
−M1
M0/parenleftbiggM∗
M∗,0/parenrightbiggδ/parenrightBigg
. (C10)
Hence,we maycombinethese twoequationstoobtaintheexpre ssionforthestellar massfunction:
dN
dlog(M∗)=φ0ln(10)/parenleftbiggM1
M0/parenrightbigg1−α/bracketleftBigg/parenleftbiggM∗
M∗,0/parenrightbiggβ/γ
+/parenleftbiggM∗
M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
×exp/parenleftBigg
−M1
M0/parenleftbiggM∗
M∗,0/parenrightbiggδ/parenrightBigg
×
β+(δ−β)/parenleftBigg
1+/parenleftbiggM∗
M∗,0/parenrightbiggβ−δ
γ/parenrightBigg−1
. (C11)26 BEHROOZI,CONROY & WECHSLER
Whilethisseemscomplicated,it maybeintuitivelydeconst ructedas:
dN
dlog(M∗)=[constant]/bracketleftbig
doublepowerlaw/bracketrightbig
×/bracketleftbig
exponentialdropoff/bracketrightbig
O(1). (C12)
As mentioned previously, this functional form is equivalen t toφdirect. To convert to the true stellar mass function φtrueor the
observed stellar mass function φmeas, it must be convolved with the scatter in stellar mass at fixed halo mass and (for φmeas)
the scatter in calculated stellar mass at fixed true stellar m ass. As such, it should be clear that—while the final form may b e
Schechter– like—there is certainly much more flexibility in the final shape of the GSMF than a Schechter function alone would
allow,asevidencedbythefiveparametersrequiredtofullys pecifyequationC11.
DATA TABLES
WereproduceherelistingsofthedatapointsinFigures5,6, and13inTables3,4,and5,respectively. Seesections4.2an d5.2
fordetailsonthedatapointsineachtable.
Table3
Stellar Mass Fractions For0 <z<1 Including Full Systematics
z=0.1 z=0.5 z=1.0
log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
11.00 −2.30+0.26
−0.23
11.25 −1.96+0.25
−0.23−2.11+0.22
−0.26
11.50 −1.67+0.24
−0.24−1.84+0.22
−0.26−2.01+0.25
−0.24
11.75 −1.53+0.23
−0.24−1.70+0.24
−0.24−1.85+0.26
−0.23
12.00 −1.54+0.24
−0.24−1.68+0.26
−0.23−1.77+0.26
−0.23
12.25 −1.62+0.24
−0.24−1.72+0.26
−0.23−1.77+0.25
−0.23
12.50 −1.74+0.24
−0.24−1.81+0.25
−0.23−1.81+0.24
−0.24
12.75 −1.87+0.23
−0.25−1.92+0.25
−0.23−1.89+0.23
−0.26
13.00 −2.02+0.23
−0.25−2.05+0.24
−0.24−1.99+0.22
−0.27
13.25 −2.18+0.22
−0.26−2.19+0.24
−0.25−2.11+0.21
−0.28
13.50 −2.35+0.22
−0.26−2.34+0.23
−0.25−2.25+0.21
−0.29
13.75 −2.52+0.22
−0.27−2.51+0.23
−0.26−2.39+0.21
−0.30
14.00 −2.70+0.21
−0.28−2.68+0.23
−0.26−2.55+0.22
−0.30
14.25 −2.88+0.21
−0.28−2.86+0.23
−0.26
14.50 −3.07+0.20
−0.29−3.04+0.23
−0.27
14.75 −3.26+0.20
−0.30
15.00 −3.45+0.20
−0.30
Note. — Halo massesare in units of M⊙. Constraints are quoted over themass range probed by the obs erved GSMF.UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 27
Table4
Stellar Mass Fractions without Systematic Errors ( µ=κ=0)
z=0.1 z=0.5 z=1.0
log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
11.00 −2.30+0.03
−0.02
11.25 −1.96+0.04
−0.01−2.10+0.04
−0.08
11.50 −1.67+0.03
−0.01−1.83+0.05
−0.06−2.02+0.10
−0.06
11.75 −1.53+0.01
−0.01−1.71+0.06
−0.03−1.85+0.10
−0.04
12.00 −1.54+0.01
−0.02−1.68+0.06
−0.02−1.78+0.09
−0.03
12.25 −1.62+0.00
−0.02−1.72+0.06
−0.01−1.77+0.08
−0.03
12.50 −1.74+0.01
−0.02−1.81+0.05
−0.02−1.81+0.06
−0.04
12.75 −1.87+0.01
−0.03−1.92+0.05
−0.02−1.88+0.04
−0.06
13.00 −2.02+0.01
−0.03−2.05+0.04
−0.03−1.98+0.03
−0.08
13.25 −2.18+0.02
−0.04−2.19+0.04
−0.04−2.10+0.04
−0.10
13.50 −2.35+0.02
−0.05−2.34+0.04
−0.05−2.24+0.04
−0.13
13.75 −2.52+0.03
−0.06−2.51+0.05
−0.06−2.38+0.05
−0.15
14.00 −2.70+0.03
−0.07−2.68+0.05
−0.07−2.54+0.06
−0.17
14.25 −2.88+0.04
−0.08−2.85+0.06
−0.08
14.50 −3.07+0.04
−0.09−3.04+0.06
−0.10
14.75 −3.25+0.05
−0.10
15.00 −3.45+0.05
−0.11
Note. — Halo massesare in units of M⊙.
Table5
Stellar Mass Fractions For 0 .8<z<4 Including Full Systematics
z=1.0 z=2.0 z=3.0 z=4.0
log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
11.50 −2.01+0.25
−0.24
11.75 −1.85+0.26
−0.23
12.00 −1.77+0.26
−0.23−1.89+0.22
−0.27−1.89+0.25
−0.27
12.25 −1.77+0.25
−0.23−1.76+0.24
−0.25−1.67+0.21
−0.29−1.58+0.22
−0.32
12.50 −1.81+0.24
−0.24−1.78+0.23
−0.26−1.63+0.21
−0.30−1.50+0.20
−0.35
12.75 −1.89+0.23
−0.26−1.87+0.20
−0.30−1.71+0.19
−0.35−1.56+0.19
−0.40
13.00 −1.99+0.22
−0.27−2.00+0.17
−0.35−1.83+0.17
−0.39−1.68+0.19
−0.44
13.25 −2.11+0.21
−0.28−2.14+0.15
−0.39−1.97+0.16
−0.44−1.82+0.19
−0.49
13.50 −2.25+0.21
−0.29−2.30+0.15
−0.42−2.13+0.17
−0.47−1.98+0.19
−0.52
13.75 −2.39+0.21
−0.30−2.47+0.17
−0.45−2.30+0.19
−0.49
14.00 −2.55+0.22
−0.30−2.64+0.20
−0.47
Note. — Halo massesare in units of M⊙.