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	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⊙.