diff --git "a/1001.0015.txt" "b/1001.0015.txt" new file mode 100644--- /dev/null +++ "b/1001.0015.txt" @@ -0,0 +1,2926 @@ +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 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∗,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 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) +00 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 01012.5M⊙. +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 +02, 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