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