Mon.Not.R.Astron.Soc.000,1–13(2007) Printed5February2008 (MNLATEXstylefilev2.2) Modelling and interpreting the dependence of clustering on the spectral energy distributions of galaxies Lan Wang1,2⋆, Cheng Li2,3,4,5, Guinevere Kauffmann2, Gabriella De Lucia2 1Department of Astronomy, Peking University,Beijing 100871, China 2Max–Planck–Institut fu¨r Astrophysik, Karl–Schwarzschild–Str. 1, D-85748 Garching, Germany 3The Partner Group of MPI fu¨rAstrophysik, Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China 4Centerfor Astrophysics, University of Science and Technology of China, Hefei, Anhui 230026, China 5Joint Institute for Galaxy and Cosmology (JOINGC) of SHAO and USTC 7 0 0 Accepted 2007??????.Received2007??????;inoriginalform2007?????? 2 n a ABSTRACT J We extend our previous physically-based halo occupation distribution models to in- 4 clude the dependence of clustering on the spectral energy distributions of galaxies. 2 The high resolution Millennium Simulation is used to specify the positions and the velocities of the model galaxies. The stellar mass of a galaxy is assumed to depend 1 only on M , the halo mass when the galaxy was last the central dominant object infall v of its halo. Star formation histories are parametrized using two additional quantities 2 thataremeasuredfromthesimulationforeachgalaxy:itsformationtime(t ),and 8 form thetimewhenitfirstbecomesasatellite(t ).Centralgalaxiesbeginformingstars 6 infall at time t with an exponential time scale τ . If the galaxy becomes a satellite, its 1 form c 0 starformationdeclinesthereafterwithanewtimescaleτs.Wecompute4000˚Abreak 7 strengths for our model galaxies using stellar population synthesis models. By fitting 0 these models to the observed abundances and projected correlations of galaxies as a / function of break strength in the Sloan Digital Sky Survey, we constrain τ and τ as h c s functionsofgalaxystellarmass.Wefindthatcentralgalaxieswithlargestellarmasses p - haveceasedformingstars.Atlowstellarmasses,centralgalaxiesdisplaya wide range o of different star formation histories, with a significant fraction experiencing recent r starbursts. Satellite galaxies of all masses have declining star formation rates, with t s similar e–folding times, τ ∼ 2.5 Gyr. One consequence of this long e–folding time is s a thatthe colour–densityrelationis predicted to flattenatredshifts >1.5,because star : v formationinthemajorityofsatelliteshasnotyetdeclinedbyasignificantfactor.This i is consistent with recent observational results from the DEEP and VVDS surveys. X r Keywords: galaxies:fundamentalparameters–galaxies:haloes–galaxies:distances a and redshifts – cosmology: theory – cosmology: dark matter – cosmology: large-scale structure 1 INTRODUCTION (Hubble 1936; Abell 1958; Davis & Geller 1976; Dressler 1980;Loveday et al.1995;Zehaviet al.2002),spectraltype The most fundamental metric of a galaxy is its luminosity, (Lovedayet al. 1999; Norberg et al. 2002; Madgwick et al. whichservesasarough indicatorofitstotalmass.Another 2003), and colour (Willmer et al. 1998; Zehaviet al. 2002; fundamentalmetricofagalaxyisitscolourorspectraltype, Budav´ari et al. 2003). On the other hand, galaxy cluster- whichisusuallyinterpretedasanindicatorofitsrecentstar ing also depends on luminosity (Hamilton 1988; Park et al. formation history (although metallicity and dust are both 1994; Willmer et al. 1998; Zehaviet al. 2002), with lumi- knowntoaffectgalaxycolours).InthelocalUniverse,galaxy nous galaxies being more strongly clustered than fainter luminositiesandcoloursareknowntobestronglycorrelated; ones. luminous elliptical galaxies are much redder than the less luminousspirals. Ithasalso longbeenknownthattheclus- Recent large surveys such as 2dfGRS (Colless et al. teringofgalaxies isastrongfunctionof morphological type 2001) andSDSS(Yorket al.2000)haveallowed thecovari- ance between galaxy luminosity and colour to be broken. Norberg et al. (2002) showed at all luminosities, galaxies ⋆ Email:[email protected] with spectral features indicative of a “passive” old stellar (cid:13)c 2007RAS 2 L.Wang, C.Li, G.Kauffmann, G. De Lucia population have higher correlation amplitude than galaxies vantageoftheHODapproachisthatitremainsunclearhow with ongoing star formation. Likewise Zehaviet al. (2005) one progresses from a purely statistical characterization of and Liet al. (2006a) measured the clustering of red and thelink between galaxies and dark matter halos, to a more blue galaxies as a function of luminosity and stellar mass, physical understanding of the galaxy formation process it- and showed that red galaxies are more strongly correlated self. and have a correlation function with a steeper slope. The In previous work (Wang et al. 2006), we attempted to strongest differences between the red and blue correlation build a physically-based HOD model that would combine functions occurred for galaxies with the lowest luminosities theadvantages of thestatistical HODmodels with thoseof and stellar masses. thesemi-analyticapproach.A largehigh resolution N-body In order to interpret these results, we need to under- simulation,theMillenniumSimulation(Springel et al.2005) stand the underlying physics that causes the dependence was used to follow the merging paths of dark matter halos of the correlation function on colour/spectral type. Semi- and their associated substructures and to specify the posi- analytic models of galaxy formation are able to provide tionsandthevelocitiesofthegalaxiesinthesimulationbox. considerable insight into the processes that determine how The properties of the galaxies (in this case, their luminosi- galaxies with different properties cluster (Kauffmann et al. ties and stellar masses) were specified using simple param- 1997, 1999; Benson et al. 2000; Croton et al. 2006). These eterized functions. Wanget al. (2006) chose to parametrize modelsuseN-bodysimulationsofthedarkmattertospecify the luminosities and masses of the galaxies in their mod- the locations of galaxies, and they invoke simple prescrip- els in terms of the quantity M , which was defined as infall tions to describe processes such as gas cooling, star forma- the mass of the dark matter halo when the galaxy was last tion and supernovafeedback. thecentralgalaxyofitsownhalo.Wanget al.(2006)tested Inthesemi-analyticmodels(SAMs),therearetworea- this parametrization using the semi-analytic galaxy cata- sonswhygalaxiestransitionfromstar-forming(blue)topas- logues of Croton et al. (2006) and and then applied it to sive (red) systems. One is a consequence of the infall of a data from the Sloan Digital Sky Survey.They were able to galaxy onto a larger halo. When this occurs, the galaxy is show that the relation between stellar mass and halo mass stripped of its supply of infalling gas and its star forma- inferredfromthecombinationofthemodelsandthecluster- tion rate declines as its cold gas reservoir is depleted. This ingdatawasingood agreementwithindependentmeasure- meansthattherewillbeapopulationofredsatellitegalaxies mentsofthisrelationusinggalaxy-galaxylensingtechniques located in groups and clusters (Diaferio et al. 2001). How- (Mandelbaum et al. 2006). ever, this process by itself is not sufficient to explain the In this paper, we extend our method to model the de- very strong observed dependence of colour on galaxy lu- pendence of clustering on the spectral energy distributions minosity. Some other process must act to terminate star ofgalaxies.Forgalaxiesofgivenstellarmass,weassumethe formation in the central galaxies of massive dark matter star formation history to depend not only on stellar mass, haloes. In recent models (Croton et al. 2006; Bower et al. butalsoonwhetherthegalaxyisacentralobjectorasatel- 2006; Cattaneo et al. 2006), feedback from active galactic liteinthesimulation.Wefitthismodeltothecolourdistri- nuclei(AGN)hasbeeninvokedasapossiblemechanism for butionsofgalaxiesofgivestellarmass,aswellastotheircor- suppressingstarformationinmassivecentralgalaxies.How- relationfunctionssplitbycolour.Wechoosetofocusonthe ever,thedetailsoftheAGNfeedbackprocessandthetrun- spectral indexD4000 rather thana moretraditional broad- cationofstarformation ininfallingsatellitegalaxiesremain band colour. TheD4000 index is defined as theratio of the poorly constrained, so in reality there is still considerable flux in two bands at the long– and short–wavelength side freedom when attemptingto specify thestar formation his- of the 4000˚A discontinuity. This 4000˚A break arises from tories of galaxies in these models. The colour distributions asum of manyabsorption linesproduced byionized metals generatedbytheSAMsshouldthusbetreatedasindicative intheatmosphereofstars.Becausetheabsorptionincreases rather than quantitativepredictions of the models. with decreasing stellar temperature, the D4000 break gets TheHaloOccupationDistribution(HOD)approachby- larger with older ages, and is largest for old and metal–rich passes any consideration of the physical processes impor- stellar populations. Therefore it is a good indicator of the tant in galaxy formation. It specifies how galaxies are re- star formation history of a galaxy. In this work we adopt lated to dark matter halos in a purely statistical fashion. the narrow definition of the 4000˚A break introduced by van den Bosch et al. (2003) was thefirst to model theclus- Balogh et al.(1999),anddenoteitasDn4000.Galaxieswith tering properties of early and late-type galaxies in the con- largeandsmallvaluesofDn4000arereferredtoas“old”and text of HOD models by using observational data from the “young”respectively.BecauseDn4000isdefinedinanarrow 2dFredshiftsurveytoconstraintheaveragenumberofearly wavelength interval,it is insensitive to the effects of dust. andlate-typegalaxies ofgivenluminosity residinginahalo Theparametrized modelsthatweconstructextendthe ofgivenmass.Morerecently,Zehaviet al.(2005)builtHOD old ones in which stellar mass M∗ is assumed to depend modelsthatwereabletoreproducethecorrelation function onlyonM byassumingthatthestarformationhistory infall ofredandbluegalaxiesasafunctionofluminosity,bydefin- of each galaxy declines exponentially after its first appear- ing a blue galaxy fraction as a function of dark halo mass. ance in the simulation with time constant τc(M∗) as long Thebluefraction also dependedonwhetherthegalaxywas asitis acentralgalaxy,andthenwith adifferenttimecon- a central or a satellite system. The results of both studies stant τs(M∗) after it becomes a satellite galaxy. Note that demonstrate that the strong clustering of faint red galaxies this strongly resembles the approach adopted in the semi- canbeexplainedifnearlyallofthemaresatellitesystemsin analyticmodels.Themaindifferenceisthatthetimescales high-mass halos. These results haverecently been extended τ and τ and their dependence on mass are not specified c s tohigherredshiftsbyPhleps et al. (2006).Themain disad- usinganyfixed“recipe”;weallowthesetimescalestobedi- (cid:13)c 2007RAS,MNRAS000,1–13 Modelling the SED dependence of galaxy clustering 3 rectlyconstrainedbytheobservationaldata.Aswewillsee, observed galaxies are randomly re-assigned. As shown in thisapproach allows ustodrawinterestingphysicalconclu- Li et al. (2006a), the 2PCFs obtained with the two meth- sionsfrom acomparison ofourmodelswiththeSDSSdata. ods are in good agreement, and here we will use the more In Sec. 2 we present observational results from the “general”method.Wehavealsocarefully correctedforpos- Sloan Digital Sky Survey (SDSS). These include the sible biases, such as the variance in mass-to-light ratio and D 4000/colour distributions of galaxies in different stellar thesmall-scaledeficiencyinthe2PCFduetofibrecollisions n mass ranges and the projected two point correlation func- (Li et al. 2006a). This ensures accurate measurements for tionofred/blueandold/younggalaxies.InSec.3,weoutline correlation functions on small scales. our basic theoretical concepts and describe how we decided To take into account the effect of “cosmic variance” to model the star formation histories of the galaxies, and on the w(r ) measurements, we have constructed a set of p calculate their spectral energy distributions. In Sec. 4 we 10 mock galaxy catalogues from the Millennium simulation describe how we fit the models to the data. Interpretation with exactly the same geometry and selection function as andtestsofourresultsaregiven inSec.5.Conclusions and the real sample. The effect of cosmic variance is modelled discussions are presented in thefinal section. byplacingavirtualobserverrandomlyinsidethesimulation boxwhen constructingthesemock catalogues. Thedetailed procedure for constructing these mock catalogues has been presented in Li et al. (2006c). For each mock catalogue, we 2 THE OBSERVATIONAL RESULTS dividegalaxiesintosubsamplesaccordingtostellarmassand Ourobservationalresultsarebasedonasampleof 200,000 g r colour, in the same way as for the real sample, and galaxies drawn from the SDSS Data Release T∼wo. The we−measure w(rp) for these subsamples. The 1σ variation galaxies have 0.01 < z < 0.3, 14.5 < r < 17.77 and betweenthesemockcatalogues isthenaddedinquadrature −pa2r3en<t mMa0g.1nrit<ude−1co6r,rwechteerdefrorisfotrheegrro-ubnadndexPtientcrotisoiann, aanpd- tboetthheebsoamotestfroarptehrerosrpsl.itNsobtyetghatrthanederbryorDsanr4e00a0ss.umedto − M0.1r isther-bandabsolutemagnitudecorrectedtoredshift Fig.1showsthedistributionsofDn4000andg−rcolour z = 0.1. This sample has formed the basis of our previous for galaxies, as well as the projected 2PCF w(rp) for the investigations of the correlation function, the power spec- “red” and “blue” subsamples in four stellar mass ranges. trum,pairwise velocitydispersion distributions,and thelu- Ascan be seen, in each stellar mass interval, Dn4000/g r − minosity and the stellar mass function of galaxies (Li et al. showsabimodaldistribution,withthefractionofgalaxiesin 2006a,b). In Wang et al. (2006), we made use of the mea- theredpeakincreasingtowardshighermasses.Older/redder surements of the projected correlation function for galaxies galaxiesofallstellarmassesaremorestronglyclusteredand in bins of luminosity and stellar mass to constrain the re- havesteeper correlation functions than theiryounger/bluer lation between galaxy luminosity/stellar mass and M . counterparts. This age/colour dependenceis much stronger infall Inthispaper,weextendthisanalysistothespectralenergy for the low mass galaxies than for the high mass galaxies, distributionsofgalaxies.Inthissection,wefocusonthedis- particularly on small scales. Notice that the clustering for tributions of Dn4000 and colour and we explore how thee subsamplessplitbyDn4000andg−r arequitesimilar,but projected two-point correlation functions split by D 4000) the distribution functions of these two quantities are quite n and colour change as a function of M∗. different, particularly at low masses. For example, for the The galaxies in our sample are divided into four sub- lowest mass bin (9.5<log(M∗/M⊙)<10), the fractions of samples according to their stellar mass. Each of the stellar red and blue galaxies are comparable, but the fraction of masssubsamplesisthendividedintotwofurthersubsamples galaxieswithlargeDn4000valuesismuchsmallerthanthat according to Dn4000 and g r, using a method similar to of the low Dn4000 population. When building our model, that adopted in Li et al. (20−06a). We fit bi-Gaussian func- wewillfirstfocusontheDn4000spectralindex,andtestto tionstothedistributionofD 4000andg rasafunctionof whatextentamodelthatreproducestheobservedtrendsas n stellarmass.ThedivisionintohighDn400−0andlowDn4000, a function of Dn4000 will also work for g−r colour. redandblue,isdefinedtobethemeanofthetwoGaussian centres in each stellar mass bin. In the computation of the D 4000andg rdistributions,wehavecorrectedforincom- n − 3 THEORETICAL CONCEPTS pletenessbyweightingeachgalaxyafactorofV /V . survey max V isthevolumeforthesample,andV isthemaxi- InthepaperofWang et al.(2006),weusedtheMillennium survey max mumvolumeoverwhichthegalaxycouldbeobservedwithin Simulation to construct a model to describe the clustering thesampleredshiftrangeandwithintherangeofr-bandap- of galaxies as a function of their luminosities and stellar parent magnitudes. masses. The positions and velocities of the galaxies in the For each subsample, the redshift-space two point cor- simulation box were obtained by following the orbits and relation function(2PCF) ξs(r ,π) is measured using the merging histories of the substructures in the simulation. p Hamilton (1993) estimator. The projected 2PCF w(r ) is Parametrized functions were then adopted to relate the lu- p then estimated by integrating ξs(r ,π) along the line-of- minositiesandstellarmassesofthegalaxies tothequantity p sight direction π, with π ranging from 0 to 40 h−1 Mpc. M , defined as the mass of the halo at the epoch when infall | | When computing 2PCFs, two different methods for con- the galaxy was last the central dominant object in its own structing random samples have been used: the “standard” halo. By fitting both the stellar mass function and the pro- method in which the redshift selection function is explic- jected correlation function w(r ) measured in five different p itly modelled using the luminosity function, and the more stellarmassbins,wewereabletousetheSDSSdatatocon- “general” method in which only the sky positions of the strainthelinkbetweengalaxystellarmassM∗ andMinfall. (cid:13)c 2007RAS,MNRAS000,1–13 4 L.Wang, C.Li, G.Kauffmann, G. De Lucia Figure 1. Observational results from SDSS in bins of stellar mass. The left two columns show Dn4000 distributions and correlation functions split by Dn4000. Red/blue lines represent subsamples with larger/smaller values of Dn4000. The right two columns are for g−r;theredandbluelineshererepresentclusteringofredandbluesubsamples. Thisrelation wasparametrized usingadoublepower-lawof Inthispaper,ouraimistoextendthismodeltodescribe theform: not only the masses and luminosities of galaxies, but also 2 theircoloursandspectralenergydistributions.Asuccessful M∗ = (Minfall)−α+(Minfall)−β×k, modelshouldbeabletoreproducethe4000˚Abreakstrength M0 M0 distribution at each stellar mass, as well as the correlation The scatter in log(M∗) at a given value of Minfall was de- functions split by Dn(4000) shown in Fig. 1. scribedusingaGaussianfunctionwithwidthσ.Ourbest-fit In standard semi-analytic models, galaxies that re- modeltotheSDSSdatahadthefollowingparameters:M = side at the centre of their dark matter halo are called 0 4.0 1011h−1M⊙, α = 0.29, β = 2.42, logk = 10.35 and central galaxies, and those that have been accreted into σ=×0.203forcentralgalaxiesandM0 =4.32 1011h−1M⊙, larger structures are termed satellite galaxies. Thesimplest × α=0.232, β =2.49, logk =10.24 and σ =0.291 for satel- model one could imagine for differentiating galaxies in the lite galaxies. 1 high/low D 4000 or red/blue peaks of the bimodal distri- n butionsshown in Fig. 1 would bethat these two peakscor- respond to these satellite and central galaxy populations. 1 Note these parameters are slightly different from those pub- Central galaxies have young stellar populations and more lished in the paper of Wangetal. (2006), because there was active star formation because of ongoing cooling and gas a small change in the definition of “first progenitor” when accretion. Satellite galaxies are older because they run out building the dark matter subhalo tree in the simulation (see of gas after they are accreted or their gas is removed by DeLucia&Blaizot (2006) for more details about this modifica- tion). Nevertheless, the best fit relation is almost the same as processes such as ram pressure stripping. In early semi- previouslyobtained, andalltheconclusions ofthe paper remain analytic models (Kauffmannet al. 1993; Cole et al. 1994; unchanged. Somerville & Primack 1999), this was indeed the case; red (cid:13)c 2007RAS,MNRAS000,1–13 Modelling the SED dependence of galaxy clustering 5 Figure 2. Leftpanel:fractionsofcentral(blue)andsatellite(red) systemsinbinsofstellarmassinthemodelsofWangetal.(2006). BlacksymbolsshowthefractionofgalaxiesinthehighDn4000peakfromSDSS.Rightpanel:theclusteringofcentral(blue)andsatellite (red)galaxiesinthemodelcomparedwithlow/highDn4000subsamplesintheSDSS(upperandlowerblackcurves). stellar masses, nearly all galaxies in the real Universe are “old”. Fig. 2 shows that the majority of these massive old galaxies must be central galaxies. In the right panel of Fig. 2, blue and red lines show theprojectedtwo-pointcorrelationfunctionsforcentraland satellitegalaxiesinthesimulationsin4differentstellarmass ranges.ResultsfromtheSDSSforthelowandhighD 4000 n subsamplesareplottedinblack.Thisagainshowsthefailure of a simple central/satellite dichotomy as a way of explain- ing the difference between the “old” and “young” galaxy populations in the SDSS. As can be seen, the difference in clustering amplitude between the two galaxy populations decreases as a function of increasing stellar mass, while the differenceinclusteringstrengthbetweencentralgalaxiesand satellite galaxies remains approximately constant. Figure 3. Distribution of infall times for satellite galaxies in different stellar mass bins. The x-axis shows time from past to present,with13.7Gyrcorrespondingtothepresentday. galaxiesweremainlysatellitegalaxiesandbluegalaxieswere Itisimportanttorememberthatsatellitegalaxieswere thecentral galaxies of their own haloes. notall“created”atthesametime.Fig.3showsdistributions Fig. 2 shows that this picture does not fit the SDSS of the times at which satellite galaxies of different stellar observations. In the left panel of Fig. 2, blue and red lines masseswerefirstaccretedbyalargerstructure.Thesatellite show thefraction of central andsatellite galaxies as afunc- infall(i.e. accretion) times t are randomly distributed infall tion of stellar mass in the model of Wang et al. (2006). For betweenthetwosimulation snapshotswhenthegalaxyfirst comparison, diamonds show the fraction of galaxies in the transitions from a being central object in its own halo to a highDn4000peakasafunctionofM∗,asmeasuredfromthe satellite system. As can be seen, high mass satellites have SDSSdata.Ascanbeseen,thefraction ofsatellite galaxies onaveragebeenaccretedmorerecentlythanlowmasssatel- inthesimulationdoesnotmatchthefractionof“old”galax- lites. This effect goes in the wrong direction to resolve the iesintherealUniverse,exceptatstellar masses 1010M⊙. discrepancies shown in Fig. 2. As we havediscussed, a sub- ∼ At lower stellar masses, the fraction of satellite galaxies is stantialnumberoflowmasssatellitegalaxiesarerequiredto higherthanthefraction of old galaxies, implyingthat some have“young”stellarpopulations,butFig.3showsthatlow low mass satellite systems are still forming stars. At high mass satellites typically become satellites quite early on. (cid:13)c 2007RAS,MNRAS000,1–13 6 L.Wang, C.Li, G.Kauffmann, G. De Lucia Figure 4. The evolution of Dn4000/g−r as a function of time for different values of the star formation timescale parameter τc for typical central galaxies. Solid lines are for solar metallicity and dashed lines show results for 0.25 solar metallicity. The coloured lines showresultsforthreedifferentvaluesofτc. Table 1.Best–fitparametervaluesforstarformationhistoriesinfourstellarmassbinsasderivedfromSDSSDn4000distributionand galaxycorrelationfunctions splitbyDn4000. log(M∗/M⊙) γk k <γc>median τc(Gyr)(50%,16%,84%) <τs>(Gyr) στs Ξ χ2dis/Ndis 9.5-10 [-1.6,2.2] 77 0.025 (39.97,-4.589,5.767) 2.312 0.992 4.214 3.126 10-10.5 [-1,1.2] 45 0.175 (5.724,69.95,2.106) 2.364 1.041 4.014 2.639 10.5-11 [-1,1] 41 0.318 (3.147,6.900,2.081) 2.491 0.534 5.267 4.239 11-11.5 [-0.8,1.2] 41 0.411 (2.435,3.845,1.953) 2.103 0.082 7.061 5.837 4 PARAMETRIZATION OF THE STAR after declines with a different e-folding time τs(M∗). The FORMATION HISTORIES OF GALAXIES IN modelthereforehastwotimescalesτc(M∗)forcentralgalax- THE MODEL ies τs(M∗) for satellite galaxies. Thestarformationrateofagalaxycanthusbewritten: 4.1 Computation of the SEDs e−t/τc ,central galaxies SFR(t)= (cid:26) e−tcentral/τce−(t−tcentral)/τs ,satellite galaxies Recent semi-analytic models (Croton et al. 2006; Bower et al. 2006; Kang et al. 2006; Cattaneo et al. where the age of a galaxy t is calculated starting from its 2006) have attempted to resolve some of the difficulties formation time t , which is assigned to a random formation outlined in the previous section by including “AGN feed- time between the simulation snapshot when the halo of the back”. The main effect of this form of feedback is to move galaxywasfirstfoundandtheimmediatelyprecedingsnap- the central galaxies of massive dark matter haloes to the shot. t = t t , is the time that the central infall formation − redsequence.Ourapproachisdifferent;ratherthanassume galaxy spendsas thecentral object of its own halo. some process that suppresses star formation in massive The resulting spectral energy distributions (SEDs) of galaxies, we parametrize the star formation histories of the the galaxies in the simulation are computed using the stel- galaxies in our simulation using simple functions, and we larpopulationsynthesismodelofBC03(Bruzual & Charlot allow theparametersof ourmodel tobeconstrained bythe 2003),assumingaChabrier(2003)IMF.Spectralproperties SDSSdata. such as the 4000 ˚A break strength and colour depend on Theformationtimeofeachgalaxyisdefinedasthetime themetallicity ofthegalaxyaswellasonitsstarformation whenthehaloofthegalaxy is firstfoundin thesimulation. history. Gallazzi et al. (2005) show that there exists a rela- We also know the infall times of all satellite galaxies, i.e. tionbetweenstellarmetallicityandstellarmassforgalaxies the times when they first became satellites. To model the in the local Universe. We use the mean relation derived in star formation histories of our galaxies, we assume that the their paper to specify the metallicity of the galaxies in our star formation rate of a galaxy declines exponentially with simulationatagivenvalueofM∗.Fig.4showstheevolution timeafteritsformationanddependsonthestellarmass(at ofD 4000/g rwithtimeforthreedifferentvaluesoftheγ n c − redshift 0) of the galaxy. We also assume that if a galaxy parameter(γ =1/τ (Gyr))foratypicalcentralgalaxyinour c c is accreted and becomes a satellite, its star formation ther- model. Results are shown for solar metallicity BC03 mod- (cid:13)c 2007RAS,MNRAS000,1–13 Modelling the SED dependence of galaxy clustering 7 Figure 5. OurbestfitstotheDn4000distributionsandthecorrelationfunctionssplitbyDn4000indifferentstellarmassbins.Inthe leftpanels,greenlinesarethefullfits,whilered/bluedashedlinesshowthecontributions fromsatellite/central galaxies.Inthemiddle panels,red/bluelinesarebest-fitcorrelationfunctionsforsubsampleswithlarger/smallervalueofDn4000.TheSDSSresultsareshown in black. The right panels show the the distributions of γc recovered by our non-parametric technique, with dashed lines showing the contributionofeachindividualGaussian. els (solid curves), as well as for 0.25 solar models (dashed In the following sections, we will concentrate on the 4000 curves). As can be seen, there is a small but significant de- ˚A break strength, because of its very weak dependence on pendence of D 4000 and g r on metallicity, particularly dust. n − forgalaxieswithshort starformation timescales. Notealso thatthiscomputationofthespectralenergydistributionne- glectstheeffectsofdustonthelight emitted bythegalaxy. (cid:13)c 2007RAS,MNRAS000,1–13 8 L.Wang, C.Li, G.Kauffmann, G. De Lucia Figure6. Upperpanels:thebestfitwhenγc isconstrainedtotakeonpositivevalues.Lowerpanels:thebestfitwhenγc isconstrained toliebelowavalueof0.5.Resultsareshownonlyfortheloweststellarmassbin:109.5−1010M⊙. Figure7. Theresultsofafullnon–parametricfittothedistributionofstarformationtimescalesofsatelliteandcentralgalaxiesinthe stellarmassrange1010−10.10.5M⊙.Intherightpanel,theblackcurveshowstheresultofthenon-parametricmethod,whilethegreen lineisfromthebest-fitGaussianofτs. 4.2 Fitting the Data resulted in rather poor fitsto the data. Furtherexperimen- tation indicatedthatit wouldbeadvantageoustoswitch to For each galaxy in the Millennium simulation, we have de- a method that would allow the distributions of τ and τ c s termined t and t . We have also built a “li- formation infall totakeoncomplexshapes.Following themethodology out- brary” of predicted present-day D 4000 values by running n lined byBlanton et al. (2003),thedistribution of γ =1/τ c c theBC03modelsfordifferentcombinationsofgalaxymetal- is parameterized by a sum of many Gaussians with mean licity z, galaxy lifetime, and the two star formation time valuesγ equally distributed in a given range: k scales τ and τ . We interpolate over the grid of parame- c s tgearlavxaileuseisnsttohreedsiminutlahteiolnib.rIanryadtodiotbiotna,inwDenk4n0o0w0tfhoer a(xll,yt,hze) P(γc)=XNk√21πσ2e−12(γc−σγk)2/XNk positionsofthegalaxies withinthesimulationvolume,and WeassumethesamescatterσforeachGaussian,butal- their stellar masses have already been specified using the low the weighting factors N to vary. For each stellar mass k modeldevelopedaspartofourpreviousstudy(Wang et al. interval, the range and number of Gaussians used in this 2006). We therefore have all the necessary ingredients to non-parametric fitting technique are listed in Table 1. The calculatetheDn4000distributionsaswellasthecorrelation central values γk of each Gaussian are equally distributed functionssplitbyDn4000fordifferentrangesinstellarmass over the range with a step of 0.05. σ for each Gaussian is andtocomparethemodelpredictionswiththeobservations. fixed to be the same as this step width. This method al- Initially weattempted toparametrize thedistributions lows thedistributionof γ valuestotakeonanyshape.This of τ and τ usingsimple Gaussian functions. However,this approach turned out to be critical for central galaxies, but c s (cid:13)c 2007RAS,MNRAS000,1–13 Modelling the SED dependence of galaxy clustering 9 resulted in little improvement when applied to the satel- 5 THE RESULTS litepopulation(seeSec.5.1). Wethereforeemploythenon- 5.1 Discussion of the Results parametricmethodonlyforthecentralgalaxies.Forsatellite galaxies, asimple Gaussian centred at <τs > and with the FromFig.5andTable.1,itisclearthatstarformationhisto- scatter of στs is used to parametrize thedistribution of τs. riesoflowmassandhighcentralgalaxiesareverydifferent. During thefittingprocedure, we also noticed that con- At large stellar masses (M∗ >1010.5M⊙), nearly all central strainingγc tobepositivedidnotallowustoreproducethe galaxies havepositive γc, and a large fraction of them have blue end of Dn4000 distribution. We therefore allowed the ceased forming stars. There is a peak in the distribution parameter γc to take on both positive and negative values. of γc at a value of around 0.45, which corresponds to an e– In practice, a negative value of γc corresponds to a galaxy foldingtimescaleof 2Gyr.Togetherwithaminorityofold ∼ that is experiencing an elevated level of star formation at andmassivesatellitegalaxies,thesecentralgalaxiesinwhich the present day (i.e. a starburst). For satellite galaxies, the star formation has shut down are necessary to explain the time scale τc is constrained to be positive, which is in any strong peak in the Dn4000 distribution at values of 1.8, ∼ case preferred by our fits. characteristic of metal-rich, evolved stellar populations. To find a best-fit to both the D 4000 distribution and Atlowstellarmasses,galaxiesdisplayamuchwiderva- n thegalaxycorrelationfunctionssplitbyD 4000,weemploy rietyinstarformation history.Thisisespecially trueinour n theLevenberg-Marquardtalgorithm,whichinterpolatesbe- lowest stellar mass bin (9.5 < log(M∗/M⊙) < 10). Nearly tween theGauss-Newton algorithm and themethod of gra- halfthegalaxies inthismassrangehavenegativeγ .Inad- c dient descent. The final results of the fitting procedure are dition,therearealso someobjects thathave“switched off’, shown in Fig. 5. For each stellar mass bin, D 4000 distri- i.e.thedistributionofγ exhibitsatailtowardlargepositive n c butionsand correlation functionsfor high/low D 4000 sub- values. n samples areshowntogether withthedistributionof γ that In the upper panel of Fig. 6 we show what happens c producesthebest fit.Intheleft panel,blueandred dashed to our fit in this stellar mass range if γ is forced to take c curves show results for central and satellite galaxies, while ononly positivevalues.Thefittingprocedureisexactly the greenlinesareforbothtypesofgalaxy.Inthemiddlepanel, same as described in Sec.4, but γ is constrained to lie in k red and blue curves refer to high and low D 4000 subsam- the range [0.05,2]. The result shows that the blue end of n ples defined using the same technique that was applied to theD 4000 distributionisnotwellfitwithoutapopulation n the SDSS data. The distribution of γ that results in these of “star-bursting” galaxies. In the BC03 model, a constant c fits is shown in the right panels. Table 1 lists the param- star formation rate will result in a D 4000 value of no less n eters of the best fit models for four stellar mass bins. For than 1.2foragalaxywithanageofafewGyrs(seegreen ∼ central galaxies, the median value of γ is listed, as well as linesin Fig. 4).It isthusnot possible toobtain low enough c the timescale τ = 1/γ , its median value and its 16 and valuesofD 4000tomatchtheobservations,unlessweallow c c n 84 percentile values. The table also lists < τ > and σ , for negative values of γ . s τs c theparametersthat describe thestar formation histories of InthelowerpanelofFig.6,wetestwhathappensifwe satellite galaxies. We estimate our parameters by minimiz- do not allow the distribution of γ to extend to very large k ing thequantity: positive values. If we truncate the distribution at a value of 0.5, we still find fairly good fit with Ξ = 4.836, which is comparable to our best-fit model. This implies that the χ2 χ2 Ξ= dis + corr existenceofalongtailofgalaxies withlargepositivevalues N N dis corr ofγ isnotstronglypreferred,i.e.ourmodelisnotsensitive k to the exact timescale over which the star formation was with truncated in these “old” central galaxies. One possibility is that these galaxies correspond to a post-starburst phase in 2 whichstarformationhasbeentemporarilyreducedfollowing P P χ2 = − SDSS exhaustionofthegasorblow-outofasignificant fractionof dis NXdis(cid:20) σ(PSDSS) (cid:21) theinterstellar medium. From the values of τ listed in Table 1, it can be seen s that unlike central galaxies, satellite galaxies of all masses and have similar e–folding timescales, with an average value of around 2 2.5 Gyr. This indicates that all satellite galax- 2 − χ2 = w(rp)−w(rp)SDSS ies experience a similar decline in star formation rate after corr NXcorr(cid:20) σ(w(rp)SDSS) (cid:21) fcaolnlicnlgusiinotno, aanldartgoesseteruicftaurGe.auTsositaenstisthsueffirocbieunsttntoesdseosfcrtihbies the dispersion in τ values, we have carried out a full non- s where χ2 is evaluated for the D 4000 distribution and parametricfittoconstraintheshapeofboththeγ andthe dis n c χ2 is evaluated for the projected correlation functions of γ (γ = 1/τ ) distributions for galaxies in the stellar mass corr s s s the high and the low Dn4000 subsamples. For each stellar range1010−10.5M⊙.TheresultsareshowninFig.7,withthe mass bin, N is the number of points along the D 4000 panelonthefarrightshowingthederiveddistributionofγ . dis n s distribution that we fit. In practice, we adopt N = 50 Thedistributionofγ obtainedusingthenon-parametricfit- dis s with D 4000 in the range [0.5,3]. N is the number of ting method is somewhat different to the simple Gaussian n corr pointson thecorrelation function used in thefit. Weadopt fit(indicated on theplot by thegreen curve).Nevertheless, N =20 2withr rangingfrom0.113to8.972h−1 Mpc. theresultingmedianvalueofτ is2.257Gyr,whichisabout corr p s × (cid:13)c 2007RAS,MNRAS000,1–13 10 L.Wang, C.Li, G.Kauffmann, G. De Lucia Figure 8. Thedistributionofspecificstarformationratespredicted byourmodel(reddashedlines)iscomparedwithresultsderived fromtheSDSS(blacksolidlines),indifferentstellarmassbins. Figure 9. g−r distributionspredictedbyourmodelcomparedtoresultsfromtheSDSS.Thegreencurvesshowthefulldistribution, whiletheredandbluedashedcurvesshowthecontributionsfromsatelliteandcentralgalaxies.BlackcurvesshowresultsfromtheSDSS. the same as that of the simple Gaussian fit. The small dif- In Fig. 8, we show the results of this test. The red ference in the distribution of τ between the two methods dashedcurvesinthefigureshowthedistributionsofthespe- s makes very little difference to the fit or to the resulting Ξ cific star formation rates predicted by our best fit model in values. fourdifferentbinsofstellarmass.Tocalculatethesevalues, wesimplyintegratethestarformationrateoverthelifetime of a galaxy to get its total stellar mass and we then divide 5.2 Consistency Checks the star formation rate at the present day by this value. Thefractionofstellarmassthatisreturnedtotheinterstel- Sofar,wehavetunedtheparametersτ andτ toreproduce c s larmediumoverthelifetimeofthegalaxyistakentobe0.5, the D 4000 distributions of SDSS galaxies as a function of n whichisthemedianvaluepredictedbytheBC03model.The stellar mass and the projected correlation functions of high blacklinesshowtheresultsobtaineddirectlyfromourSDSS D 4000andlowD 4000galaxysubsamples.Wehavechosen n n sample, which have beed corrected for volume incomplete- tofocusonD 4000becauseitisrelativelyinsensitivetothe n ness with the 1/V weighting scheme (the same method effectsofdust.However,D 4000isonlyoneofmanypossible max n as for computing D 4000 and g r distributions described age-indicators,soitisinterestingtocheckwhetherweobtain n − inSec.2).Fromtheplotweseethattheagreementbetween consistent results for other measures of the star formation our model “predictions” and the observations is reasonably history of a galaxy. good. The qualitative trends as a function of stellar mass Brinchmann et al.(2004)computedspecificstarforma- are reproduced quite well, but the absolute values of the tionrates(i.e.thestarformation rateperunitstellarmass) specific star formation rate predicted by the model tend to forSDSSgalaxies.Forstar-forminggalaxies,thevalueswere besomewhat lower than in thedata. derived using emission line fluxessuitably corrected for the effectsofdust,sothisindicatorofrecentstarformationhis- Wenowtestwhetherthesamemodelsallowustorepro- tory is independent of D 4000. For galaxies with absent or ducethecolour distributionsof galaxies. Aswehavenoted, n very weak emission lines, the specific star formation rates galaxycoloursarequitesensitivetotheeffectsofdust,soit wereinfact estimatedusingtheD 4000index,sothisisno would besurprising tous if this were the case! n longer an independent measure. Nevertheless it is interest- Fig. 9 shows thepredicted g r colour distributionsin − ingtotestwhetherthedistributionofSFR/M∗ predictedby the absence of dust and indeed, it appears that the model our best-fit model is in agreement with the results derived doesnotfitverywell. Atlowstellar masses, themodel pro- directly from theSDSS data. duces too many extremely blue galaxies and not enough (cid:13)c 2007RAS,MNRAS000,1–13