Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2) The Non-Parametric Model for Linking Galaxy Luminosity with Halo/Subhalo Mass: Are First Brightest Galaxies Special? A. Vale1,2⋆ and J. P. Ostriker1,3 7 1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom 0 2CENTRA, Departamento de F´ısica, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal 0 3Princeton UniversityObservatory, Princeton University,Princeton NJ 08544, USA 2 n a 5February2008 J 4 ABSTRACT 1 v 6 Werevisitthelongstandingquestionofwhetherfirstbrightestclustergalaxiesare 9 statisticallydrawnfromthesamedistributionasotherclustergalaxiesorare“special”, 0 using the new non-parametric,empirically based,model presentedin Vale & Ostriker 1 (2006) for associating galaxy luminosity with halo/subhalo masses. 0 We introduce scatter in galaxy luminosity at fixed halo mass into this model, 7 buildingaconditionalluminosityfunction(CLF)byconsideringtwopossiblemodels:a 0 simplelognormalandamodelbasedonthedistributionofconcentrationinhaloesofa / h givenmass.Weshowthatthismodelnaturallyallowsanidentificationofhalo/subhalo p systems with groups and clusters of galaxies, giving rise to a clear central/satellite - galaxy distinction, obtaining a special distribution for the brightest cluster galaxies o (BCGs). r t Finally, we use these results to build up the dependence of BCG magnitudes s on cluster luminosity, focusing on two statistical indicators, the dispersion in BCG a : magnitude and the magnitude difference between first and second brightest galaxies. v We compare our results with two simple models for BCGs: a statistical hypothesis i X that the BCGs are drawn from a universal distribution, and a cannibalism scenario merging two galaxies from this distribution. The statistical model is known to fail r a from work as far back as Tremaine & Richstone (1977). We show that neither the statistical model nor the simplest possibility of cannibalism provide a good match for observations, while a more realistic cannibalism scenario works better. Our CLF models both give similar results, in good agreement with observations. Specifically, we find < m1 > between -25 and -25.5 in the K-band, σ(m1) ∼ 0.25 and < ∆12 > between 0.6 and 0.8, for cluster luminosities in the range of 1012 to 1013h−2L⊙. Key words: galaxies:haloes–galaxies:fundamentalparameters–galaxies:clusters: general – dark matter – methods: statistical 1 INTRODUCTION On the theoretical side, there has been renewed interest in this subject with recent studies of the relation The nature of brightest cluster galaxies (BCGs) has long between galaxies and their dark matter haloes from a been a subject of interest and much debate (Peebles 1968; theoretical, statistical point of view, involving the study Sandage 1972; Dressler 1978). In particular, investigators of the distribution of the galaxy population through dif- have asked whether their origin is statistical or special in ferent haloes while bypassing the complications of the nature, that is, whether they follow a special distribution physicsofgalaxyformation(e.g.,Berlind & Weinberg2002; independent of the fainter galaxies in the cluster, or on the Vale& Ostriker 2004; Tasitsiomi et al. 2004; Yang et al. contrary, they are merely the extreme values of the same 2005; Zehavi et al. 2005; Zhenget al. 2005; Cooray 2006; global distribution derived for all cluster galaxies. Conroy, Wechsler& Kravtsov 2006; van den Bosch et al. 2006). Since these involve populating dark matter haloes with galaxies, they usually lead to a distinction between ⋆ E-mail:avale@fisica.ist.utl.pt 2 A. Vale and J. P. Ostriker central and satellite galaxies. This in turn has lead to, in a combination of the two, probably depending on the type many of these works, central galaxies being treated sepa- of cluster (Bhavsar 1989; Bernstein & Bhavsar 2000). rately from the rest, and therefore having a distinct dis- The other property studied is the ratio r = ∆ /σ , 12 1 tribution, with consequences visible, for example, in the where ∆ is the average magnitude difference between the 12 luminosity function. Some of these studies have in fact firstand second brightest galaxies, and σ thedispersion in 1 looked at some specific BCG-related properties of clus- themagnitude of the first brightest galaxy. It is possible to ters, like the magnitude gap (e.g., Milosavljević et al. 2006; provethepowerfulconclusion(Tremaine & Richstone1977) van den Bosch et al. 2006). that, if all galaxies are drawn from the same statistical dis- In the past, observational studies which have tribution,regardless ofits exact form, thenr 1. Observa- ≤ focused on this issue (Tremaine & Richstone 1977; tionalresultsgiveavalueforraround1.5(e.g.,Lin & Mohr Hoessel, Gunn & Thuan 1980; Schneider, Gunn,& Hoessel 2004;Loh & Strauss2006),which would excludethispossi- 1983; Bhavsar & Barrow 1985; Hoessel & Schneider 1985; bility. Bhavsar1989;Postman & Lauer1995;Bernstein & Bhavsar This has led to the study of possible alternative 2000) have been hindered by the limited numbers of high scenarios for the formation of BCGs, in order to ac- luminosity galaxy observations available, since the strongly count for their special nature. One such is galactic can- decliningnatureofthebrightendoftheluminosityfunction nibalism, initially proposed by Ostriker and collabora- requires having very large samples to obtain significant tors(Ostriker& Tremaine1975;Ostriker& Hausman1977; numbers of high luminosity galaxies. Due to this, these Hausman & Ostriker 1978). Such a scenario is akin to tak- studies were mostly inconclusive when it came to answer ing the above case of having all galaxies drawn from the the question of whether BCGs were statistical or special same distribution, but then merging the brightest of them in nature, although many works hinted at the latter. with one or more of the others. From this simplistic model More recently, the advent of large scale surveys such as of the process, it is easy to see that this mechanism would the 2dF Galaxy Redshift Survey or the Sloan Digital help to solve the above problem, mostly by increasing the Sky Survey (SDSS), has motivated plentiful, ongoing value of ∆ as the luminosity of the first brightest galaxy 12 work on this subject (e.g., Lin, Mohr & Stanford 2004; is driven up by the mergers and the brightness of the sur- Lin & Mohr 2004; Loh & Strauss 2006; Bernardi et al. vivingsecondbrightestgalaxydeclinesasluminousgalaxies 2006;von derLinden et al. 2006). are merged out of existence. This issue is in large part motivated by the fact that, In the present paper, we explore this issue in light observationally,BCGsdolookdifferentfromothergalaxies. of the non-parametric model for the mass luminosity rela- They usually sit at the centre of the cluster, and tend to tion presented in Vale & Ostriker (2006) (hereafter paper be considerably brighter than the remaining cluster mem- I; see also Vale & Ostriker 2004). The basic idea behind bers. The most striking case is cD galaxies, found in the the non-parametric model is to adopt the simple proposi- centre of rich clusters and which dominate their satellites tion that more luminous galaxies are hosted in more mas- in both size and brightness, while having a characteristi- sive haloes/subhaloes. No attempt at physical modelling is cally distinct morphology and surface brightness distribu- made and the association is made simply by matching one- tion (e.g., Binggeli, Sandage & Tammann 1988). Likewise, to-one the rank ordered observational list of galaxies with cD galaxies tend to be brighter than what would be ex- therankorderedcomputedlistofhaloes/subhaloes.Wehere pected from the bright end of the cluster galaxy luminos- extendthismodelbyintroducingscatterintoit,andalsoby ity function. In fact, it has been observed that, when an- considering possible effects on the total disruption of some alyzing composite luminosity functions of cluster galaxies, subhaloes into thetotal luminosity related to thehalo. the most luminous of them form a hump at the bright AsisthecaseinHODmodels(e.g.,Berlind & Weinberg end (e.g., Colless 1989; Yagi et al. 2002; Eke et al. 2004). 2002;Zehaviet al.2005;Zheng et al.2005),thismodelnat- Yet,at thesame time,thereislittle variation in magnitude urallygivesrisetoaseparationbetweencentralandsatellite among them (Hoessel & Schneider 1985; Postman & Lauer galaxies, by associating the former with the parent halo it- 1995;Bernardi et al.2006;von derLinden et al.2006).This selfandthelatterwiththesubhaloesassociatedwithit.We ties in with the fact that the luminosity of BCGs is ex- analyzethisissueinmoredetail,studyinghowitaffectsthe pectedtovaryonlyslowlywithincreasingclusterluminosity clustergalaxy luminosityfunctionandgivesrisetoabright (Lin & Mohr2004;seealsotheresultsforthemassluminos- endbumpcausedbythecentralgalaxies.Wethendevelopa ity relation of central galaxies in Vale & Ostriker 2006). modelfortheBCGluminositydistribution.Sincethehaloin In order to try to answer this problem from available fact arises from the union of subhaloes this non-parametric observational data, two different indicators have been con- model is a statistically well defined variant of the cannibal- sidered. One is the shape of the overall distribution of the ism scenario. magnitude of BCGs. If BCGs are merely the extreme cases Thispaperisorganisedasfollows:insection2,wegivea of a general distribution applicable to all cluster galaxies, briefsummary of thenon-parametricmodel relating galaxy then it is expected that results from extreme value the- luminositywithhalo/subhalomasspresentedinpaperI,and ory in statistics apply, predicting a resulting distribution introduceasimplerecipeforcheckingthecontributionofde- shaped like the Gumbel distribution (Bhavsar & Barrow stroyed subhaloes to the halo mass, and how this changes 1985; Bernstein & Bhavsar 2000). On the other hand, if our estimate of the total luminosity. In section 3, we intro- BCGsareconsideredaspecial,distincttypeofgalaxy,then duce scatter into the non-parametric model by building a some particular distribution is to be expected, such as a conditionalluminosityfunction,whereweconsidertwopos- Gaussian(Postman & Lauer1995)orlognormal.Somestud- sibilities for it, either a simple lognormal shape or a better ies have also raised the possibility that it could be actually motivatedapproachinvolvingthedistributionofconcentra- Are First Brightest Galaxies Special? 3 tion for haloes of a given mass. In section 4, we explore mass of the parent (otherwise it would, by definition, moreindepthhowthemodelgivesrisetoacentral/satellite be the parent). The slope α = 1.9 is set to the same galaxy separation, and show how this impacts the cluster value as is generally found for the present day sub- galaxyluminosityfunction.Insection5,webuildupamodel halo mass function in simulations (e.g., Gao et al. 2004; for the distribution of cluster galaxies, based on the mass- Weller et al. 2005; van den Bosch, Tormen & Giocoli 2005; luminosityrelationandthehalo/subhaloesseparationwhich Zentneret al. 2005; Shaw et al. 2006). The normalization underpins it. In section 6, we present simple models to ac- A(M) = 1/β[Γ(2 α) Γ(2 α,1)] is set so that the to- − − − count for another two possible origins for the BCG distri- tal mass originally in subhaloes corresponds to the present bution:first,weconsider thatall clustergalaxies aredrawn day mass (where the integration is done to an upper limit fromthesamedistribution;thenwetakeasimplemodelfor of 0.5M). This approximation potentially ignores the prob- cannibalism, by merging two of the galaxies (the brightest lem of total disruption of some of themerged subhaloes, as plusoneother)inthefirstexample.Finally,insection 7we can occur for example in the case of major mergers, by as- present the results of all models for the average magnitude sumingthatall ofthesesubhaloesarestill presentandthat of first and second brightest galaxies as well the dispersion therefore the total fraction of mass originally in subhaloes of the former as a function of cluster luminosity. We then isone.InVale & Ostriker(2006),weshowedthataslongas compare these results with observations. this fraction is close to one, then the resulting mass lumi- Throughout we have used a concordance cosmological nosity relation is similar, with both number of satellites in model,withΩ =0.24,Ω =0.76,h=0.735andσ =0.74 a halo and their total luminosity decreasing slightly. From m Λ 8 (Spergel et al. 2006). thestudyofsimulationresultsitisstillnotcompletelyclear how to treat this complex issue, and no simple analytical models are available, so we explore a simple recipe to bet- 2 THE MASS-LUMINOSITY RELATION ter account for this problem in the context of our model in section 2.1. The work presented below is based on the non-parametric Thegalaxydistributionisgivenbytheluminosityfunc- modelforrelatinggalaxyluminositywithhalo/subhalomass tion. This is given by theusual Schechterfunction fit: presented in paper I. The basic idea is that more massive hacacloreetse/smubohrealgoaessahnadvesudbeseepqeurenptoltyenwtiilallhwaveellsmaonrdelwumillinthouuss φobs(L)dL=φ∗(cid:16)LL∗(cid:17)αexp(cid:16)− LL∗(cid:17)dLL∗ . (3) galaxiesformingwithinthem.Ineffect,wetaketherelation The values of the parameters will depend on the wave- betweengalaxyluminosityandhalo/subhalomasstobeone band used. In this paper, we use mostly the K-band lu- tooneandmonotonic.Anadditionalextraingredientisnec- minosity function from the 2MASS survey, with parame- essary to maintain this approximation in the framework of ters given by α = 1.09, φ∗ = 1.16 10−2h3Mpc−3 and themodel,sincesubhaloeslosemasstotheparenthaloafter M∗ 5logh= 23.3−9 (Kochanek et al.×2001). For compari- accretionduetotidalinteractions.Alternativelyput,ahalo son,−wealsoob−taintheb -band2dFsurvey,withα= 1.21, J is not simply the sum of the identifiable subhaloes within φ∗ = 1.61 10−2h3Mpc−3 and M∗ 5logh = −19.66 it due to tidal stripping. Therefore, we need to account for (Norberg et×al. 2002).Also notethat we−are in fact e−xtend- the mass of the subhaloes not at present, but that which ing these fits as necessary, including beyond the magnitude they had at the time of their merger into the parent. The intervalin which theywere obtained. relation between mass and luminosity is then obtained sta- The basic mass-luminosity relation can then be ob- tisticallybymatchingthenumbersofgalaxieswiththetotal tainedfrom theseingredientsbyacountingprocess,match- numberofhosts,thatis,haloesplussubhaloesthroughtheir ingthenumbersofgalaxiesatagivenluminositytothetotal distributions. numberof hosts at a given mass: ThehaloabundanceisgivenbytheusualSheth-Tormen ∞ ∞ mass function (Sheth & Tormen 1999): φ(L)dL= (n (M)+n (M))dm, (4) h sh Z Z 1 2ρ dν ν2 L M nh(M)dM =A 1+ ν2q rπ MmdMexp − 2 dM, (1) where the host contribution is separated into a halo term, (cid:16) (cid:17) (cid:16) (cid:17) n (M),andasubhalotermobtainedbysummingupallthe h with ν = √a δc , a = 0.707, A 0.322 and q = 0.3; subhaloes at that mass, n (m) = ∞N(mM)n (M)dM. D(z)σ(M) ≈ sh 0 | h asusual,σ(M)isthevarianceonthemassscaleM,D(z)is An average relation between hostRmass and galaxy lumi- thegrowthfactor,andδ isthelinearthresholdforspherical nosity can then be built through this process, with results c collapse, which in the case of a flat universe is δ = 1.686, thatmatchwellwithobservations(seepaperIforadetailed c with a small correction dependent on Ω (δ = 1.673 for analysis). The resulting relation can be well fit by a double m c Ω =0.24). power law of thetype: m Following the discussion in paper I, we will assume a (M/M )a very simple model for thesubhalo mass distribution within Lref(M)=L0[1+(M/M0)bk]1/k , (5) theparent.Intermsoftheiroriginal,pre-accretionmass,we 0 assume that the subhalo distribution is given by a simple wherethedifferentsparametersareshownintable2;massis Schechterfunction: in units of h−1M⊙, luminosity in h−2L⊙. The fit was done N(m|M)dm=A(M)(m/βM)−αexp(−m/βM)dm/βM,(2) 3in th10e15mhas−s1Mra⊙ng.e 1011 (3×1010 in the bj band case) to × where the cutoff parameter β = 0.5 serves to in- Figure1showstheresultsfortheluminosityofasingle sure that no subhalo was larger than half the present galaxyasafunctionofthemassofthehostinghalo/subhalo, 4 A. Vale and J. P. Ostriker K-band bJ-band together with the corresponding mass-to-light ratio. Shown L0 1.37×1010 4.12×109 are curves for both K- and bJ bands. We caution that the M0 6.14×109 1.66×1010 results for the latter band should be treated with some re- a 21.03 6.653 serve.This countingmethod is not entirely adequateto get b 20.74 6.373 the mass-luminosity relation in the blue, due to complica- k 0.0363 0.111 tions arising from recent star formation, although it is still interesting to compare the differences obtained from using Table 1.Fitparametersforthemass-luminosityrelation. two different luminosity functions. 2.1 Destroyed subhaloes and the subhalo mass fraction Asmentioned,apotentiallyimportantcorrectiontothenon- parametric model in paper I is to account for subhaloes which have been completely destroyed. The study of this evolution of the subhalos and their eventual destruction, with the subsequent merger (or not) of their galaxy with thecentralone,isaveryinterestingtopicbyitself, whichis still not completely understood but which is essential to a completeunderstandingoftheformationofBCGs.However, such a detailed look at this question is beyond the scope of thepresentpaper;here,wearemerelyinterestedinasimple modeltoaccountforhowmuchmasswasinthesedestroyed subhalos, to correct the normalization of our original subhalo mass function. Our scheme is based on the fact that most of the luminosity of the central galaxy is built up by merging with thesatellitegalaxiesbroughtinbythesesubhaloes.Inother words, the central (BCG) optical galaxy is made up of the galaxies that havebeen ”merged away” – disappeared from the original distributions. This is consistent with what is known of the size, shape and colour properties of central galaxies. We therefore assume that the fraction of mass in these destroyed subhalos (with respect to the total halo mass),isgivenbytheratioofthecentralgalaxy luminosity to thetotal luminosity of thehalo: m L f = dest = cent . (6) dest M L total For a given L(M) relation, which sets the luminosity of both the central and satellite galaxies as a function of the halo/subhalomass,thepreviousequationcanthenbesolved for f as a function of halo mass, since the total lumi- dest nosity is going to be a function of only it and the total mass: L = L (M)+(1 f )L (M), where total cent dest sat,max − L is the maximum contribution of the satellites for sat,max when f =0. This is given by: dest 0.5M L (M)= L (m)N(mM)dm. (7) sat,max ref Z | 0 Theupperpannelof figure2 showsthedestroyedmass fraction as a function of halo mass for our base mass- luminosityrelation,givenbyequation(5),whilethebottom Figure 1. Mass-luminosity relation as obtained using the non- pannel shows the effect on the total luminosity. As can be parametricmodel,intheK-andbJ−bands.Upperpannelshows seen, this is most pronounced at the lower end of the mass galaxyluminositynormalizedtothecharacteristicluminosity,L∗, scale shown, and becomes small enough to havelittle effect ofeachband;lowerpannelshowsthecorrespondingmass-to-light at high mass. ratio. Therearetwoadditionalfactorsthatneedtobenoted. First, the introduction of this term can also have an effect ontheactualmass-luminosity relation,sinceweareusinga counting method to obtain it. However, for the mass range we are interested in, the number counts are dominated by Are First Brightest Galaxies Special? 5 Finally, it needs to be stressed that this is just a very simpleapproximation.Thecalculatedfactorisappliedtothe whole subhalo mass fraction as a correction to the normalization, without taking into account a possible dependence on subhalo mass. In particular, the situation with very low mass subhaloes is very uncertain in this scheme, since they are expected to be very faint, and havetherefore verylittle weight in the sum of the total luminosity, while they can contribute an important fraction of the mass. Another im- portantpointisthatinprincipletheluminosityofthegalax- ies that were contained in the destroyed subhaloes should be added to the central galaxy luminosity, since under this schemeweareassumingthatthesearemerging.Inpractice, though,thelightinthesedestroyedsubhaloesisgoingtobe small in comparison with the BCG in this model, since the largest fraction of destroyed subhaloes occurs for less massive haloes where the BCG is dominant. For simplicity, we will hereignore this contribution. 3 INTRODUCING SCATTER 3.1 The conditional luminosity function In the context of the present paper, we need a more detailed model than the one described previously. Most im- portantly, it needs to include some kind of scatter in the mass-luminosity relation. Naturally, we expect that not all galaxies in hostsof thesame mass will havethesame luminosity. To capture this, we introduce a dispersion around the average relation describe above. We use the conditional luminosity function (CLF) formalism introduced by Yanget al. (2003) (see also vanden Bosch et al. 2006 and references therein) and by Cooray and collaborators (e.g., Cooray & Milosavljević 2005a; Cooray 2006 and references therein). This consists of replacing a deterministic mass- luminosity relation, liketheonein equation (5),with adis- tributionofluminosityaroundanaveragevalueforanygiven halomass, φ (LM)dL, which representstheprobability CLF | ofhavingagalaxyofluminosityLinahaloofmassM.Note that here we are only applying thisto thecentral galaxy in anygivenhalo,sincethatistheimportantoneforthestudy of BCGs; the distribution of satellite galaxies we draw di- rectly from thedistribution of subhalo masses. Figure 2. Upper pannel: total mass in subhaloes which have AnimportantpointisthatthisCLFmust,bydefinition, been completely disrupted, as a fraction of the total halo mass. matchtheobservedluminosityfunctionwhenitisintegrated Bottompannel:totalluminosityasafunctionofhalomasswithor ∞ overall haloes, i.e. φ (LM)n(M)dM =φ(L),where withoutusingthefractionofmassindestroyedsubhaloesshown 0 CLF | in the upper pannel. Results are for the K-band, using the base n(M)isthehalomaRssfunction,φ(L)theobservedluminos- ity function and L should only be considered in the range mass-luminosityrelationofequation(5). wherethehaloesdominatethenumberofhosts(i.e.,athigh luminosity,whichisprecisely therangeweare interestedin the central galaxies and will therefore not be affected (see whenlookingatBCGs;otherwise, wewouldalsoneedtoac- section4).Likewise,theonlyhaloescapableofhostingsub- countforsubhalocontribution).Theintroductionofscatter haloeslargeenoughtobecountedinthisrangearethemost then leads to a problem with the mass-luminosity relation massiveones,forwhichtheeffectissmallest (seesection3). derived from the counting method, however. As noted by Secondly,inprinciple,thisapproachwillalsodependonthe Tasitsiomi et al. (2004), the fact that the mass function is exactformofthemass-luminosityrelation.However,forthe decreasing with increasing mass causes an effect similar to small deviations from thebase relation we will beconsider- the Malmquist bias: for any given mass bin, more objects inginthispaper,theeffectonthetotalluminosityissmall, arescatteredintoitfromlowermassbinsthanarescattered since the variation to the base relation will be greatest at out of it. If we then take our base mass-luminosity relation highermass,wherethiseffectissmallest.Wewilltherefore, to be the average one in the CLF distribution, because of for simplicity, usethis one result throughout thepaper. thiseffectwewillendupwithacalculated luminosityfunc- 6 A. Vale and J. P. Ostriker tionthatgreatlyoverestimatestheabundanceofverybright be defined in some way in order to determine the latter. galaxies when compared to theobserved one. We do this by determining which value of the dispersion Togetthecorrect matchingtotheobservedluminosity leads to an average luminosity as a function of mass which function, it is then necessary to modify the average mass- bestfitsobservationalvalues.Forsimplicity,wewillconsider luminosity function we take for the basis of the CLF. This thatthevalueofthedispersion,σ,isconstantandindepen- is achieved byintroducing an additional term, of theform: dentofmass.Sinceweareonlyinterestedinthebrightend, where we expect central galaxies to dominate, and these L(M)=L (M)(1+M/M )a, (8) ref s are known to have only a small scatter in luminosity (e.g., where L (M) refers to the base mass-luminosity relation Postman & Lauer 1995; Bernardi et al. 2006), this is quite ref of equation (5). In practice, a is going to be negative since likelyagoodapproximation.Semi-analyticalmodellingalso weneedtolowertheluminositycorresponding toanygiven showsthatscatterinstellarmassisonlyaweakfunctionof high-mass halo in orderto drivethevalue of ourcalculated halo mass (Wanget al. 2006). luminosity function down. The luminosity L of a galaxy in a host of mass M, is There is one final, potentially important point about then given by a lognormal distribution of thetype: this issue: once scatter is introduced, care must be taken 1 (ln[L/L (M)]+σ2 /2)2 wlahtieonn.loInokoiunrgaaptptrhoeaccha,lctuhleaatevderaavgeerlaugmeimnoassist-yluamtfiinxoesditmyarses- φCLF(L|M)dL= √2πσLNLexp(cid:16)− 0 2σL2N LN (cid:17)dL,(9) needstogodown,relativetothescatter-lesscaseor,looking whereL (M) is some average luminosity for a host of mass 0 at it the opposite way, the same average luminosity is ob- M (discussed further below), and σ is thedispersion in the tained for higher mass haloes. This is due to the fact that, normallogarithmoftheluminosity.Notethatthereissome whenconsideringtheCLF,wearedoingthebinningbymass confusionintheliteratureovertheexactformdefinedforthe (ormoreprecisely,takingtheconditionalvariableinthedis- lognormaldistribution.First,itisnecessarytopayattention tributiontobethemass).Ifwehadinsteadbinnedbylumi- to whether the distribution is in the natural logarithm or nosity,theeffectofintroducingscatterwouldhavebeenthe base 10 logarithm of the variable; the quoted value of σ opposite:theaveragemasscorrespodingtoagivenluminos- will be different in the two cases for the same distribution. ity would instead have gone down. This is to be expected Secondly, the way we have defined it in equation (9), the and is just a statistical effect of the two different ways in average luminosity is given by L (M). This is due to the 0 which theconditional function can bedefined.It doeshow- second term in the denominator of the exponential term, evermean that care must betaken when comparing results whichnot allauthorsinclude;if itisomitted,L wouldnot 0 of different authors to look at how the binningwas donein bethe average luminosity. each case. Thedifficultywithusinganapproachsuchasthisisthat In this paper we will consider two different models for weareleftwithtwounknownsweneedtodetermine,theref- the CLF of the central galaxy: a simpler model where we erenceluminosityfunctionL andthedispersionσ.Theonly 0 assume the distribution is lognormal, but where we are left condition we can impose on this distribution is that, when withafreeparameterinthescatterintroduced;andamore integrated over all hosts, the resulting luminosity function complicatedonebasedonthedistributionofconcentrations must match the observed one. Fitting this calculated lumi- at a given halo mass, which fully motivates the introduc- nosity function totheobserved one then allows us torelate tionofscatterintheCLFwithout anyfreeparameters.For thetwoparameterswehavewhenwetakeL =L(M)from 0 the satellite galaxies we will use the same modified mass- equation(8),aandM ,tothescatterσ.However,thisstill s luminosityrelation aswell,sincethesewerecentralgalaxies leaves us with one free parameter which we cannot other- within theirown independenthaloes prior tomerging, so it wisespecify.Inordertoaddressthisproblem,wedetermine is reasonable to expect the same effects to apply to them. whichvalueofthescattergivesusanaverage luminosity as Fromsemi-analyticalmodelling,ithasbeenshownthatthis afunctionofmassthatbestfitstheobserveddata.Sinceour isagoodapproximation,althoughamorecarefultreatment original mass-luminosity relation was already quite a good showsaslightlydifferentrelationforsattelitesthanforcen- fittothedata(seepaperI),wemustnecessarilyhaveonlya tralgalaxies (Wanget al. 2006).Butnotethat,whendoing smallcorrectiontoit(i.e.,asmallvalueofa),whichinturn analytical calculations, using the subhalo mass function al- implies a small value for the scatter, which is qualitatively readyintroducesaform of distribution forthesubhaloes as ingood agreementwithobservations(seefurtherdiscussion well(inthatthemassofagivensubhalocanbedrawnfrom in section 7). it, see section 5.1 for furtherdiscussion). 3.3 Concentration model 3.2 Lognormal model TheothermodelfortheCLFofBCGsweconsiderisbased The simplest CLF model we consider is to assume it has a onthevariationoftheconcentrationofhaloeswiththesame lognormalform.Thisissimilartowhatwasdonepreviously mass. The basic idea behind this is that the distribution by other authors (Cooray & Milosavljević 2005a; Cooray of concentration in same mass haloes will lead to different 2006), and such a form seems a good match to the distri- massintheinnerregionofthehalowherethegalaxywillbe bution of stellar mass obtained in semi-analytic modelling present; the luminosity of the hosted galaxy will then sim- (Wanget al.2006).Theproblem withthisapproachisthat ply be proportional to this mass. In practice, we calculate there is no a priori reason to assume any specific value for the mass-to-light ratio of this inner region, for the average thedispersion.Furthermore,thisvalueislinkedtothemod- concentrationandwithanaverageluminositygivenbyequa- ifiedmass-luminosityrelation ofequation(8),soitneedsto tion (8),and thencalculate thechangein luminosity as the Are First Brightest Galaxies Special? 7 concentration changes by assuming that the mass-to-light r = (3M/4π∆ ρ¯)1/3 and ∆ = 387; ρ is normalized vir vir vir 1 ratioisfixed(observationally,ithasbeennotedthatthedy- to givethe halo mass at thevirial radius. namicalmass-to-lightratioofBCGsisalmostconstant,e.g. For the concentration distribution, we take the model von derLinden et al.2006).ThisthengivesustheBCGlu- ofBullock et al.(2001)(seealsoMacciò et al.2006,whoget minosityasafunctionofbothconcentrationandhalomass. similar results). This relates the average concentration of a Although the model is conceptually simple, the details haloofmassM withthescalefactoratitscollapse,a ,given c are problematic. The main issue is to determine what ex- by: actlyisthisinnerregionandhowtocalculateitsmass.The δ most obvious solution, taking quoted values from the liter- σ(fM)= c , (15) D(a ) ature for BCG radius and its dependence on luminosity, is c notreallysatisfactorysincethesearemostoftendetermined whereσis,asusual,thevarianceofthelinearlyextrapolated fromisophotallimitsandinthecaseofcDgalaxiesitwould powerspectrumofperturbations,D(a)isthegrowth factor be necessary to further consider whether to include the en- and δ = 1.673 the linear threshold for collapse; f = 0.001 c velopes;also,itisnaturaltoassumethattheactualregionof is a parameter. The concentration is then given by c = vir influenceforthedarkmatterismoreextensivethanthevis- k/a , with theparameter k=3. Finally, thedistribution of c iblegalaxy.Otheroptionssuchassomeparameterfrom the the concentration is given by a lognormal distribution with dark matter structure, run into the problem of motivating average c and variance σ[log(c )] 0.18. vir vir ∼ what exactly it should be. The BCG luminosity distribution is then obtained Intheend,aftercheckingtheresultsofseveraldifferent from the concentration distribution by φ(LM) = | possiblemodels,weconcludedthattheonethathasthebest f(cM)/(dL/dc), where f(cM) is the concentration distri- | | motivationandalsogivesthebestresultsistocalculatethe bution as a function of halo mass. As mentioned, the lumi- inner region mass by using a weighting function based on nosity for any given concentration and halo mass is given theluminosity profile of the BCGs. by Sincewejustrequireittomakeourweightingfunction, M forsimplicityweassumethattheluminosityprofileofBCGs L(c,M)= inner , (16) (M/L) can beuniversally fit by a Sersic profile: 0 where(M/L) isthemass-to-lightratiowiththeinnermass I(r)=Aexp( b [(r/r )1/n 1]), (10) 0 − n e − calculated at the average concentration and the luminos- where A is a normalization factor, and Γ(2n)=2γ(2n,b ), itygivenbyourmass-luminosityrelation,equation(8),and n whereΓ(2n)isthegammafunctionandγ(2n,x)theincom- Minner is given by equation (13). Finally, we fit our calcu- plete gamma function; this can be well approximated by lated luminosity function to the observed one, in order to b 2n 0.327(Capaccioli 1989).Itisknowthatthereisa determine the parameters that go into the modified mass- n cor≈relati−onbetweentheprofileparametersnandr ,andalso luminosity relation (see table 2). e between r and the galaxy luminosity L, although the cor- Note that both Bullock et al. (2001) and Macciò et al. e relation between n and L is very weak (e.g., Graham et al. (2006) find that subhaloes tend to have higher concentra- 1996).Forsimplicity,wewillassumethatwecanrelateboth tionsthanparenthaloesofthesamemass.Althoughitgoes parameters to the galaxy luminosity; although in practice beyondthescopeofthepresentwork,itiswortwhiletomen- this is not really true, for our purposes here it is a suffi- tionthatintheframework ofthemodeljustpresented,this cient, if rough, approximation. Based on results from the can possibly lead to slightly different distributions of lumi- literature (Graham et al. 1996; Lin & Mohr 2004), we use nosity as function of mass for the subhaloes, although this thefollowing relations: is most likely complicated by the fact that we need to take thesubhalo properties at accretion rather than at present. n=2.8694log(re)+2.0661, (11) Table2showsthevaluesweobtainfortheparametersof the modified mass-luminosity relation of equation (8). Fig- log(r )=0.9523log(L) 9.1447, (12) e − ure 3 shows examples of the actual distribution we obtain whereListheK-bandluminosity;thesearesimilartowhat for the BCG luminosity, both for the lognormal model and is reported by other authors (e.g., Bernardi et al. 2006). for theconcentration model. Our weighting function is then given not by the actual profile, but rather the integration factor for the luminosity, w(r) = rI(r), and the normalization A chosen so that rvirw(r)dr = 1. Finally, the inner region mass is simply 4 CENTRAL VS SATELLITE GALAXIES 0 Robtained by integrating the mass density times the weight- As was briefly mentioned above, the way we build up the ing function: mass-luminosity relation naturally gives rise to a model for rvir clusters,featuringadistinctseparationbetweencentraland M = 4πr2w(r)ρ(r)dr. (13) satellitegalaxies. Thiscomesfromthefactthatweconsider inner Z 0 that galaxies are hosted by both the parent halo and the We use the usual NFW profile for the density subhaloes.Sinceweconsiderthatthesamemass-luminosity (Navarro, Frenk,& White 1997), relation applies for both, and the former will be, by definition,considerably moremassivethanthelatter,thisresults ρ ρ= 1 , (14) in there being a central, very luminous galaxy, hosted by x(x+1)2 the parent halo, while fainter satellite galaxies are spread where x=r/r , with r =r /c and the virial radius is throughout in thesubhaloes. s s vir vir 8 A. Vale and J. P. Ostriker model σLN a Ms concentration N/A -0.08 1013.5 lognormal 0.265 -0.07 1013 Table 2. Fit parameters for the modified mass-luminosity relation of equation (8), for the concentration and lognormal models. The dispersioninthelatterisfixedtothevalueshown,andvariableintheformer. mass and its distribution, without the need to account for survivingsubhaloes. Itisinfactpossibletoderivethedifferentcontributions tothegloballuminosityfunctionfromthecentralandsatel- litegalaxies, byassociatingthemwiththehaloandsubhalo distributions, respectively, and then using the CLF formalism presented in theprevious section: ∞ φ (L)dL= φ (LM)n (M)dM, (17) i CLF i Z | 0 wheretheiindexesrefertothehaloesandsubhaloes,respectively.Thederivedluminosityfunctionsforcentralandsatel- litegalaxiesareshowninfigure4.Unsurprisingly,thecentral galaxiescompletelydominatetheoverallluminosityfunction athighluminosity,withtheirnumbersbecomingcomparable to thesatellites only at low luminosity. This simply reflects the trends seen in the halo and subhalo numbers (see pa- perI). The expected relative contributions of both typesof galaxies are still uncertain: while Benson et al. (2003a) using their semi-analytical modelling find satellite galaxies to dominate at the faint end, Cooray & Milosavljević (2005b) using a conditional luminosity function formalism find that Figure 3. Examples of the calculated distribution for the BCG central galaxies dominate throughout the range (likewise, luminosity,forboththelognormalandconcentrationmodels,and Zhenget al. (2005) find that central galaxies dominate the fortwodifferentvaluesofthehalomass. stellar mass function on any mass scale). Of particular importance for the question of whether 4.1 Definition of cluster threshold mass the first brightest galaxies are special, this separation im- A necessary first step before continuing this analysis is to plies that these galaxies should indeed have a special lumi- define precisely what we mean by a ”cluster”. We will opt nositydistribution,independentfrom thatoftheremaining for a simple choice, following the standard Abell definition galaxiesinthecluster.Thiscomesfromthefactthatthedis- of rich cluster, namely that it must haveupwards of 30 ob- tributionfunctionsofthesetwotypesofgalaxieswillbedif- jects brighter than m +2m, where m is themagnitude of ferent in origin: the central galaxies one will be determined 3 3 thethirdbrightest galaxy in thecluster.Itis thenpossible, by the halo mass function, while the satellite galaxies one following ourmodel,totranslate thisintoaminimummass will depend on the subhalo mass function. This dichotomy threshold for a halo to host a cluster, as follows. is also found in HODmodels, for instance when accounting Thethirdbrightestgalaxywillcorrespondtothesecond for total galaxy occupation number (e.g., Yang et al. 2005; most massive subhalo (since the brightest galaxy is hosted Zhenget al. 2005): while P(N M), the probability that a | bytheparenthaloitself),andtheprobabilityofthishaving halo of mass M hosts N galaxies, is Poissonian at high N, a mass m is then: where satellite galaxy numbers dominate, it is significantly 2 sub-Poissonian atlowN,indicatingthatthedistributionof P (m ,M )=N(m M )e−, (18) 2 s,2 h s,2 h thecentral galaxies is much more deterministic. | This has an important consequence, derived from the where N(ms,2 Mh) is the mass distribution function of the fact that, at high mass, and therefore also at high luminos- subhaloes, equ|ation (2), and < N >= m∞s,2N(m′|Mh)dm′ ity,thetotal mass function is dominated by thehaloes, not istheaveragenumberofsubhaloesmorRemassivethanmin subhaloes. This means that, when analysing the luminos- aparenthaloofmassM .Thisexpressionassumesthatthe h ity function of galaxies in clusters, the brightest region will distribution of subhalo masses is Poissonian with average be dominated by the central galaxies, which will actually , as expected for the subhaloes (e.g., Kravtsov et al. be more abundant overall than the brightest of the satel- 2004; since we are looking at cluster sized haloes, < N > lite galaxies. We then expect that this will cause a feature will be large in this case), and it is simply the product of in the cluster galaxy luminosity function at the bright end; the probability of having a subhalo with mass m , given 2 thispointwillbeexaminedinfurtherdetailbelow.Another by the first term on the right hand side, by the Poisson consequence is that we expect theluminosity of the central probabilityofhavingexactlyonesubhalomoremassivethan galaxy in the cluster to be completely determined by halo m . Using the fact that d/dm= N(m M ), it is 2 s h − | Are First Brightest Galaxies Special? 9 Figure 4. Contribution to the high-end luminosity function of Figure5.ProbabilitythatahaloofmassM containsmorethan central andsatellitegalaxies.Theseparticularcurves areforthe 30galaxiesbrighterthanm3+2mandisconsideredarichcluster K-band,lognormalmodel,buttheresultsaresimilarforthecon- accordingtotheusualAbelldefinition.UsingeitherthebJ orthe centration model. It is very noticeable that the halo numbers K bands todothecountingresultsinthetwodifferentcurves. dominate in this luminosity range. The relative satellite contri- butiontothetotal luminosityfunctionisqualitativelysimilarto whatisfoundbyother authors(vandenBoschetal.2006). easy to check that this probability is well normalized to 1. The average value of the magnitude corresponding to this subhalo, m , can then be calculated from the distribution 3 by: ∞ = m(m )P (m ,M )dm , (19) 3 h s 2 s h s Z 0 where m(m ) represents the corresponding magnitude as a s function of the subhalo mass, calculated using the mass- luminosityrelationfrom section2.Inthisinstance,wehave used the simpler relation of equation (5), since it greatly simplifies the calculations and using thefull CLF results in only a slight difference. Usingthismagnitudewecanthenobtainm +2m,and 3 thenconvertthisbackintoamassthreshold,m (M ),which t h will be dependent on the parent halo mass. Finally, we can findtheprobability,asafunctionofM ,thatN(m M ) h t h | ≥ 29(givingmorethan30objectsabovethemagnitudelimit, when including the central galaxy). Since we are assuming that the subhalo distribution is Poissonian, this will simply ∞ Figure6.Luminosityfunctionofgalaxiesinrichclusters,inthe be given by P(n,µ), where P(n,µ) is the normal n=29 K-band. The two curves correspond to the two models for the Poisson probaPbility with average µ =< N(mt|Mh) >. This dispersion used, described in the previous section. Also shown, gives a smooth transition for the mass of cluster hosting for comparison, is the global LF. Only galaxies above the mass haloes, shown in figure 5, starting around a halo mass of thresholdshowninfigure5wereconsidered. Mh = 1014h−1M⊙, but which depends on the luminosity function considered. this threshold with the mass function of all the subhaloes hosted by them (i.e., use equation (2), but further multi- 4.2 Cluster galaxy luminosity function pliedbyaterm toreflect theprobability thatthehalo does Once we have a mass threshold for haloes hosting clusters, indeedhostarichcluster,givenbytheresultshowninfigure obtainingtheclustergalaxy luminosityfunctionisstraight- 5).Then,wetransformthisintoaluminosityfunctionusing forward:wesimplysumupthemassfunctionofhaloesabove theCLF. Ourresult is shown in figure 6. 10 A. Vale and J. P. Ostriker Qualitatively, we obtain a good agreement with can be described by combining a Schechter function repre- observed luminosity functions (such as the one of senting the satellite galaxies contribution with a high mass dePropris et al.2003,although in thisparticularcase adi- gaussian due to the central galaxy. Likewise, Zehavi et al. rect comparison is difficult because these results are in a (2005) build up HOD models from SDSS results, and ana- different band; redoing our analysis in the same band pro- lyze the central/satellite galaxy split; from this, they build duces a good match), particularly in the lower luminosity conditionalluminosityfunctions,andshowthattheirresults range.Atthebrightend,thereissomedisagreementcaused implythatthecentralgalaxiesliefaraboveaSchechterfunc- by a particular feature of our result, a bump in the lumi- tion extrapolation of the satellite population. The observa- nosityfunctionatthebrightend.Itissimpletounderstand tional work also hints at similar features: for example, it is that this bump is caused by the central galaxies. The dis- known that cD galaxies are brighter than what is given by crepancy in numbers between central and satellite galaxies thebrightendoftheclustergalaxyluminosityfunction(e.g., comes from the fact that, at high luminosity, the contribu- Binggeli, Sandage & Tammann1988).Thisisalsopresentin tion from parent haloes to thetotal luminosity distribution studiesoftheluminosityfunctionofgalaxiesinclusters(e.g., (shown in figure 4) completely dominates over the subhalo Ekeet al. 2004). All this once again reinforces the notion one.Thisisareflectionofthefactthathaloesaremuchmore that central galaxies form a special distribution, essentially abundant at the high mass end than subhaloes (see paper separate from thesatellite galaxy one. I). In fact, it can be seen from the figure that these central galaxies essentially correspond to the high luminosity end of the global luminosity function. This is hardly surprising, 5 BUILDING THE BCG LUMINOSITY sincewecanexpectthemostluminousgalaxiestolieatthe DISTRIBUTION centreof themost massive clusters. Such a feature is thus a natural consequence of the As discussed above, the non-parametric model used natu- model: very luminous galaxies will predominantely be cen- rallybuildsapictureofgalaxyclusters.Thistranslatesitself tralgalaxiesofhighmasshaloes,whichwillthereforedomi- into a procedure to build up the total luminosity distribu- nateinnumberoversatellitegalaxiesofthesameluminosity; tion of galaxies within a halo of a given mass. This will be at the same time, since we introduce a lower mass limit to composedoftwosteps,onedealingwiththesatellitegalaxies richclusterhostinghaloes,thefaintendcontributiontothe in the subhaloes, anotherwith the central galaxy. luminosityfunctionofgalaxiesinsuchclusterswillcomeen- tirelyfromsatellites. Itisimportanttonotethatthisisnot 5.1 Satellites a particularity of this specific model: any model associat- ingcentralgalaxieswithparenthaloesandsatellitegalaxies In this step, we need to sum up the total luminosity in the with subhaloes will show a similar feature, due to the dis- satellite galaxies contained in the halo. We start by taking crepancyinnumbersbetweenthetwoathighmass(though thetotalnumberof subhaloesinagiven halo,ascalculated it may also require that this be associated with high lumi- from theSHMF (that is, the occupation number;see paper nosity in both cases; or, more particularly, that the same I) , as an average number for a parent halo of this mass, mass-luminosity relation is used for both haloes and sub- taking into account the effect of destroyed subhaloes as in- haloes, as is thecase in themodel used here). troducedinsection2.1.Inthisstepitisnecessarytospecify Anoteofcautioncomesfromthefactthattheshapeof aminimummass:wetakealowenoughvaluetoensurethat thebumpdependsonthedispersionintheCLF:thesmaller weaccountforallsubhaloesmassiveenoughtogiveanotice- itis,thesharperthebumpwill be.Furthermore,theactual ablecontributiontothetotalluminosityofthehalo.Wethen shape of the bump is determined by the cluster definition assumethatthetotalnumberofsubhaloesfollowsaPoisson beingused,through theclusterthreshold mass discussed in distribution(asdiscussedabove;e.g.,Kravtsov et al.2004). theprevious section. This cutoff mass is responsible for the For each subhalo, up to a total as calculated from the decreasing values of the cluster galaxy luminosity function Poisson distribution,wedetermineamass,byassumingthe on the left side of the bump; a lower threshold mass would subhaloesfollowarandomdistributiongivenbythesubhalo result in a wider bump. Taking this threshold to lower and mass function (2). Then we convert this to the luminosity lowervalues(beyondtherangewhereitwouldbereasonable of thehosted galaxy using the mass luminosity relation. toassumethepresenceofclusters)resultsintheprogressive Finally,oncewehavethetotalnumberofsubhaloes(as disappearanceofthebumpaswenaturallyregaintheoverall determined initially from the Poisson distribution), we can global luminosity function. It should be stressed, however, sum all of their calculated luminosities to obtain the total that the presence of a bump is a fundamental prediction of in satellite galaxies. themodel,independentofthepreciseclusterdefinitionbeing Atthesametime,theaverageluminosityofthebright- used, since it is a direct consequence of the discrepancy in est satellite galaxy can be calculated in a more direct fash- numbersbetween haloes and subhaloes at high mass. ion. Using the subhalo mass distribution, we can get the Thiskindoffeatureisalsopresentinsomerecentwork probabilitydistributionofthemassofthemostmassivesub- dealing with HOD models and the central/satellite galaxy halo. Analogously towhat was donein theprevioussection separation.InZheng et al.(2005)(seealsoZhuet al.2006), for thesecond most massive subhalo (see equation 18),this theauthorsuseasemi-analyticalmodelofgalaxyformation will begiven by toobtaintheconditionalgalaxybaryonicmassfunction.For P (m ,M )=N(m M )e−, (20) high masshaloes in therangeweareconsidering here,they 1 1 h 1| h alsoobtainahighmassbumpinthisfunctioncausedbythe where = ∞N(m′M )dm′ as before. Used together m | h central galaxy. They show that the baryonic mass function withthemassluRminosityrelation,theaverageluminosityof

The Non-Parametric Model for Linking Galaxy Luminosity with Halo/Subhalo Mass: Are First Brightest Galaxies Special?

Similar The Non-Parametric Model for Linking Galaxy Luminosity with Halo/Subhalo Mass: Are First Brightest Galaxies Special?