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

0.46 MB·English

by  A. Vale

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 T´ecnico, 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 in-
terest 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., Milosavljevi´c 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 sub-
halo mass function.
Our scheme is based on the fact that most of the lu-
minosity 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 normal-
ization, 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 mas-
sive 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 de-
tailed 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 lumi-
nosity. To capture this, we introduce a dispersion around
the average relation describe above. We use the condi-
tional 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 & Milosavljevi´c 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 & Milosavljevi´c 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)(seealsoMacci`o 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 Macci`o 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 luminos-
ity, 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 defini-
tion,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 formal-
ism presented in theprevious section:
∞
φ (L)dL= φ (LM)n (M)dM, (17)
i CLF i
Z |
0
wheretheiindexesrefertothehaloesandsubhaloes,respec-
tively.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) us-
ing their semi-analytical modelling find satellite galaxies to
dominate at the faint end, Cooray & Milosavljevi´c (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 )<N >e−<N>, (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 <N >, 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<N >/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 )>= 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−<N>, (20)
high masshaloes in therangeweareconsidering here,they 1 1 h 1| h
alsoobtainahighmassbumpinthisfunctioncausedbythe where <N >= ∞N(m′M )dm′ as before. Used together
m | h
central galaxy. They show that the baryonic mass function withthemassluRminosityrelation,theaverageluminosityof