Modelling galaxy clustering in a high resolution simulation of structure formation

0.42 MB·English

by  Lan Wang

Mon.Not.R.Astron.Soc.000,1–12(2005) Printed3February2008 (MNLATEXstylefilev2.2)
Modelling galaxy clustering in a high resolution simulation
of structure formation
Lan Wang1,2⋆, Cheng Li2,3,4, Guinevere Kauffmann2, Gabriella De Lucia2
1Department of Astronomy, Peking University,Beijing 100871, China
2Max–Planck–Institut fu¨r Astrophysik, Karl–Schwarzschild–Str. 1, D-85748 Garching, Germany
3Centerfor Astrophysics, University of Science and Technology of China, Hefei, Anhui 230026, China
4The Partner Group of MPI fu¨rAstrophysik, Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China
7
0 Accepted 2006??????.Received2006??????;inoriginalform2006??????
0
2

n ABSTRACT
a We use the Millennium Simulation, a 10 billion particle simulation of the growth of
J cosmic structure, to constructa new model of galaxyclustering. We adopt a method-

4 ology that falls midway between the traditional semi-analytic approach and the halo
2 occupationdistribution(HOD)approach.Inourmodel,weadoptthepositionsandve-
locitiesofthe galaxiesthatarepredictedbyfollowingtheorbitsandmerginghistories

2 ofthe substructuresinthe simulation.Ratherthanusingstarformationandfeedback
v
‘recipes’tospecifythephysicalpropertiesofthegalaxies,weadoptparametrizedfunc-
6
tionstorelatethesepropertiestothequantityM ,definedasthemassofthehalo
4 infall
attheepochwhenthegalaxywaslastthecentraldominantobjectinitsownhalo.We
5
testwhethertheseparametrizedrelationsallowustorecoverthebasicstatisticalprop-
3
0 erties of galaxies in the semi-analytic catalogues, including the luminosity function,
6 the stellar mass function and the shape and amplitude of the two-point correlation
0 function evaluated in different stellar mass and luminosity ranges. We then use our
/ model to interpret recent measurements of these quantities from Sloan Digital Sky
h
Survey data. We derive relations between the luminosities and the stellar masses of
p
galaxies in the local Universe and their host halo masses. Our results are in excellent
-
o agreement with recent determinations of these relations by Mandelbaum et al using
r galaxy-galaxyweak lensing measurements from the SDSS.
t
s
a Keywords: galaxies:fundamentalparameters–galaxies:haloes–galaxies:distances
: and redshifts – cosmology: theory – cosmology: dark matter – cosmology: large-scale
v
i structure
X
r
a
1 INTRODUCTION whichaimstoprovideapurelystatisticaldescriptionofhow
dark matter haloes are populated bygalaxies.
According to the current standard paradigm, galaxies form
and reside inside extended dark matter haloes. Three dif- TypicalHODmodels areconstructedbyspecifying the
ferent approaches have been used to model the link be- number of galaxies N that populate a dark matter halo
tween the properties of galaxies and the dark matter of mass M as well as the distribution of galaxies within
haloes in which they are found. One approach is to carry thesehaloes(Kauffmann et al.1997;Peacock & Smith2000;
out N-body + hydrodynamical simulations that include Seljak 2000; Benson et al. 2000; Berlind & Weinberg 2002;
both gas and dark matter(Katz et al. 1996; Pearce et al. Berlind et al. 2003). More recent models haveconcentrated
2001; Whiteet al. 2001; Yoshikawa et al. 2001). Another ontheso-calledconditionalluminosityfunctionΦ(L|M)dL,
approach is to combine N-body simulations with simple which gives the number of galaxies of luminosity L that
prescriptions, taken directly from semi-analytic models of reside in a halo of mass M(Yang et al. 2003). Most HOD
galaxyformation(Kauffmann et al.1999),totrackgascool- models also distinguish between “central” galaxies, located
ing andstar formation in galaxies. Thethird method is the at the centres of dark matter haloes and “satellite” galax-
so called Halo Occupation Distribution (HOD) approach, ies, which are usually assumed to have the same density
profile as the dark matter within the halo. Physically, this
is supposed to reflect the fact that gas cools and accu-
mulates at the halo centres until the halo merges with a
⋆ Email:[e-mail] larger structure. With this approach, the models can be
(cid:13)c 2005RAS
2 L.Wang, C.Li, G.Kauffmann, G. De Lucia
used to explore the parameters that are required to match butforsatellitegalaxies,itisthemassofthehalowhenthe
simultaneouslythegalaxyluminosityfunctionaswellasthe galaxy was first accreted by a larger structure.
the luminosity, colour and morphology dependences of the This approach has the advantage of the semi-analytic
correlationfunction(van den Bosch et al.2003;Zehavi et al. models in that it provides very accurate positions and ve-
2005;Yanget al.2005).OtherpapershaveusedHODmod- locities for all the galaxies in the simulation. It maintains
els to explore the detailed shape of two-point correlation the simplicity of the HOD approach, because it bypasses
function(Zehaviet al. 2004) as well as higher order correla- theneedtoincorporatedetailedtreatmentofstarformation
tion functions(Wang et al. 2004). and feedback processes. Our aim in developing these mod-
N-body simulations can now be carried out with els is to use them as a means of constraining the relation
high enough resolution to track the histories of indi- between galaxy physical properties and halo mass directly
vidual substructures (subhaloes) within the surrounding from observational data, not as a means of understanding
halo(Springel et al. 2001). It is thus becomes possible to thephysicsof galaxy formation.
specifythepositionsandvelocitiesofgalaxies withinahalo The paper is organized as follow: we first introduce
in a dynamically consistent way, rather than assuming a the Millennium Run and the methodology used for iden-
profileor form for thevelocity distribution. Galaxy cluster- tifyinghaloes,subhaloesandgalaxiesinthissimulation.We
ing statistics that are computed using the full information thenstudytherelationbetweenluminosity/stellarmassand
available from these high resolution simulations should in M in mock galaxy catalogues constructed using these
infall
principle bemore accurate and robust. simulations. In Sec. 4 we introduce a parametrization for
Caution must be exercised, however, when only using these relations and show that we are able to recover ba-
subhaloes as tracers of galaxies in high resolution simula- sicstatisticalquantitiessuchasthegalaxyluminosity/mass
tions, as has been recently done by Vale& Ostriker (2004, functionandtheshapeandamplitudeofthetwo-pointcor-
2005); Conroy et al. (2005). In standard models of galaxy relation function in differentluminosity/mass bins.Wealso
formation, when a galaxy is accreted by a larger system investigate the the effect of changing the parameters of the
suchasacluster,itssurroundinggasisshock–heatedtohigh relation on the luminosity function and correlation func-
temperatures. Star formation then terminates as the inter- tion. In Sec. 5 we apply the method to real data on the
nalgassupplyofthegalaxyisusedup.Thestellarmassesof clustering of galaxies as a function of luminosity and stel-
satellite galaxies only change by a small amount after they lar mass(Li et al. 2006) derived from the Sloan Digital Sky
are accreted, while their luminosities dim due to aging of Survey (SDSS). Finally, we discuss our results and present
theirstars. Incontrast,thedark matterhaloes surrounding ourconclusions.
the satellites gradually lose mass as their outer regions are
tidally stripped (DeLucia et al. 2004). Near the centres of
the halos, most of the substructures have been completely
2 THE SIMULATION
destroyed.Gao et al.(2004)haveshownthattheradialdis-
tribution of subhaloes is much less centrally concentrated The Millennium Simulation(Springelet al. 2005) used in
than the radial distribution of galaxies predicted by simu- this study, is the largest simulation of cosmic structure
lations that follow the full orbital and merging histories of growthcarried outsofar.Thecosmological parametersval-
these systems. 1 ues in the simulation are consistent with recent determina-
In this paper, we make use of the Millennium Simula- tionsfromacombinedanalysisofthe2dFGRS(Colless et al.
tion,a10billionparticlesimulationofthegrowthofcosmic 2001)andfirstyearWMAPdata(Spergel et al.2003).Aflat
structure,toconstructanewmodelofgalaxyclustering.We ΛCDM cosmology is assumed with Ωm =0.25, Ωb =0.045,
adoptamethodologythatfallsinbetweenthesemi-analytic h = 0.73, ΩΛ = 0.75, n = 1, and σ8 = 0.9. The simulation
approach, which tracks galaxy formation ‘ab initio’ within follows N =21603 particles of mass 8.6×108h−1M from
⊙
thesimulation,andtheHODapproach,whichonlyprovides redshift z =127 to the present day, within a comoving box
astatisticaldescriptionofhowgalaxiesarerelatedtotheun- of 500h−1Mpc on a side.
derlying dark matter density distribution. In our approach, Full particle data are stored at 64 output times. For
weadopt thepositions and velocities of thegalaxies aspre- each output, haloes are identified using a friends-of-friends
dicted by following the orbits and merging histories of the (FOF) group-finder. Substructures (or subhaloes) within a
substructuresin thesimulation. Ratherthanusingstarfor- FOF halo are located using the SUBFIND algorithm of
mation and feedback ‘recipes’ tocalculate howthephysical Springel et al.(2001).Afterfindingallhaloesandsubhaloes
propertiesofthegalaxiessuchastheirluminositiesorstellar at all output snapshots, merging trees are built describing
masses evolve with time, we adopt parametrized functions indetailhowthesesystemsmergeandgrow astheuniverse
to relate these properties to the quantity Minfall, defined evolves. Since structures merge hierarchically in CDM uni-
as the mass of the halo at the epoch when the galaxy was verses, for any given halo, there can be several progenitors,
lastthecentraldominantobject.Forcentralgalaxies atthe butingeneraleachhaloorsubhaloonlyhasonedescendant.
present day, Minfall is simply the present day halo mass, Merger trees are thus constructed by defining a unique de-
scendantforeachhaloandsubhalo.Throughthosemerging
trees, we are able to follow thehistory of haloes/subhaloes,
1 NotethatthesimulationsanalyzedbyConroyetal.(2005)are
as well as thegalaxies inside them.
significantlyhigherresolutionthantheones analyzedinthispa-
Once a halo appears in the simulation, it is assumed
per,butaremuchsmallerinvolume.Asdiscussedintheirpaper,
the problem of disrupted subhaloes is not likely to be a signifi- that a galaxy begins to form within it. As the simulation
cantproblemforgalaxies intherangeofluminositiesconsidered evolves,thehalomaymergewithalargerstructureandbe-
intheiranalysis. comeasubhalo,whilethegalaxybecomesasatellitegalaxy.
(cid:13)c 2005RAS,MNRAS000,1–12
Modelling galaxy clustering 3
Figure1. Relationsbetweenstellarmass,baryonicmassandMinfallcalculatedfromthesemi-analyticgalaxycatalogues.Opencircles
represent central galaxies and triangles are for satellite galaxies. Error bars indicate the 95 percentiles of the mass distribution at the
givenvalueofMinfall.Dashedlinesshowthedoublepowerlawparametrizedfittothemedianvalueofrelationsforthegalaxysample
aswhole.
The galaxy’s position and velocity are specified by the po- (http://www.mpa-garching.mpg.de/galform/agnpaper/) to
sition and velocity of the most bound particle of its host study how galaxy properties such as stellar mass, baryonic
halo/subhalo. Even if the subhalo hosting the galaxy is mass (i.e. stellar mass+ cold gas mass) and luminosity
tidally disrupted, the position and velocity of the galaxy is depend on M , the mass of the halo in which the
infall
still traced through this most bound particle. We will refer galaxywaslastacentralobject.Weconstructparametrized
tothesegalaxies withoutsubhaloesas“orphaned”systems. relations between these quantities and M that match
infall
Galaxies thus only disappear from the simulation if they the relations found in the mock catalogue. We then show
mergewithanothergalaxy.Thetimetakenforanorphaned that our parametrization allows us to recover both the
galaxytomergewiththecentralobjectisgivenbythetime luminosity/mass functions of the simulated galaxies and
taken for dynamical friction to erode its orbit, causing it the shape, amplitude and mass/luminosity dependence of
to spiral into the centre and merge. The satellite orbits are the two-point correlation functions. Croton et al. (2006)
thus tracked directly until the subhalo is disrupted; there- have shown that their catalogues provide a good match to
after, the time taken for the galaxy to reach the centre is the observed galaxy luminosity function and the clustering
calculated using thestandard Chandrasekhar formula. properties of galaxies, so we believe that it is a reasonable
In this paper, we will parameterize quantities such as to use these catalogues as a way of motivating and testing
galaxyluminosityandstellarmassasafunctionofthequan- oursimple parametrizations.
tity M , which is defined as the virial mass of the halo
infall In Fig. 1 we plot the relations between M and
hostingthegalaxyattheepochwhenitwaslastthecentral infall
galaxy of its own halo. The Millennium simulation cata- galaxy stellar mass (Mstars) and baryonic mass (Mbaryon).
Weshowresultsforpresent-daycentralgalaxiesinblueand
logues includehaloes downtoaresolution limit of20 parti-
cles,whichyieldsaminimumhalomassof2×1010h−1M⊙. satellite galaxies in red. Error bars indicate the 95th per-
centiles of the distributions. As we will show, the relations
Inourstudy,weonlyconsidergalaxieswithM greater
than 1010.5h−1M⊙.(Note that M is simpinlyfatlhle virial between Minfall and Mstars/ Mbaryon are well described
infall by a double power law. The crossover point between the
massofthehosthaloforcentralgalaxiesatthepresentday.) two power laws is at a halo mass of ∼ 3 × 1011h−1M ,
This results in a total sample of 11761178 galaxies within ⊙
which corresponds to a galaxy with stellar mass of around
thesimulation volume. 1010h−1M .Inlessmassivehaloes,supernovafeedbackacts
⊙
to prevent gas from cooling and forming stars as efficiently
asinhigh masshaloes. Inmassivehaloes, thecooling times
3 THE RELATIONS BETWEEN MINFALL, becomelongerandasmallerfractionofthebaryonsarepre-
STELLAR MASS AND LUMINOSITY IN dicted to cool and form stars. In addition, in the models of
THE SEMI-ANALYTIC GALAXY Croton et al. (2006), heating from AGN also acts to sup-
CATALOGUES press cooling onto high mass galaxies.
In the following two sections we use the semi- Fig. 2 shows that the distribution of Mstars at a
analytic galaxy catalogues constructed from the given value of M is well-described by a log-normal
infall
Millennium simulation by (Croton et al. 2006) function. The width of the lognormal depends weakly on
(cid:13)c 2005RAS,MNRAS000,1–12
4 L.Wang, C.Li, G.Kauffmann, G. De Lucia
Figure 2. The distribution in stellar mass for different Minfall
binsincreasingfrom1010.5h−1M⊙(left)to1014.0h−1M⊙(right).
Solidanddashedlinesareforcentralandsatellitegalaxies(note Figure 3. The same as in Fig. 1, but luminosity (represented
that they lie on top of each other for three lower mass bins). by magnitude of bj band) is plotted as a function of Minfall.
Dotted lines indicate the Gaussian fits to the distributions that Dashed(dotted) linesshow thedoublepowerlawfits totherela-
areusedinourparametrizedmodel. tionforcentral(satellite)galaxies.
4 PARAMETRIZATIONS AND TESTS
halo mass with a maximum dispersion σ of 0.2 dex at
M = 1010.5h−1M and a minimum σ of 0.1 dex at 4.1 Functional form
infall ⊙
M = 1011.5h−1M . The relations depend very little
infall ⊙ We use a two–power–law model of the following form to fit
on whether the galaxy is a central or satellite system. The themedian valueof therelations between Mstars, Mbaryon,
dispersion around the relations is also similar for the two L and M :
infall
types(in Fig. 2, the solid and dashed lines for central and
2
satellite galaxies lie almost ontop of each other).Thesimi- x= ×k,
larity in theMinfall- Mstars relations between satellite and (MiMnf0all)−α+(MiMnf0all)−β
centralsmayberegardedassomethingofacoincidence.Al-
though thereis little changein thestellar/baryonic compo- where x denotes Mstars,Mbaryon or L, and the relation be-
nentofthegalaxyafteritfalls intoalargerstructure,halos tween luminosity L and bj band magnitude is given by:
of the same mass at different times have different circular Mbj −5logh=−2.5logL
velocities and hence different cooling and and star forma-
We fit these relations for central and satellite galaxies sep-
tion efficiencies. Aswewill show later, weobtain betterfits
arately,as well asfor thegalaxy population asawhole. We
to the observational data if we allow the relations between
will later test whether separate fits to the central galaxies
central and satellites galaxies to differ.
and satellites make significant difference to our results. We
Fig. 3 shows the relation between luminosity and
alsoassumethatthedispersionaroundthemedianvaluehas
M .Italsocanbefitbyadoublepowerlaw,butthedif-
infall a lognormal form.
ference between central and satellite galaxies is much more
Table 1 lists the parameters of the best–fitting models
obvious.AtagivenvalueofM centralgalaxiesaremore
luminous than satellites becaiunsfealtlhey are forming stars at for the relations between Minfall and Mstars, Mbaryon and
L. The models havefiveparameters.
higher rates and their stellar mass-to-light ratios are lower.
The difference between central and satellite galaxies be- (i) M0 is thecritical mass/luminosity at which the slope
comes very small at large values of M . This can be of the relation changes. When we fit satellite and central
infall
understood as a simple consequence of hierarchical struc- galaxies separately, we find almost exactly the same values
ture formation: massive haloes were formed more recently forthisparameter(evenforluminosity,thedifferenceisless
than less massive haloes and subhaloes with large masses than20%).WethereforefixM0 at thebest-fitvaluefor the
are likely to havebeen accreted relatively recently.Massive galaxy sample as a whole.
satellite galaxies have therefore not been satellites for long (ii) α and β describe the slope of the relations at high
and thus have mass-to-light ratios that are more similar to and low valuesof M .
infall
theircentralcounterparts.InadditiontheCrotonetalmod- (iii) k is a normalization constant.
els include a “radio AGN mode” of feedback, which acts to (iv) We have calculated the interval in logMstars,
suppress cooling onto the most massive galaxies. This also logM and logL that encloses the central 68% of the
baryon
acts to reduce the difference between central and satellite probability distribution for 8 different values of logM
infall
galaxy colours and mass-to-light ratios. from 1010.5h−1M⊙ to 1014h−1M⊙, with step 0.5 dex. We
(cid:13)c 2005RAS,MNRAS000,1–12
Modelling galaxy clustering 5
Table 1. Best-fit parameter values for the relations between Minfall and Mstars, Mbaryon and L as derived from the semi-analytic
galaxycatalogues ofCrotonetal.(2006).
M0(h−1M⊙) α β log(k) σ χ˜2
Mstars total 3.16×1011 0.39 1.92 10.35 0.156 0.0146
central 3.16×1011 0.39 1.96 10.35 0.148 0.0240
satellite 3.16×1011 0.39 1.83 10.34 0.167 0.0057
Mbaryon total 3.61×1011 0.36 1.59 10.44 0.147 0.0415
central 3.61×1011 0.35 1.59 10.46 0.133 0.0542
satellite 3.61×1011 0.37 1.59 10.40 0.162 0.0273
L(Mbj) total 1.49×1011 0.36 1.90 7.14 0.215 0.0360
central 1.49×1011 0.31 1.99 7.25 0.169 0.1250
satellite 1.49×1011 0.46 1.81 6.90 0.189 0.0359
thencalculatetheaverageofthese8valuesandthevalueof relationforbothkindsofgalaxies,weobtainthered-dashed
σ quoted in Table 1 is 0.5 times this number. curve in Fig. 5, which is even more discrepant. Our results
suggestthatinordertoreproducetheclusteringdependence
Theresultingmodelfitsareplottedasdashedanddot-
onluminosityinamoreexactway,onewouldneedtointro-
ted lines in Fig. 1 and Fig. 3. Thequality of thefit is given
duceanadditionaldependenceoftheL−M relationon
infall
in thelast column of Table 1 and is calculated as:
theparametert ,thetimewhenthegalaxywaslastthe
infall
χ˜2 = (xfit−xSAM)2 central object of its own halo. This does not appear to be
X xSAM necessaryinordertoreproducethestellarmassdependence
where x represents Mstars, Mbaryon, L for each re- ofgalaxyclustering.Thereasonforthisdifferenceisbecause
lation, and the sum is over the 8 mass bins with the optical light from galaxies, unlike their stellar mass, is
1010.5h−1M 6M 61014.0h−1M . heavily influenced by the contribution from the youngest
⊙ infall ⊙
stars,which havelifetimes whichareshort compared tothe
age of the Universe. Once a galaxy becomes a satellite, it
4.2 Tests will fade in luminosity even thoughits stellar mass remains
approximately constant. For the sake of simplicity, we will
Thenextstepistoseewhethertheseparametrizedrelations
not introduce t as an additional parameter in this pa-
allowustorecoverthebasicstatisticalpropertiesofthesim- infall
per, but we will come back to this in future work in which
ulated galaxy catalogue, such as the mass/luminosity func-
we consider thecolour-dependence of galaxy clustering.
tion and the mass/luminosity dependence of the two point
correlation function. When fitting to the quantities Mstar
and M , we do not distinguish between central and
baryon 4.3 The effect of “Orphan” Galaxies
satellite galaxies because the relations are almost the same
for both.When fittingto galaxy luminosity, we doallow α, ThemajorityofHODmodelsintheliteratureonlyconsider
β,kandσtovarybetweencentralandsatellitegalaxies,but dark matter haloes and subhaloes that can be identified at
M0 remains fixed for both. Note that the positions and the thepresenttime.Satellitegalaxieswithoutsurroundingsub-
velocities of the galaxies are exactly the same as specified haloes are thus omitted from the analysis. We now explore
inthesemi-analyticgalaxycatalogues;theparametrizedre- theeffect of these ’orphan’ satellite galaxies on our results.
lationsbetweengalaxymass/luminosityandM simply The left panel of Fig. 6 compares the L−M rela-
infall infall
provideuswithaalternativewayofspecifyingtheproperties tion for orphan satellites with the results obtained for cen-
of the galaxies. tral galaxies and satellite galaxies that have retained their
Fig. 4 and Fig. 5 show the results of our test. Sym- subhaloes.TherightpanelofFig.6showstherelativecontri-
bols show results calculated directly from thesemi-analytic bution of central galaxies, satellite galaxies with subhaloes
galaxy catalogues and lines are from our parametrized and orphan satellites without subhaloes to the luminosity
model. The stellar mass function is well reproduced, and function of the galaxies in the semi-analytic catalogue. As
wecan also recoverthecorrelation fordifferent stellarmass can be seen, orphan satellites have lower luminosities at a
bins: 109h−1M , 1010.5h−1M and 1011.5h−1M (Fig. 4). givenvalueofM thaneithercentralgalaxiesorsatellite
⊙ ⊙ ⊙ infall
For luminosity, the parametrized model is not quiteas suc- galaxies withsubhaloes–i.e.orphangalaxies aretheoldest
cessful.Althoughtheluminosityfunctioniswell-reproduced, galaxies with thehighest mass-to-light ratios in thesimula-
there are some discrepancies in the dependence of the tion. In addition, we see that the contribution of these or-
clustering amplitude on luminosity (solid-blue curve in phansatellitesishighestatthefaintestlumninosities.Fig.7
Fig. 5). Part of the reason for this discrepancy is that our exploresthecontributionoftheorphansatellitestothecor-
parametrization of the L−M relation has somewhat relation function in three different bins of absolute magni-
infall
larger χ2 than the Mstars−Minfall relation (see Table.1). tude. The solid curves show the result for all the galaxies
Inaddition,inordertoreproducetheclusteringtrendsasa while the dashed red curves show the result when the or-
functionofluminosity,itiscriticaltofittherelationforcen- phan satellites are omitted. As can be seen, the orphaned
tralandforsatellitesgalaxiesseparately.Ifweapplyasingle satellites contribute heavily to the correlation function of
(cid:13)c 2005RAS,MNRAS000,1–12
6 L.Wang, C.Li, G.Kauffmann, G. De Lucia
Figure 4. Results from the comparison of the parametrized model and the semi-analytic galaxy catalogue. The left panel shows the
stellar mass function. The right panel shows correlation functions for three different stellar mass bins: 109h−1M⊙, 1010.5h−1M⊙ and
1011.5h−1M⊙.Symbolsareforthesemi–analyticgalaxycatalogue. Solidlinesareforourparametrizedmodels.
Figure 5. Same as Fig. 4, except for the luminosity function (left) and the correlation function as a function of absolute magnitude
(right). Symbols are the semi-analytic results, error bars in the right panel are the boot strap error of correlation length for the semi-
analyticmodel. Solidlinesareforparametrized modelswhererelations forcentral galaxies andsatellitegalaxies arefitseparately. The
dashedlinesareformodelswherethefitisforthegalaxypopulationasawhole.
faintgalaxiesonscalesoflessthan1Mpc.Omissionofthese chosentoassumethatthevisiblegalaxiessurviveevenafter
systems causes the amplitude of the correlation function to their subhalo falls below the resolution limit of the simu-
be underestimated by more than an order of magnitude at lation. It is possible that we over-estimate the number of
separations of0.1Mpcforgalaxies with−18<M <−17. these objects because we do not include tidal stripping on
bj
the stellar component. However, we believe that ”orphan”
We note that there are uncertainties in our treat- galaxies (at least part of them) are needed in order to ex-
ment of orphan galaxies in the simulation. Some of these plainobservationalresults.FromFig.7weseethatwhen”or-
galaxies may indeed be destroyed or significantly reduced phan”galaxiesareexcluded,thecorrelationsignaldecreases
in mass by tidal stripping effects. Indeed, the existence at small scales, at odds with observational results(see later
of a significant intra-cluster light component does sug- in Fig.9 and Fig.10). In this work we consider all the ’or-
gest tidal effects or mergers do unbind some of the stars phan’systems as part of satellite subsamples.
in satellite galaxies(Arnaboldi 2004; Feldmeier et al. 2004;
Zibetti et al. 2005). In face of these uncertainties, we have
(cid:13)c 2005RAS,MNRAS000,1–12
Modelling galaxy clustering 7
Figure 6. L−Minfall relations(left) and luminosity functions(right) for different types of galaxies from the semi-analytic galaxy
catalogue: central galaxies (solid lines), satellite galaxies with subhaloes (dashed lines), satellite galaxies without subhaloes (dotted
lines).Thedashed-dotted lineintheright-handpanel showsthetotalluminosityfunctionforallgalaxies.
Figure7.Correlationfunctionsforthreeluminositybinsincluding(solidlines)andnotincluding(dashedlines)orphansatellitegalaxies.
4.4 Changes in the input parameters end of the mass function. A change in scatter σ has simi-
lar effect to a change in α, and influences the amplitude of
Oneadvantageofourparametrizedapproachisthatwecan
themassfunctionatthehighmassend.Thisisbecausethe
understand the effect of changing each different parameter
mass function is relatively flat at low masses and declines
and thus gain intuition about what changes are necessary
steeply at high masses, so an increasing amount of scatter
tobringthemodelsintotheclosestpossibleagreementwith
in theMstars−Minfall relation will havea strongeffect on
theobservationaldata.Thisisdifferentinspirittoexploring
thenumberof high mass galaxies.
parameter space in the semi-analytic models, because the
parameters in these models are tied to the physical recipes
forstarformationandfeedbackratherthantherelationbe-
tween halo mass and galaxy properties, which is the focus
of our approach. The lower panels in Fig. 8 show the effect of the
In the upper panel of Fig. 8, we show how changing same parameter changes on the amplitude of the correla-
each of the parameters affects the stellar mass function. tion function evaluated on scales of r = 0.33h−1 Mpc and
Note that the normalization constant k is always adjusted r = 5.30h−1 Mpc. We see that a parameter change that
in order to keep the amplitude of the mass function at causes an increase in the number of galaxies of given mass,
Mstars=1011M⊙ fixed.ChangingM0affectsthemassscale will cause a corresponding decrease in theclustering ampli-
of thetransition between the two power laws as well as the tude of these systems. This is easy to understand. In order
amplitude of the mass function at both low and at high tohavemoregalaxiesofagivenmassinthesimulation,they
masses. Changes to α affect the shape of themass function must be shifted into lower mass haloes and these low mass
atthehighmassend,whilechangestoβaffectthelowmass haloes are more weakly clustered.
(cid:13)c 2005RAS,MNRAS000,1–12
8 L.Wang, C.Li, G.Kauffmann, G. De Lucia
Figure 8. The effect of changing parameters on the stellar mass function(upper panels) and correlation at scales of r=0.33h−1Mpc
andr=5.30h−1Mpc(lowerpanels). ThesolidlinesrepresentthebestfitmodelfortheMstars−Minfall relation.
5 APPLICATION TO SDSS Tocompare ourmodels with theobservations, we need
toeitherconvertw(rp)totherealspacecorrelationfunction
In this section, we apply our models to observational data
ξ(r),ortocalculate w(rp)from ourmodelgalaxy catalogue
from the Sloan Digital Sky Survey. Recent large scale red-
directly.Wetested themethod presented byHawkins et al.
shift surveyssuchas2dfGRS(Colless et al.2001)andSloan
(2003)forconvertingw(rp)toξ(r)onscaleslessthanaround
DigitalSkySurvey(SDSS;York et al.(2000))providegalaxy 30h−1Mpc. We find that the conversion amplifies the error
samples that are large enough to measure the luminosity
andtheresultsforthelowluminosityandlowmassbinsare
dependence of galaxy clustering accurately (Norberg et al.
then too noisy to provide good constraints on our models.
2002a,b; Zehaviet al. 2005). In this paper, we make use of
Therefore,wederivew(rp)fromourcataloguebyintegrating
the recent measurements of the projected correlation func-
thereal space correlation function ξ(r):
tion w(rp) by Li et al. (2006). These authors calculated
swt(erllpa)rnmoatsosnulsyinagsaasfaumncptlieonofogfaglaaxlaiexsyclounmstirnuocstietyd,fbroumt atlhsoe w(rp)=2Z0∞ξ(prp2+rk2)drk =2Zrp∞ξ(r) r2rd−rrp2
SDSS Data Release 2 (DR2) data. The methods for esti- p
matingthestellar massesaredescribedinKauffmann et al. We truncate the integration at r = 60h−1Mpc and the re-
(2003). Here we make use of these measurements to con- sulting w(rp) is reliable up to a scale of ∼10h−1Mpc.
strain the relation between galaxy luminosity, stellar mass We now generate a grid of models by systematically
andM .Totakeaccounttheeffectof”cosmicvariance” varyingthe5parameterslistedinTable1.Wecompareeach
infall
ontheobservationalresults,wehaveconstructedasetof16 model with the galaxy luminosity function(Blanton et al.
mockgalaxycataloguesfromthesimulationwithexactlythe 2003b) and the w(rp) measurements in different ranges in
samegeometryandselectionfunctionasintheobservational luminosity. We define the best fitting model to be the one
sample. The effect of cosmic variance is modelled by plac- giving a minimum χ2 defined as follows:
iwnhgena vcoirntsutarlucotbisnegrvtehresreanmdoomcklycaintasildogeutehse. Fsiomruelaactihonmboockx χ2 = χ2(Φ) + χ2corr
catalogue, we measure w(rp) for galaxies in thesame inter- NΦ Ncorr
vals of luminosity/stellar mass as in the observations. The with
1−σvariationbetweenthesemockcataloguesisthenadded χ2(Φ)= [Φ−ΦSDSS]2
asanadditionalerrorinquadraturetothebootstrap errors X σ(ΦSDSS)
givenbyLi et al.(2006).Thecosmicvarianceerrorsbecome NΦ
significant forthelowluminosityandlowmasssubsamples, and
particularlyatlargevaluesofrp.Thedetailedprocedurefor
constructing these mock catalogues will be presented in a χ2 = [w(rp)−w(rp)SDSS]2
separate paper(Li et al., in preparation). corr X σ(w(rp)SDSS)
Ncorr
(cid:13)c 2005RAS,MNRAS000,1–12
Modelling galaxy clustering 9
Figure 9.Bestfitmodeltotheluminosityfunctionandthecorrelationfunctionevaluatedindifferentluminositybinsusingdatafrom
theSDSS.SolidlineswitherrorbarsaretheSDSSresults,andreddashedlinesarefromourparametrizedmodel.Greendashed/dotted
linesareresultsforcentral/non-central subsamplesofourparametrizedmodel.
Figure 10. Best fit model to the stellar mass function and the correlation function evaluated in different stellar mass bins using data
from the SDSS. Symbols with error bars are the SDSS results, and dashed red lines are from our parametrized model. Dotted blue
lines show the results obtained when central and satellite galaxies are treated separately. Green dashed/dotted lines are results for
central/non-central subsamplesoftheparametrizedmodelwhencentralandsatellitegalaxiesaretreatedseparately.
NΦ is the number of points over which the luminosity E-correction, each galaxy is assigned a redshift by placing
function is measured (NΦ = 102 for the r-band absolute a virtual observer at the centre of the simulation box. The
magnitude ranging from −18 to −23). Ncorr is the number redshiftas”seen”bytheobserveristhusdeterminedbythe
of points over which the correlation function is measured ( comovingdistancetotheobserverandthepeculiarvelocity
Ncorr =93,rangingfrom0.11to8.97h−1Mpcforluminosity ofthegalaxy.Thecorrected r-bandmagnitudeisgivenby:
bins[−19,−18],[−20,−19],[−21,−20],[−22,−21] and from
0.57 to 8.97h−1 Mpc for the most luminous bin [−23,−22] M0.1r =−2.5×logL+Kcorrection+Ecorrection−5logh
in ther-band). Our best fit model has the parameters: M0 = 3.41×
1011h−1M , α = 0.221, β = 1.67, k = 8.13 and σ = 0.440
⊙
To compare with the SDSS observations, where the for the central galaxies and M0 = 2.58 × 1011h−1M⊙,
median galaxy redshift is around 0.1, we correct the r- α = 0.345, β = 3.83, k = 7.71 and σ = 0.742 for the satel-
band absolute magnitude Mr of each model galaxy to its litegalaxies(seeTable2).Theresultingluminosityfunction
z=0.1valueM0.1r usingtheK−correctioncode(kcorrect and correlation functions are shown in Fig. 9. χ2(Φ)/NΦ is
v3 1b) of Blanton et al. (2003a) and the luminosity evolu- 3.348andthetotalχ2is6.115.Alsoplottedaretheresultsof
tionmodelofBlantonetal.(2003b).TocalculatetheK-and centralandsatellite subsamplesofourparametrizedmodel,
(cid:13)c 2005RAS,MNRAS000,1–12
10 L.Wang, C.Li, G.Kauffmann, G. De Lucia
Figure 11.BestfitL−Minfall andMstars−Minfall relationsasconstrainedbytheSDSSdata. Bluecirclesarethecentral galaxies
andredtrianglesaresatellites.Greenlinesarethebestfittingrelationsfromthesemi-analyticcatalogue ofCrotonetal.(2006).Filled
circlesshowthecentralhalomassfromthegalaxy-galaxylensingresultsofMandelbaum etal.(2006);errorbarsarethe95%confidence
limits.Theresultsshownarethecombinedsampleofearlyandlate-typegalaxies(Mandelbaum, privatecommunication).
Table 2.Best-fitparametervaluesfortherelationsbetween Minfall andMstars andL(Mr)asderivedfromtheSDSSdata.
M0(h−1M⊙) α β log(k) σ χ2 χ2(Φ)/NΦ
Mstars(M⊙) total 3.15×1011 0.118 2.87 10.26 0.326 16.96 2.487
central 3.33×1011 0.276 2.59 10.27 0.241 5.351 1.850
satellite 4.64×1011 0.122 2.48 10.26 0.334
L(Mr) central 3.41×1011 0.221 1.67 8.13 0.440 6.115 3.348
satellite 2.58×1011 0.345 3.83 7.71 0.742
shown by green dashed and dotted lines. The drop in the parameters of the best-fit models are listed in Table 2. For
correlation function on scales larger than ∼ 10h−1 Mpc is the stellar mass function, the errors due to sample size are
notcausedbyapoorfit;itisduetothetruncationofourin- muchsmaller than thesystematic errors in thestellar mass
tegrationoftherealspacecorrelationfunctionatr=60h−1 estimates themselves. We therefore assign the same error
Mpc−1. to all points at stellar masses less than 1011.5h−2M (the
⊙
errorisequaltothevalueatthatmass).Inourfirstattempt
We now carry out the same analysis for stellar mass,
rather than luminosity. We have constructed the stellar at fitting the data, we assumed that the Mstars−Minfall
wouldbethesameforcentralandsatellitegalaxies,because
mass function directly from the SDSS DR2 data (Fig. 10;
the relations are very similar in the semi-analytic galaxy
left panel) and use this, in conjunction with the measure-
catalogues. The red dashed lines in Fig. 10 show the best
ments of w(rp) as a function of stellar mass published by
fittingresults.Themodelclearlyover-predictstheclustering
(Li et al. 2006), to constrain the Mstars-Minfall relation.
ofthemoremassivegalaxiesonsmallscales.Ifweallowthe
In the computation of stellar mass function, we have cor-
rected for the volume effect by weighting each galaxy by relationbetweenMstarsandMinfalltodifferforcentraland
satellite galaxies, we obtain the results shown by the blue
a factor of Vsurvey/Vmax, where Vsurvey is the volume for
dotted lines, which are considerably better.
the sample and Vmax is the maximum volume over which
the galaxy could be observed within the sample redshift The best-fit r-band luminosity – Minfall and Mstars−
range (0.01 < z < 0.3) and within the range of r−band M relations derived from ourmodels are illustrated in
infall
apparent magnitude (14.5 < r < 17.77). A Schechter func- Fig. 11. Results are shown separately for central galaxies
tion provides a good fit to our measurement at stellar (blue) and satellite galaxies (red). In our models, satellite
mass Mstars < 1011.5h−2M⊙. We find best-fit parameters: galaxies have lower luminosities and smaller stellar masses
Φ∗ = (0.0204±0.0001)h3Mpc−3, α = −1.073±0.003 and thancentralgalaxiesatagivenvalueofM .Thiseffect
infall
M∗ =(4.11±0.02)×1010h−2M .Thiscorrespondstoa islargerforluminositythanforstellarmass,particularlyat
stars ⊙
stellar mass densityof (8.779±0.067)×108hM Mpc−3. low values of M .
⊙ infall
We fit our models to 30 points along the stellar mass For comparison, we also plot the L-M and
infall
functionand20pointsalongthecorrelationfunctionforfive Mstars − Minfall relations from the semi-analytic galaxy
differentstellarmassbinsrangingfrom109 to1012M⊙.The catalogue(Croton et al. 2006). We transform the bj
(cid:13)c 2005RAS,MNRAS000,1–12