Large scale bias and the peak background split

8 Pages·0.28 MB·English

by  Ravi K. Sheth

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Large scale bias and the peak background split
Ravi K. Sheth1 & Giuseppe Tormen2
1 Max-Planck Institut fu¨r Astrophysik, 85740 Garching, Germany
2 Dipartimento di Astronomia, 35122 Padova, Italy
Email: [email protected], [email protected]
Submitted20December1998
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9
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ABSTRACT
1
Dark matter haloes are biased tracers of the underlying dark matter distribution.

n We use a simple model to provide a relation between the abundance of dark matter
a haloesand their spatialdistributiononlargescales.Ourmodel showsthat knowledge
J
of the unconditional mass function alone is sufficient to provide an accurate estimate

3 of the large scale bias factor. Then we use the mass function measured in numerical
1
simulations of SCDM, OCDM and ΛCDM to compute this bias. Comparison with

these simulations shows that this simple way of estimating the bias relation and its
2
evolution is accurate for less massive haloes as well as massive ones. In particular, we
v
2 showthathaloeswhichareless/moremassivethantypicalM∗ haloesatthetimethey
2 form are more/less strongly clustered than formulae based on the standard Press–
1 Schechter mass function predict.
1
Key words: galaxies: clustering – cosmology:theory – dark matter.
0
9
9
/
h
p 1 INTRODUCTION mass function and the large scale bias factor. We then use
- themassfunctionmeasuredinthesimulationstoshowthat
o There has been considerable interest recently in developing
thisrelationisaccurate.Section3showsthatourmodelpro-
r models for the shape and evolution of the mass function of
t videsareaonablygoodfittothebiasrelationoflessmassive
s collapseddarkmatterhaloes(Press&Schechter1974;Bond
haloes as well as massive ones.
a
etal.1991;Lacey&Cole1993)aswellasfortheevolutionof
:
v the spatial distribution of these haloes (Mo & White 1996;
i Catelanetal.1997;Sheth&Lemson1999).Inthesemodels,
X
the haloes are biased tracers of the underlyingdark matter
r 2 MASS FUNCTIONS AND LARGE SCALE
distribution.Ingeneral,thisbiasdependsonthehalomass,
a BIASING
and,for agiven massrange, itisanonlinearandstochastic
function of theunderlying dark matter density field.
Let f(m,δ)dm, where δ = δ(z) is a function of redshift z,
Theshapeofthemassfunction(anditsevolution)pre- denote the fraction of mass that is contained in collapsed
dictedbythesemodelsisinreasonableagreementwithwhat haloes that at z have mass in the range dm about m. The
is measured in numerical simulations of hierarchical clus- associated unconditional mass function is
tering from Gaussian initial conditions (e.g. Lacey & Cole ρ¯
1994). Althoughthisagreement isbynomeansperfect (see n(m,δ)dm= f(m,δ)dm (1)
m
Fig. 2 below), to date, the emphasis has been on how well
the models fit the simulations (but see Tormen 1998). On whereρ¯is thebackgrounddensity.Let f(m1,δ1M0,δ0) de-
|
theotherhand,recentworkhasshownthat,whilethemodel notethefraction ofthemassofahaloM0 atz0 thatwasin
predictionsforthebiasrelationareinreasonableagreement subhaloes of mass m1 at z1, where z1 >z0. The associated
conditional mass function is
withwhatismeasuredinnumericalsimulationsformassive
hleaslsoeasn,til-ebsisasmeda,sstihveanhathloeesmaordeelms oprreedsitcrton(Jgilnygc1lu9s9t8e;reSdh,etohr N(1|0)= Mm10 f(m1,δ1|M0,δ0). (2)
& Lemson 1999; Porciani, Catelan & Lacey 1998).
In this paper, following a suggestion by Sheth & Lem- Finally,letN¯(m1,δ1|M,V,z0)denotetheaveragenumberof
son(1999),wearguethatthisdiscrepancybetweenthebias (m1,δ1) haloes that are within cells of size V that contain
model and simulation results arises primarily because the mass M at z0. The halo bias relation is definedby
mtioondse.lImnaSsescftuionnct2iownesaprreovdiidffeeraensitmfrpolmertehlaotsieoninbtehtewesiemnutlhae- δh(1|0)≡ N¯(mn(1m,δ11,|Mδ1),VV,z0) −1. (3)
(cid:13)c 0000RAS
2 Ravi K. Sheth & Giuseppe Tormen
Since M/V ρ¯(1+δ), this expression provides a relation independent. Therefore, the ratios of the correlation func-
betweenthe≡overdensityofthehalodistributionandthatof tions themselves should also be b2 , independent of scale.
Eul
the matter distribution. To compute this quantity,we need However, correlation functions and their volume integrals
modelsforthenumeratorandthedenominator.Thiscanbe are just differently weighted integrals over the associated
doneas follows. power spectrum. If b is independent of scale on large
Eul
An M0 halo which collapsed at z0 initially occupied a scales, then the ratio of the halo and dark matter power
certain volume V0 in the initial Lagrangian space. Let δ0 spectra should also be b2Eul, independent of wavenumber k
denote the initial overdensity within this V0. If δ0 1, for small wavenumbers. Thus, in the large scale limit, we
| | ≪
then M0 = ρ¯(1+δ0)V0 ρ¯V0. So the mean Lagrangian expectthebiasfactortobethesame,regardless ofwhether
≈
space bias between haloes and mass is given by setting it was computed using the counts-in-cells method, the cor-
N¯(m1,δ1M0,V,z0) = (10) and V = V0 in equation (3) relation functions themselves, or power spectra. Moreover,
| N |
above. Thus, in this limit, the Eulerian bias is trivial to compute if the
Lagrangian space bias is known.
δhL(1|0)= N(nm(m1,1δ,1δ|M1)0V,0z0) −1. (4) TheLagrangianbiasfactorisdifferentforhaloesofdif-
ferent masses. The work of Mo & White (1996) shows that
This expression was first derivedbyMo & White(1996). It theexactformofthisdependencecanbecomputedprovided
can beexpanded formally as a series in δ0: thatboth theconditional and theunconditional mass func-
tions are known (equation 4). Our contribution is to add
δhL(1|0)= bLk(m1,δ1)δ0k, (5) onemoresimplesteptotheargument.Namely,onthelarge
Xk scales discussed above, the peak background split should
in which form it will appear later. be a good approximation (e.g. Bardeen et al. 1986; Cole
To compute the bias relation in the evolved Eulerian & Kaiser 1989). In this limit, when expressed in appropri-
space they suggested the following procedure. Continue to atevariables,theconditionalmassfunctioniseasily related
assume that N¯(m1,δ1 M0,V,z0) can be written as some to the unconditional mass function. Therefore, on the large
|
(10), but now provide a new relation between the Eule- scales where the peak background split should apply, the
N |
rian variables M0, V, and z0, and the Lagrangian variables biasrelationcanbecomputedevenifonlytheunconditional
M0, V0 and δ0. For any given z0, Mo & White used the mass function is known.
spherical collapse model to write δ0(M0/ρ¯V). Recall that Tosummarize:wehaveusedthepeakbackgroundsplit
M0/ρ¯V ≡(1+δ).Thus,δ0 isafunctionofδ,and,ingeneral to argue that the dependence of bLag (and so bEul also) on
δ0(δ) is nonlinear. However, when δ 1, then δ0 δ. On halo mass should be sensitive to the shape of the uncondi-
| |≪ ≈
largescalesinEulerianspace,thermsfluctuationaboutthe tional mass function;different models for theunconditional
meandensityissmall.Therefore,onsufficientlylargescales, mass function will produce different bias relations. In par-
δ 1 almost surely.So, on large scales in Eulerian space, ticular, if the unconditional mass function is different from
| |≪
δhE(1|0) ≡ N¯(mn(1m,δ11,|Mδ1),VV,z0) −1 tehnet.oInnetihnesnimexutlasteioctniso,nthweepwreildlicutseedthbeiasunwciollnadlistoiobnealdmiffaesrs-
function measured in simulations as input to the formulae
≈ (1+δ) 1+δhL(1|0) −1 abovetocomputetheassociated halotomassbiasrelation.
1+bLa(cid:2)g(m1,δ1) δ(cid:3), (6)
≈
where the fina(cid:2)l line follows from(cid:3) using equation (5), setting
δ0 δ,andsoexpandingtolowestorderinδ(since δ 1), 3 THE SIMULATIONS
and≈writingbLag insteadofbL1.Thisprovidesasimp|le|l≪inear In what follows, we will study the halo distributions which
relation between δE and δ, where the constant of propor-
h formed in what are known as the GIF simulations, which
tionality is
have been kindly made available by the GIF/Virgo collab-
bEul(m1,δ1) 1+bLag(m1,δ1). (7) oration (e.g. Kauffmann et al. 1998a). We will show results
≡
for three choices of the initial fluctuation distribution be-
Furthermore, Mo & White (1996) argued that one can
longing to the cold dark matter family: a standard model
write ⋆
(SCDM:Ω0 =1,ΩΛ =0,h=0.5 ),anopenmodel(OCDM:
Dw(cid:0)hδehEre(cid:1)2tEhVe ≈avebr2Eaugl(emi1s,oδv1)erDδc2eEllsV,of size V placed random(8ly) hzΩe0=ro=0c.o70s).m.3,TolΩhoeΛgsiec=aslim0c,ounhlastt=iaonnt0s.7(wΛ)eCraeDndpMea:rfΩoflra0mt=emdo0w.d3iet,lhΩw2Λi5t6h=3np0oa.n7r--,
in Eulerian space. This expression neglects the effects of ticles each, in a box of size L = 85 Mpc/h for the SCDM
stochasticityonthebiasrelation(seeSheth&Lemson1999 run, and L= 141 Mpc/h for the OCDM and ΛCDM runs.
for details). The left hand side is the volume average (over Themodelsandresolutionaresimilartothosepresentedby
cellsofsizeV)ofthehalo-halocorrelationfunction,whereas Jing (1998).
δ2 isthevolumeaveraged correlation functionofthedark We measure mass functions in the simulations in the
h i
matter.Thus,inthislimit,theratioofthevolumeaveraged usual way, using a spherical overdensity group finder (see
correlation functions gives the square of the Eulerian bias Tormen 1998 for details). Previous authors have computed
factor. the large scale bias factor by using the ratio of the volume
Now, bEul(m1,δ1) is a function of halo mass only; it
does not depend on scale. So, in this large scale limit, the
ratio of the volume averaged correlation functions is scale ⋆ TheHubbleconstant isH0=100hkms−1Mpc−1
(cid:13)c 0000RAS,MNRAS000,000–000
Large scale bias and the peak background split 3
tion is correct, then we should be able to use the shape of
thehalomassfunctiontomodelthisdependenceaccurately.
We turn, therefore, to the shape of the halo mass function
in the GIFsimulations.
The shape of the unconditional mass function is ex-
pectedtodependontheinitialfluctuationdistribution(e.g.
Press & Schechter 1974). If the initial distribution is Gaus-
sianwithascalefreespectrum,thentheunconditionalmass
function can beexpressed in unitsin which it has auniver-
salformthatisindependentofredshiftandpowerspectrum.
Namely, let ν [δc(z)/σ(m)]2, where δc(z) is that critical
≡
valueoftheinitialoverdensitywhichisrequiredforcollapse
atz,computedusingthesphericalcollapsemodel,andσ(m)
isthevalueofthermsfluctuationinsphereswhichonaver-
age contain mass m at the initial time, extrapolated using
linear theoryto z.Forexample, in models with Ω0 =1 and
Λ0 = 0, δc(z) = 1.68647, and σ(m) (1+ z)−1. Then,
∝
for initially scale free spectra in models with Ω0 = 1 and
Λ0 =0,
n(m,z) dlogm
νf(ν) m2 (9)
≡ ρ¯ dlogν
has a universal shape (Press–Schechter 1974; Peebles 1980;
Figure1.Thesquarerootb(k)oftheratioofthepowerspectra
Bond et al. 1991). It is not obvious that this scaling holds
of dark matter haloes to that of the dark matter, computed at
for more general initial power spectra, such as those of the
the time the haloes were identified: z = 0, z = 2 and z = 4 for
GIFsimulations.
thetop,middleandbottom rows,respectively.Thedifferentline
The different symbols in Fig. 2 show theunconditional
typesineachpanelcorrespondtohaloesofdifferentmassranges
(mass binsincrease insizebyfactors of two). The moremassive mass function at different output times in the GIF simula-
haloesatagivenredshifttypicallyhavelargervalues ofb(k). tionsplottedasafunctionofthescaledvariableν.Thedot-
ted line shows the Press–Schechter formula for νf(ν); note
the discrepancy at both high and low ν. The dot-dashed
line,whichfitsthedataslightlybetterthanthedashedline,
averaged halo and mass correlation functions, obtained us-
is the mass function computed using the Zeldovich approx-
ing a counts-in-cells procedure (Mo & White 1996; Sheth
imation by Lee& Shandarin (1998) (see AppendixA).The
& Lemson 1999). Others haveused the ratio of thecorrela-
solid line through the data points shows a modification to
tionfunctionsdirectly(Jing1998;Porciani,Catelan&Lacey
thePress–Schechter function. It has the form:
1998). Here, we will compute the bias factor by measuring
the ratio of the power spectra of the haloes and the dark 1 ν′ 1/2 e−ν′/2
matter. On the large scales on which the peak background νf(ν)=A 1+ ν′p 2 √π , (10)
split should apply, all these measures should be equivalent. (cid:16) (cid:17) (cid:18) (cid:19)
Indeed, the extent to which these procedures all agree is a whereν′ =aν,witha=0.707andν =(δc/σ)2asdescribed
measureofthelinearityofthelargescalebiasrelation(equa- in theprevious paragraph, p=0.3, and A is determined by
tion 6),andtheaccuracy oftheassumption thattheeffects requiringthattheintegraloff(ν)overallν giveunity.The
of stochasticity on thebias relation are trivial toestimate. original Press–Schechter formula has a = 1, p = 0, and
Wewillbeginourcomparisonwithsimulationsbyshow- A=1/2.
ing the ratios of the halo and dark matter power spectra. Recallthat,inprinciple,theshapeofνf(ν)candepend
ThevariouspanelsinFig.1showthesquarerootofb2(k) ontheshapeoftheinitialpowerspectrum,andonredshift.
≡
Phalo(k)/Pmatter(k)forhaloesidentifiedatzform =0,2,and Sincethedifferentsymbolsineachpaneloverlapeachother
4, at which time both Phalo(k) and Pmatter(k) were com- reasonablywell,Fig.2showsthat,intheGIFsimulationsat
puted. The various line types in each panel correspond to least, it isa good approximation tosay thatthemass func-
haloes with mass in different mass ranges. The lowest mass tions scale similarly to the scale free case. So, although in
halo we consider has twenty particles, and the mass bins principlewecouldhavealloweda(z,Ω0,Λ0)andp(z,Ω0,Λ0)
increase in size by factors of two. Thus, the lower limit of in equation (10) (in fact, we could even have allowed a dif-
the mth bin is at 10 2m particles per halo (m > 0). One ferent functional form for each output time of each model),
×
consequence of this is that the highest redshift outputs of Fig. 2 shows that, in practice, equation (10) with the same
the GIF simulations are only able to probe high values of valueofpprovidesareasonablygooddescriptionofthemass
b(k), whereas lower redshift outputs primarily probe lower functionatalloutputtimes.Thissuggeststhatthedynam-
values of b(k). ics of collapse is sensitive to ν, and not to the mass scale
Anoptimistwouldarguethatthefigureshowsthatb(k) itself. That the same value of p is a good approximation to
is approximately independent of k for small k, though this νf(ν) in all three panels is presumably a consequence of
approximation is more accurate when b(k) 1 than other- thefactthattheshapesoftheinitialpowerspectrainthese
∼
wise. Atagivenredshift,thesmall k valueofb(k)increases models are not very different. Therefore, in what follows,
with halo mass. If the model presented in the previous sec- rather than findingbest fit shapes to νf(ν) at each output
(cid:13)c 0000RAS,MNRAS000,000–000
4 Ravi K. Sheth & Giuseppe Tormen
Figure3.ThelargescaleEulerianbiasrelationatzobs=zform betweenhaloeswhichareidentifiedatzform,andthemass.Solidcurves
showtherelationwepredictusingtheGIFmassfunction,dottedcurvesshowtherelationswhichfollowfromthePress–Schechtermass
function,anddot-dashedcurvesshowtherelationsassociatedwiththeZeldovichmassfunction.
timeofeachmodel,wewillsimplyusethisonefunctionwith exceptfor thefactor of a.Sincea<1, ourformula predicts
thegiven constants a and p. that massive haloes are slightly less biased than the origi-
This fortuitous rescaling of the mass function is use- nalPress–Schechterbasedformulapredicts.Forlessmassive
ful because it means that we need compute the peak- haloes the extra term cannot be ignored. It is positive for
background split limit associated with the unconditional all ν1, so our formula predicts that less massive haloes are
mass function only once, rather than having to compute it more positively biased (or less anti-biased) than the Mo &
for each output time of each model. Simply set ν ν10 in White formula suggests. The bias in Eulerian space is got
→
theexpression above, where bysubstituting this expression intoequation (7):
ν10 ≡ σ[δ2c((mz11))−−δσc2((zM0)]02) ≈ (δ1−σ12δ0)2 ≈ν1(cid:16)1− 2δδ10(cid:17), bGEuIFl(m1,δ1)=1+ aν1δ1−1 + 1+2p(/aδν11)p. (12)
with δ1 δc(Ω1), where Ω1 Ω(z1), σ1 σ(m1) scaled Both the trends discussed above remain true in Eulerian
using lin≡ear theory to z1 and≡ν1 (δ1/σ1≡)2, and require space.
≡
that f(m1 M0)dm1=f(ν10)dν10. ThedifferentsymbolsinFig.3showthelargescalebias
|
Oncetheunconditionalandconditional massfunctions relation for haloes that are identified at z and are ob-
form
are known, the peak background split bias in Lagrangian servedatz =z .Thatis,theyshowthesmallkvalues
obs form
space is easy to compute. It is of b2(k) obtained as the value predicted at k = 2π/L by a
linear least-square fit to the12 left-most data points. Error
δhL(1|0)≈ aν1−1+ 1+(2apν1)p δδ01 =bGLaIgF(m1,δ1)δ0,(11) barsshowtheformalleast-squareerror.Inthiswaywetried
(cid:20) (cid:21) toestimatetheactuallarge-scalebiasevenwhenb(k)onthe
where the final expression defines bGIF. When a = 1 and largestscalesprobedbythesimulationsshowsadependence
Lag
p=0 the mass function has the Press–Schechter form, and on k. One might argue that the weak scale dependence of
this formula reduces to the one given by Cole & Kaiser b(k) in Fig. 1 means that the symbols in Fig. 3 probably
(1989) and Mo & White (1996). However, the GIF mass slightly under/overestimate the true large scale (small k)
function has a=0.707 and p=0.3. Since ν1 increases with valuefor small/large values of b(k).
increasinghalomass, thefinaltermintheexpressionabove Each panel shows four sets of curves, corresponding to
isnegligible formassive haloes. Inthislimit,theexpression haloes which formed at z = 4, 2, 1, and 0. The three
for
aboveresemblestheoriginalPress–Schechterbasedformula, curves for each redshift show how the bias relation com-
(cid:13)c 0000RAS,MNRAS000,000–000
Large scale bias and the peak background split 5
Figure 2. The unconditional mass functions from five different Figure 4.The largescaleEulerianbias relationof the previous
outputtimes(filledtriangles,opentriangles,opensquares,filled figure,rescaledasindicatedbytheaxislabels.Solidcurvesshow
circles,open circlesshow results forz=0, z=0.5,z=1,z =2 therelationwepredictusingtheGIFmassfunction, dot-dashed
and z = 4) in the GIF simulations plotted as a function of the curves show the corresponding relation predicted using the Zel-
scaledvariableν.DottedcurveshowsthePress–Schechterpredic- dovichmassfunction,anddottedcurvesshowtherelationwhich
tion, dot-dashed curve shows the mass function associated with followsfromthePress–Schechter massfunction.
theZeldovichapproximation,andsolidcurveshowsourmodified
fittingfunction.
Sincethesediscrepancies aresmall, andsinceourmeasured
values of b(M) at small M when z = 0 are very similar
puted using our peak background split model depends on to those in Fig. 3 of Jing (1998), it appears that using the
the unconditional mass function. Solid and dotted curves ratio of the power spectra to define b gives nearly the
Eul
show equation (12) for the GIF (a = 0.707 and p = 0.3) sameresultasusingtheratioofthecorrelationfunctions.As
and the Press–Schechter (a= 1, p= 0) mass functions, re- discussedattheendoftheprevioussection,thisisexpected
spectively. The dot-dashed curves show the bias associated if thebias is independentof scale.
with the Zeldovich mass function (equation A3). As with Supposethattheunconditionalmassfunctionνf(ν)re-
themassfunction,thedot-dashedcurvesfitthedatabetter ally was independent of z (recall that our Fig. 2 shows
form
than the dotted curves, and slightly worse than the solid this is a good approximation). Then our peak background
ones. The solid curves fit the data reasonably well for all splitargumentsaysthat,afterappropriatetransformations,
masses and redshifts. it should bepossible to present thelarge scale bias relation
There are small discrepancies between our GIF based as function of ν only. Namely, equation (12) suggests that
predictions andthesymbols at large and small b(M). Since we should be able to scale all the results for the different
theyareinthesensediscussedearlier,wesuspectsomeofthe zform above to one plot of the product (δ1bLag) versus ν.
discrepancy arises from the weak scale dependence of b(k). Fig. 4 shows the result. The symbols show the simulation
(cid:13)c 0000RAS,MNRAS000,000–000
6 Ravi K. Sheth & Giuseppe Tormen
Figure 5. The square root of the ratio of the halo and matter
power spectra for haloes that formed at zform = 4 (top panels)
andzform=2(bottom panels)butwereobservedatzobs=0.
data from the same output times as before. They were ob-
tainedbytransformingthelargescaleEulerianbiasrelation
asindicated bythelabels on theaxes. Thecurvesshow the
theoreticalpredictionsforthethreedifferentmassfunctions
we havebeen considering.
There are two points to be made about this plot. The
first is that our peak background split model (solid curves)
provides a reasonable description of the results. The main
reason for including this plot is to show that the scatter
aboutthesolidcurvesinourplotsisapproximatelythesame
as the scatter about the mass function fits in Fig. 2. As we
argued above, we have no a priori reason to expect such a
scalingifourrelation betweentheunconditionalmassfunc-
tion and theEulerian bias were not accurate.
Finally,wecanalsostudythecaseinwhichhaloesform
at high redshift, but are observed at lower redshift: z >
form
z . In this case, the large scale Eulerian bias should be
obs
described byour equations (7) and (12), butwith Figure 6. The large scale Eulerian bias relation at zobs = 0
δ2(Ω ) D(z ) for haloes that formed at zform. Solid curves show the relation
ν1 ≡ cσ2(fmorm) and δ1 ≡ D(zfoobrms)δc(Ωform) wcuervpersedsihcotwustihnegrtehlaetiGonIFs wmhaiscshffuonllcotwionfr,odmotttheedPanredssd–oStc-hdeacshhteedr
andZeldovichmassfunctions,respectively.
where D(z) is the linear theory growth factor, and σ(m) is
scaled usinglinear theory to its value at z .
form
Fig. 5 shows the ratio of the halo and matter power therefore,thatourformulaforthebiasrelationisreasonably
spectraforhaloeswhichformed atzform >0butwerestud- accurate in thistwo epoch context as well.
iedatz =0.Thus,itissimilartoFig.1,exceptthatthere
obs
z = z . There are a few obvious differences: some of
obs form
theb(k)curvesarenotparticularly flat,evenon thelargest 4 DISCUSSION
scales. Those curves that do asymptote nicely (at small k)
do so at a value of b(k) that is closer to unity than when Collapsedhaloesarebiasedtracersofthedarkmatterdistri-
zobs = zform. Moreover, the scale dependence of b(k) is dif- bution.Thisbiasdependsonhalomass.Ingeneral,toquan-
ferent from before: whereas previously b(k) for these haloes tifythisbiasrequiresadetailedknowledgeofthemergerhis-
increased with increasing k, now it decreases. tories of dark matter haloes. We used thepeak background
Estimatingthelarge-scalevalueofb(k)byaleast-square splittoarguethat,onlargescales,thisdetailedinformation
fit as done for Fig. 3 yields the symbols shown in Fig. 6. is unnecessary: knowledge of only the shape of the uncon-
Most of the symbols are well fit by our simple formula for ditional mass function is sufficient to compute a good ap-
the large scale bias relation. The symbols that are not well proximation to the large scale bias relation. This fact was
fitbyourformula(typicallythoseinthehighestmassbins) anticipated by Sheth & Lemson (1999), but seems to have
almostalwayscorrespondtohaloesinamassrangeinwhich been unnoticed before.
the power spectra were not particularly flat. We conclude, Forexample,ourapproachisdifferentfrom,butconsis-
(cid:13)c 0000RAS,MNRAS000,000–000
Large scale bias and the peak background split 7
tentwith,therecentworkofJing(1998)andPorciani,Cate- shape, and are in the process of developing and testing the
lan & Lacey (1998). Following Sheth & Lemson’s demon- detailedpredictionsofsuchamovingbarriermodelfurther.
stration that the bias in Lagrangian space was not well de-
scribed by standard models, Porciani et al. quantified the
discrepancy. They showed that using equation (7) to trans-
ACKNOWLEDGMENTS
form Jing’s (1998) fittingformula fortheEulerian biaspro-
videsareasonablygoodfittotheLagrangianbiasfactor.In
GTacknowledges financialsupport from apostdoctoral fel-
otherwords,theirresultsshowthatequation(7)isrelatively
lowship at the Astronomy Department of the University of
accurate. They then provided fitting functions for the La-
Padova. We thank Simon White and the Virgo consortium
grangian bias relation. However, since we have argued that
for providing usaccess to theGIF simulations.
theLagrangian biasisrelatedtothehalomassfunction,we
havechosen to providea fit to themass function instead.
In so doing, we showed that a simple modification
(equation 10) of the Press–Schechter formula provides a REFERENCES
good fit to the unconditional halo mass function in the
Bardeen J.M., Bond J.R., Kaiser N., Szalay A., 1986, ApJ, 304,
SCDM, OCDM and ΛCDM models we studied (Fig. 2).
15
This allowed us to derive a single, simple, physically moti-
BaughC.M.,BensonA.J.,ColeS.,FrenkC.S.,LaceyC.G.,1998,
vatedformula forthelarge scale biasrelation (equation 12)
MNRAS,submitted, astro-ph/9811222
that can be used for haloes of all masses at all times in Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379,
all three cosmologies. Comparison of the predictions of our 440
model with numerical simulations showed good agreement. CatelanP.,LucchinF.,MatarreseS.,PorcianiC.,1998,MNRAS,
We showed that this was true for the case in which haloes 297,692
are observed at the same time as when they are first iden- Efstathiou G., Frenk C., White S. D. M., Davis M., 1988, MN-
tified (z = z ), and also for the case in which haloes RAS,235,715
obs form
HaehneltM.G.,NatarajanP.,ReesM.J.,MNRAS,300,817
form at z , evolve, and are observed at some later time
form
HaimanZ.,LoebA.,1998,astro-ph/9811395
z <z .
obs form JingY.,1998,ApJL,503,L9
If haloes are significantly more/less massive than the
KauffmannG.,ColbergJ.M.,DiaferioA.,White S.D.M.,1998a,
typicalM∗ haloatthetimeofformation,thenourformulae MNRAS,submitted, astro-ph/9805283
predict that they should be less/more biased than results KauffmannG.,ColbergJ.M.,DiaferioA.,WhiteS.D.M.,1998b,
based on the Press–Schechter formula would predict. Since MNRAS,submitted, astro-ph/9809168
galaxiesarethoughttoforminsmall,i.e.,M/M∗(zform)<1, LaceyC.,ColeS.,1993,MNRAS,262,627
haloes (Kauffmannet al. 1998b; Baugh et al. 1998), our re- LaceyC.,ColeS.,1994,MNRAS,271,676
sults are useful for galaxy formation models. The first gen- LeeJ.,ShandarinS.,1998,ApJ,500,14
eration of stars and high redshift quasars are thoughtto be MoH.J.,WhiteS.D.M.,1996, MNRAS,282,347
MoH.J.,JingY.,WhiteS.D.M.,1997,MNRAS,284,189
associated withhaloeshavingM/M∗(zform) 1(Haehnelt,
≫ PorcianiC.,CatelanP.,LaceyC.,1998,ApJL,submitted,astro-
Natarajan & Rees 1998; Haiman & Loeb 1998), so our re-
ph/9811477
sults(forthenumberdensityaswellasthebiasfactor)may
PressW.,Schechter P.,1974, ApJ,187,425
also be useful for studies of the reionization history of our
Sheth R. K., Lemson G., 1999, MNRAS, accepted, astro-
Universe. ph/9808138
Although we showed how the bias relation depends on TormenG.,1998,MNRAS,297,648
the unconditional mass function (for example, the bias re-
lation associated with the simple Press–Schechter spherical
collapsemassfunctionisdifferentfromthatassociatedwith
APPENDIX A: LARGE SCALE BIAS AND THE
the Zeldovich mass function), we did not provide a deriva-
ZELDOVICH APPROXIMATION
tion oftheshapeofthemassfunction itself. Sincethemass
function in the GIF simulations is different from those pre-
The universal unconditional mass function associated with
dictedbysimpleapplicationsofthesphericalcollapsemodel
theZeldovich approximation is given by equation (9) with
or by the Zeldovich approximation, our model is still far
′ ′ ′
fromcomplete.Wehopethatourresultswillstimulatework νf(ν)=Za(ν )+Zb(ν )+Zc(ν ),
aimed at rectifying this.
where
In this regard, we think it important to stress the fol-
lcorwosisnigngfamcto.dBeolntoddeteraivl.e(t1h9e91P)reussse–dScahreacnhdteormunwcaolnkdbitairorniearl Za(ν′) = 245 10πν′ 53ν′ − 112 exp −52ν′ erfc √2ν′
mass function from the statistics of the initial fluctuation r (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
field. In their model, the Press–Schechter mass function is ′ 25 15ν′ 15ν′ 3ν′
Zb(ν ) = exp − erfc
associated withthefirstcrossingdistribution ofabarrier of 16 π 4 4
r (cid:18) (cid:19) r !
constant height. The barrier height is constant because, in
the spherical collapse model, there is a critical overdensity Zc(ν′) = 125√5 ν′ exp −9ν′ ,
δc requiredforcollapse,andthisvalueisindependentofthe − 12 π (cid:18) 2 (cid:19)
mass of the collapsed object. If this same barrier crossing
and
model is to yield the GIF mass function, then the barrier
shapemustdependonmass.Wehavesolvedforthisbarrier ν′=(λc/δc)2 ν =(λc/δc)2 (δc/σ)2 (A1)
(cid:13)c 0000RAS,MNRAS000,000–000
8 Ravi K. Sheth & Giuseppe Tormen
(Lee & Shandarin 1998). In these variables, the mass func-
tion has two free parameters: the usual spherical collapse
δc, and the ratio (λc/δc). The dot–dashed curve in Fig. 2
showsthisfunctionwiththevalueusedbyLee&Shandarin:
λc=0.38.
One way to approximate the associated conditional
massfunction istousethepeakbackgroundsplit discussed
in themain text.In this approximation, one simply sets
ν′= (λσc12−λσc20)2 ≈ν1′(1−δc0/δc1)2,
1− 0
intotheexpressionforf(ν)dν above.Althoughwehavenot
donesohere,inprinciple,onecouldalsoallowλc(z,Ω0,Λ0).
In our case, the associated Lagrangian large scale bias is
given by equation (4), so
′
δhL(1|0) ≈ δδ01 "5ν1′ −1−2Za(ν1′)202ν01′ν−1 1
5 ′ ′ ′
+4 2ν1Zb(ν1)−Zc(ν1)
#
(cid:16) (cid:17)
≡ bZLealg(m1,δ1)δ0. (A2)
TheEulerianspacepeakbackgroundsplitbiasfactorforthe
Zeldovichapproximationisgivenbyinsertingthisexpression
for bLag in equation (7):
bZEeull(m1,δ1)=1+bZLealg(m1,δ1). (A3)
(cid:13)c 0000RAS,MNRAS000,000–000