Table Of ContentLA-UR-05-9198
DRAFTVERSIONFEBRUARY4,2008
PreprinttypesetusingLATEXstyleemulateapjv.26/01/00
CAPTURINGHALOSATHIGHREDSHIFTS
KATRINHEITMANN1,ZARIJALUKIC´2,SALMANHABIB3,ANDPAULM.RICKER2,4
1ISR-1,ISRDivision,TheUniversityofCalifornia,LosAlamosNationalLaboratory,LosAlamos,NM87545
2Dept.ofAstronomy,UniversityofIllinois,Urbana,IL61801
3T-8,TheoreticalDivision,TheUniversityofCalifornia,LosAlamosNationalLaboratory,LosAlamos,NM87545
4NationalCenterforSupercomputingApplications,Urbana,IL61801
DraftversionFebruary4,2008
ABSTRACT
6 WestudytheevolutionofthemassfunctionofdarkmatterhalosintheconcordanceΛCDMmodelathighred-
0 shift. Weemployoverlapping(multiple-realization)numericalsimulationstocoverawiderangeofhalomasses,
0 107- 1015h- 1M⊙,withredshiftcoveragebeginningatz=20. ThePress-Schechtermassfunctionissignificantly
2 discrepantfromthesimulationresultsathighredshifts.Ofthemorerecentlyproposedmassfunctions,ourresults
areinbestagreementwithWarrenetal. (2005). Thestatisticsofthesimulations–alongwithgoodcontrolover
n
a systematics–allowforfitsaccuratetothelevelof20%atallredshifts.Weprovideaconcisediscussionofvarious
J issuesindefiningandcomputingthehalomassfunction,andhowtheseareaddressedinoursimulations.
1 Subjectheadings:methods:N-bodysimulations—cosmology:halomassfunction
1
1 1. INTRODUCTION carried out over differentmass and redshiftranges. The clos-
v est to the presentwork are Reed et al. (2003)and Springelet
Dark matter halos occupy a central place in the paradigm
3 al. (2005);incomparisontotheirresults, ourhalomassrange
of gravitationally-driven structure formation arising from the
3 goesdeeperbythreeordersofmagnitude,withgoodstatistics
nonlinearevolutionofprimordialGaussiandensityfluctuations.
2 and control of systematics out to z=20, substantially higher
Gascondensation,resultantstarformation,andeventualgalaxy
1 than in these papers. (We review results from other work be-
0 formation occurswithin halos. Consequently, the halo profile
low.)Essentially,theearlierresultsareinverygoodagreement
6 andmassfunctionarecentralingredientsinphenomenological
withtheSheth-Tormenmassfunction(Sheth&Tormen1999),
0 modelsofnonlinearclusteringofgalaxies. Thedistributionof
atredshiftsz≤10.Asweshowbelow,variousfittingformulae
/ halomasses–thehalomassfunction–anditstimeevolution,
h givenintheliterature–mosttunedtosimulationresultsatz=0
arealsosensitiveprobesofcosmology.
p – can differsubstantially in their predictionsathigh redshifts,
The halo mass function at the high-mass end (cluster mass
- byasmuchasafactoroftwo.Therefore,itisimportanttocarry
o scales)isexponentiallysensitivetotheamplitudeoftheinitial
r density perturbations,the meanmatter density parameter,Ω , outsimulationsofsufficientdynamicrangeandaccuracytotest
t m thesepredictions.
s andtothedarkenergycontrolledlate-timeevolutionoftheden-
a In order to extractthe mass function from simulations, dif-
sity field. Thelastfeature,particularlyatlowredshifts,z<2,
: ferentquestionshavetobeaddressed,suchas:Howisthemass
v allowsclusterobservationstoconstrainthedarkenergycontent,
Xi ΩΛ,andtheequationofstateparameter,w(Holderetal. 2001). fuulantciotino?ntWohbeendmefiunsetdt?heWsihmeunldatoiotnhebfierssttahrtaeldosinfoorrmdeirntoasciamp--
Thehalomassfunctionisalsoofconsiderableinterestathigh
r ture these halos? What force and mass resolution is required
a redshift, relating to questions such as predictions for quasar
to capture halos of a certain mass at a specific redshift? We
abundanceandformationsites(Haiman&Loeb2001),thefor-
havederivedandtestedcertaincriteriatoensurethatoursimu-
mation history of collapsed baryonic halos, and the reion-
lationscapturethehalosofinterest; detailswillbegivenelse-
ization history of the Universe (Furlanettoetal. 2005). Re-
where(Lukic´etal. 2006).
cent results from the Wilkinson Microwave Anisotropy Probe
The paper is organizedas follows. In Section 2 we discuss
(WMAP)(Kogutetal. 2003;Spergeletal. 2003)indicatethat
popularmassfunctionformulae,previouswork,andstrategies
reionization could have begun at redshifts as high as z∼20.
for determiningthe halo mass function at high redshifts. The
Much of the work on possible reionization scenarios is based
simulations and mass function results are discussed in Sec-
on the simple Press-Schechter (PS) mass function (Press &
tion3.Criteriaformassandforceresolutionandinitialredshift
Schechter 1974, Bond et al. 1991) the use of which can lead
needed to span the desired mass and redshift range are given
toimprecisepredictionsforthereionizationhistory.
here.WepresentourconclusionsinSection4.
Simulations play a dual role in characterization of the halo
mass function. If only a few fixed sets of cosmological pa-
2. THEMASSFUNCTION
rameters and a finite dynamic range are required, simulations
can producevaluable results. In order to investigatea variety Overthelastthreedecadesdifferentfittingfunctionsforthe
of cosmologiesanddifferentscenariosfor physicalprocesses, mass function have been suggested. The first analytic model
e.g.,reionization,itisneverthelessveryconvenient,ifnotnec- for the mass function was developed by Press & Schechter
essary, to have accurate analytic fitting relations. Simulations (1974). Their theoryconsidersa sphericallyoverdenseregion
canbeusedtovalidatethesefitsoverawide(albeit,discretely in an otherwise smooth background density field. The over-
sampled)rangeofparameters. density evolves as a Friedmann universe with positive curva-
Various numerical studies of the mass function have been ture. Initially, the overdensity expands, but at a slower rate
thanthebackgrounduniverse(thusenhancingthedensitycon-
1
2 CapturingHalosatHighRedshifts
trast), until it reaches the ‘turnaround’ density, after which it The determination of mass functions at high redshifts is a
collapses. Althoughformally this collapse ends with a singu- nontrivialtask. Highredshifthaloshaveverylowmasses,plac-
larity,itisassumedthatinrealitytheoverdenseregionwillviri- ingheavydemandsonthemassandforceresolutionneededto
alize. ForanEinstein-deSitteruniverse,thedensityofsuchan resolvethem.Theserequirementscanbeachievedintwoways.
overdenseregionatthevirializationredshiftisρ≈180ρ (z).At First, a simulation with a very large number of particles and
c
thispoint,thedensitycontrastfromthelineartheoryofpertur- highforceresolutioncanbeperformed. Thisisexpensive,and
bationgrowth[δ(~x,z)=D+(z)δ(~x,0)]isδ (z)≈1.686(1+z).For onlyaverylimitednumberofsuchsimulationscanbecarried
c
Ω <1,δ (z)evolvesdifferently(seeLacey&Cole1993),but out. Second, since determining the mass function is simply a
m c
the dependenceoncosmologyis weak(see e.g., Jenkinsetal. questionofstatistics, manyrelativelymodestsimulationswith
2001).Thus,weadoptδ =1.686=δ (0). moderateparticleloadingcanbeperformed:thisisthestrategy
c c
Followingtheabovereasoningandwiththeassumptionthat weadopthere. Assimulationscanonlybetrusteduntila red-
theinitialdensityperturbationsaregivenbyaGaussianrandom shiftatwhichthelargestmodeisclosetobecomingnonlinear,
field,thePSmassfunctionisgivenby: multipleoverlappingboxsizesmustbeused.
Springel et al. (2005)have recently followed the evolution
f (σ)= 2δcexp - δc2 , (1) of 21603 particles in a 500h- 1Mpc box. The high mass and
PS rπ σ (cid:20) 2σ2(cid:21) forceresolutionallowthemtostudythemassfunctionreliably
where σ is the variance of the linear density field, f(σ,z) ≡ out to a redshift of z=10, covering a mass range of roughly
(MU/ρsibn)g(denm/pdilrnicσa-l1a)r,gaunmdeρnbtsisSthheethba&ckTgorromunend(d1e9n9s9it,yh.ereafter l1a0ti1o0hn-s1iMnc⊙lutdoe1J0a1n6gh--C1Mon⊙d.elElx&amHperlensqoufissti(n2g0l0e1s)m(a1lhl-- b1Moxpscimboux-
ST)proposedanimprovedfitofthefollowingform: with 1283 particles evolved to z=10) and Cen et al. (2004)
(4h- 1Mpc box, 5123 particles, evolved to z = 6). Results in
f (σ)=0.3222 2aδcexp - aδc2 1+ σ2 p , (2) both papers are claimed to be consistent with PS but without
ST rπ σ (cid:20) 2σ2(cid:21)(cid:20) (cid:18)aδ2(cid:19) (cid:21) detailed quantification. The simulation of Reed et al. (2003)
c
is a compromise between the two extremes: a box size of
witha=0.707,and p=0.3. Shethetal. (2001)interpretedthis
50h- 1Mpc with 4323 particles and a concomitant halo mass
fittheoreticallybyextendingthePSapproachtoanellipsoidal
rangeof roughly1010h- 1M to 1014.5h- 1M . Reed etal. find
collapsemodel.Inthismodel,thecollapseofaregiondepends ⊙ ⊙
goodagreement(betterthan20%)withtheSTfituptoz≃10.
notonlyon its initialoverdensity,butalso on the surrounding
For higher redshifts they find that the ST fit overpredicts the
shearfield.ThedependenceischosentorecovertheZel’dovich
number of halos, at z=15 up to 50%. At this high redshift,
approximation(Zel’dovich1970) in the linear regime. A halo
however,theirresultsbecomestatistics-limited,themassreso-
isconsideredvirializedwhenthethirdaxiscollapses(seealso
lutionbeinginsufficienttoresolvetheverysmallhalos.
Lee&Shandarin1997).
In this paper we analyze a suite of 50 N-body simulations
Jenkins et al. (2001, hereafter Jenkins) combine high res-
withvaryingboxsizesbetween4h- 1Mpcand126h- 1Mpcwith
olutionsimulationsfordifferentcosmologiesspanninga mass
rangeofoverthreeordersofmagnitude[∼(1012- 1015)h- 1M⊙], multiplerealizationsofallboxestostudythemassfunctionat
and includingseveralredshiftsbetween z=5 and z=0. They
findthatthefollowingfittingformulaworksexceptionallywell
0.2 Warren et al.
(within20%),independentoftheunderlyingcosmology:
Jenkins et al.
0
fJenkins(σ)=0.315exp - |lnσ- 1+0.61|3.8 . (3) -0.2 PSrheestsh--STcohremchenter z=0
Theaboveformulaisveryclose(cid:2)tothenominalST(cid:3)fit. 9 10 11 12 13 14 15
By performing 16 nested-volume simulations Warren et al. 0.4
0.2
(2005,hereafterWarren)obtainsignificanthalostatisticsspan- 0
ning a mass range of five orders of magnitude [∼ (1010- -0.2 z=5
1015)h- 1M⊙]. Theirbestfitemploysafunctionalformsimilar uals-0.49 10 11 12 13
toanimprovedversionofST(Sheth&Tormen2002): d
si 0
e
fWarren(σ)=0.7234 σ- 1.625+0.2538 exp(cid:20)- 1.1σ9282(cid:21). (4) ve R--00..48 z=10
(cid:0) (cid:1) ati 7 8 9 10 11 12
ThediscrepancybetweenPSandthemoreaccuratefitsisev- el
R
identinFigure1wheretheredshiftevolutionofthemassfunc- 0
tion is shown. The redshift dependence in the analytic mass -0.4
z=15
functionsentersonlythroughσ(z)=σ(0)d(z),whered(z)isthe -0.8
growthfactornormalizedsuchthatd(0)=1. Asthefunctional 7 8 9 10 11
dependenceon σ is differentin the differentfits, this leads to 0
substantialvariationacrossthefitsasafunctionofredshift.For
-0.4
z=0 the Warren fit agrees – especially in the low mass range z=20
-0.8
below1013M –tobetterthan5%withtheSTfit. Atthehigh
⊙ 7 8 9 10
mass end the difference increases up to 20%. The Jenkins fit log(M [M./h])
O
leads to similar results over the considered mass range. Note
FIG. 1.—RelativeresidualsofthePS,Jenkins,andWarrenmassfunction
that at higher redshifts and intermediate mass ranges around
fitswithrespecttoSTforfivedifferentredshifts. Notethattherangesofthe
109M⊙ the disagreement between the Warren and ST fits in- axesaredifferentinthedifferentpanels.WedonotshowtheJenkinsfitbelow
creasesupto40%. massesof1011h- 1M⊙sinceitisnotvalidinthismassrange.
Heitmann,Lukic´,Habib,Ricker 3
redshiftsuptoz=20andtocoveralargemassrangebetween 16h- 1Mpcboxes.Thehalogrowthfunctionisparticularlyvalu-
107h- 1M and1015h- 1M evenathighredshifts.Significantly, able for determining when the halos at a certain mass should
⊙ ⊙
atz=20,gasinhaloswithamassscaleabove∼107h- 1M⊙can firstform. Thisisagoodtestforproblemsinsimulationsaim-
coolviaatomiclinecooling(Tegmarketal. 1997). ingto capturehaloswitha givenminimummassatsome red-
shift. Anexampleofthisisinsufficientforceresolutioninthe
3. SIMULATIONSANDMASSFUNCTIONRESULTS base-gridsofadaptive-mesh-refinement(AMR)codes.
Allsimulationsinthispaperarecarriedoutwiththeparticle- Oncethenumberofparticlesforasimulationanda desired
mesh code MC2 (Mesh-based Cosmology Code). MC2 has massforthesmallesthaloaredecided,therequiredboxsizeis
been extensively tested against other cosmologicalsimulation fixed.Theforceresolutionneededtoresolvethesmallesthalos
codes(Heitmannetal. 2005).Thechosenvaluesofcosmolog- has then to be determined. Our aim here is not to precisely
icalparametersare: measurethe haloprofilebutsimply to becertain thatthe total
Ω =1.0, Ω =0.253, Ω =0.048, halomass is correct. As shownin Heitmannet al. (2005)the
tot m baryon
halomassisarelativelyrobustquantityanda simpleestimate
σ =0.9, H =70km/s/Mpc, (5)
8 0 oftheforceresolutionisallthatisneeded.Theforceresolution
as set by the latest cosmic microwave background and large mustbesmallcomparedtothecomovinghalovirialradiusr∆
scalestructureobservations(MacTavishetal. 2005). Themass (withtheoverdensityrelativetothecriticaldensity,∆∼200)at
transferfunctionsaregeneratedwithCMBFAST(Seljak&Zal- allredshifts.Theresultinginequalitycanbestatedintheform
darriaga 1996). We summarize the different runs, including
δ n Ω(z) 1/3
theirforceandmassresolutioninTable1. f <0.62 h , (6)
Weidentifyhaloswiththefriends-of-friendsalgorithm(FOF), ∆p (cid:18) ∆ (cid:19)
basedonfindingneighborsofparticlesatacertaindistance(see whereδ istheforceresolutionandn isthenumberofparticles
f h
e.g., Einasto et al. 1984; Davis et al. 1985). The halo mass per halo. In the simulations performedhere we use a ratio of
is defined simply by the sum of particles which are members oneparticleper64gridcells,whichallowshaloswithroughly
of the halo. (For connections between differentdefinitions of 50particlestobe captured. Ithasbeenshownin Heitmannet
halo masses, see White 2001.) Despite several shortcomings al. (2005)thatthisratiodoesnotcausecollisionaleffectsand
of the FOF halo finder, e.g., halo-bridging (see, e.g., Gelb & leadstoconsistentresultsincomparisonwithothercodes.Mass
Bertschinger 1994, Summers et al. 1995) or statistical biases functionconvergencetests with differentforce resolutionsare
foundbyWarrenetal.(2005),theFOFalgorithmitselfiswell- nicely consistent with the above estimate as shown in Lukic´
definedandveryfast. etal. (2006); time-stepcriteria andconvergencetests are also
There are two sourcesof possible biases in determiningin- describedthere.
dividual halo masses using FOF. First, the halo may be sam- Thelargesetofsimulationswehavecarriedoutallowsusto
pledwithaninsufficientnumberofparticles(seeWarrenetal. study the mass functionat redshiftsbetween z=20 and z=0.
2005). Second, the effective slope of the halo density profile The main results are shown in Figure 3, where the simulation
close to the virialradiusr , atfixed particlenumber,also in- data for the mass function are shown along with the Warren,
vir
fluences the FOF mass. If the force resolution of the N-body PS,andSTfitsatdifferentredshifts.AtallredshiftstheWarren
code affectsthe profile, this too, addsa systematic bias. Here fithasthebestagreementwiththesimulationswithascatterof
werecordthemassfunctionforthelinkinglengthb=0.2FOF approximately 20% and is a numerically significant improve-
massincludingonlythecorrectionofWarrenetal. (2005).Ina mentoverST.Suchaclosematchisquitegratifyinggiventhe
follow-uppaper(Lukic´etal.2006)wewilladdresssystematics overalldynamicrangeoftheinvestigation.ThePSfit,overthe
issuesindetermininghalomassesindetail. massrangeconsidered,isapoorfitatz≥10,deviatingbymore
Wenowdiscusscriteriafoundtobeveryimportantfordemon- thanafactoroftwofromthenumericalresults.
stratingtheconvergenceandrobustnessofourresults. Details
willbepresentedinLukic´ etal. (2006). Thefirstissuerelates
totheinitialredshiftofthesimulation. Twoconditionsareim- 10000
portant:(i)thesimulationmustbeginsufficientlyearlythatthe 109 - 1010MO./h
initialZel’dovichdisplacementisasmallenoughfractionofthe 1010 - 1011MO./h
meaninterparticleseparation∆p; onaverageaparticleshould 1000 1011 - 1012MO./h
notmovemorethan∼∆ /3;(ii)thehighestredshiftwherethe
p
mass functionis to be evaluatedmust be sufficiently removed halos 100
fifrrosmt cthhaenrceedstohifftoormffihrsatl-ocsr.osTsihnegsztcrrionssgewnhceyreopfatrhteicsleescrhiatevreiathies ber of
m
such that the small boxes require very high starting redshifts, Nu 10
e.g.,the4h- 1Mpcboxhadaninitialredshiftz =500.Thisisa
in
muchearlierstartingredshiftthanthoseusedinprevioussimu-
lations;theconventionalrequirementthatallmodesinthebox 1
be linear at the initial redshiftprovesto be much weaker, and
thereforeinadequate,asaconvergencecriterion.
0.1
A simple test of how well the simulations track the mass 5 10 15 20
Redshift
function formulae is to follow the number of halos in a spec-
ified mass bin at a given redshift. For this purpose we con- FIG. 2.—Halogrowthfunctionforthreemassbinsforthe16h- 1Mpcbox.
TheWarren(solid),ST(long-dashed),andPS(short-dashed)fitsarecompared
vertthemassfunctionfitintoafunctionofz,definingthehalo
tosimulationdatawithPoissonerrorbars. Notethequalityoftheagreement
growth functionas shown in Figure 2. The evolution of three withtheWarrenfit.
massbinsisshownasafunctionofzalongwithresultsfromthe
4 CapturingHalosatHighRedshifts
TABLE1
SUMMARYOFTHEPERFORMEDRUNS
Mesh BoxSize Resolution z z ParticleMass SmallestHalo #ofRealizations
in final
10243 126h- 1Mpc 120h- 1kpc 50 0 9.94×109h- 1M⊙ 3.98×1011h- 1M⊙ 10
10243 64h- 1Mpc 62.5h- 1kpc 80 0 1.30×109h- 1M⊙ 5.2×1010h- 1M⊙ 5
10243 32h- 1Mpc 31.25h- 1kpc 150 5 1.63×108h- 1M⊙ 6.52×109h- 1M⊙ 5
10243 16h- 1Mpc 15.63h- 1kpc 200 5 2.04×107h- 1M⊙ 8.16×108h- 1M⊙ 5
10243 8h- 1Mpc 7.81h- 1kpc 250 10 2.55×106h- 1M⊙ 1.02×108h- 1M⊙ 20
10243 4h- 1Mpc 3.91h- 1kpc 500 10 3.19×105h- 1M⊙ 1.27×107h- 1M⊙ 5
Note.—Massandforceresolutionsofthedifferentruns.Thesmallesthalosweconsidercontain40particles.Allsimulationsarerunwith2563particles.
3
quired,asWhite(2002)pointsout,“itmaynotbesufficientto
useasimpleparametrizedform”inconstrainingcosmological
Warren fit
2 4 Mpc/h parameterswiththemassfunction.
8 Mpc/h Theerrorcontrolcriteriadevelopedinthisworkhaveanat-
1 16 Mpc/h uralapplicationinhigh-resolutionAMRsimulationsintheset-
-3h)]) 0 3624 MMppcc//hh ttiinognoisfirnefipnroegmreesnst.anderrorcontrolcriteria.Workinthisdirec-
c/ 126 Mpc/h
p
M PS fit
M[( -1 ST fit WethankKevAbazajian,DanHolz,LamHui,GerardJung-
man,SavvasKoushiappas,AdamLidz,SergeiShandarin,Ravi
g
o
dl-2 Sheth, and Mike Warren for useful discussions. The authors
dn/ acknowledge support from IGPP, LANL. S.H. and K.H. ac-
log( -3 knowledge support from the DOE via the LDRD program at
LANL. P.M.R. and Z.L. acknowledge UIUC, NCSA, and a
DOE/NNSA PECASE award (LLNL B532720). S.H., K.H.,
-4
and P.M.R. acknowledge the hospitality of the Aspen Center
forPhysicswherepartofthisworkwascarriedout. We espe-
-5 z=20 z=15 z=10 z=5 z=0 cially acknowledge supercomputing support under the LANL
InstitutionalComputingInitiative.
7 8 9 10 11 12 13 14 15
log(M./h]))
O
FIG.3.—Themassfunctionat5differentredshiftswithPoissonerrorbars. REFERENCES
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