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Merger rates in hierarchical models of galaxy formation

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Merger rates in hierarchical models of galaxy formation
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    a  s   t  r  o  -  p   h   /   9   4   0   2   0   6   9   2   4   M  a  y   9   4 Mon.Not.R.Astron.Soc. 000 ,000{000(1994) Mergerratesinhierarchicalmodelsofgalaxyformation.II.ComparisonwithN-bodysimulations  CedricLacey  1 ; 3 andShaunCole  2 ; 4  1 PhysicsDepartment,OxfordUniversity,Nuclear&AstrophysicsBuilding,KebleRd,OxfordOX13RH  2 DepartmentofPhysics,UniversityofDurham,ScienceLaboratories,SouthRd,DurhamDH13LE  3 cgl@thphys.ox.ac.uk 4 Shaun.Cole@durham.ac.uk 24May1994 ABSTRACT  WehavemadeadetailedcomparisonoftheresultsofN-bodysimulationswiththeanalyticaldescriptionofthemerginghistoriesofdarkmatterhalospresentedinLacey& Cole(1993),whichisbasedonanextensionofthePress-Schechtermethod(Bond  etal. 1991;Bower1991).Wendtheanalyticalpredictionsforthehalomassfunction,mergerratesandformationtimestoberemarkablyaccurate.TheN-bodysimulationsused128 3 particlesandwereofself-similarclustering,with=1andinitialpowerspectra  P  ( k  ) /  k  n ,withspectralindices n  =    2 ;   1 ; 0.Theanalyticalmodelishoweverexpectedto applyforabitraryandmoregeneralpowerspectra.Darkmatterhaloswereidentied inthesimulationsusingtwodierentmethodsandatarangeofoverdensities.Forhalosselectedatmeanoverdensities   100   200,theanalyticalmassfunctionwasfoundto provideagoodttothesimulationswithacollapsethresholdclosetothatpredicted bythesphericalcollapsemodel,withatypicalerrorof   <  30%overarangeof10 3 in mass,whichisthefulldynamicrangeofourN-bodysimulations.Thiswasinsensitivetothetypeoflteringused.Overarangeof10 2   10 3 inmass,therewasalsogood agreementwiththeanalyticalpredictionsformergerrates,includingtheirdependenceonthemassesofthetwohalosinvolvedandthetimeintervalbeingconsidered,andforformationtimes,includingthedependenceonhalomassandformationepoch.TheanalyticalPress-Schechtermassfunctionanditsextensiontohalolifetimesand mergerratesthusprovideaveryusefuldescriptionofthegrowthofdarkmatterhalosthroughhierarchicalclustering,andshouldprovideavaluabletoolinstudiesoftheformationandevolutionofgalaxiesandgalaxyclusters. Keywords: galaxies:clustering{galaxies:evolution{galaxies:formation{cosmol-ogy:theory{darkmatter 1INTRODUCTION  Inthestandardcosmologicalpicture,themassdensityoftheuniverseisdominatedbycollisionlessdarkmatter,and structureinthiscomponentformsbyhierarchicalgravita-tionalclusteringstartingfromlowamplitudeseeductua-tions,withsmallerobjectscollapsingrst,andthenmerg-ingtogethertoformlargerandlargerobjects.Thehalosofdarkmatterformedinthisway,objectswhichareinapprox-imatedynamicalequilibrium,formthegravitationalpoten-tialwellsinwhichgascollectsandformsstarstoproducevisiblegalaxies.Subsequentmergingofthedarkhalosleadstoformationofgroupsandclustersofgalaxiesboundto-getherbyacommondarkhalo,andisaccompaniedbysomemergingofthevisiblegalaxieswitheachother.Itisobvi-ouslyofgreatinteresttounderstandthisprocessofstructureformationviamerginginmoredetail.Oneapproach,begun byAarseth,Gott&Turner(1979)andEfstathiou&East-wood(1981),istocalculatethenon-linearevolutionofthedarkmatternumerically,usinglargeN-bodysimulations.A secondapproach,complementarytotherst,istodevelop approximateanalyticaldescriptionswhichrelatenon-linearpropertiessuchasthemassdistributionandmergingprob-abilitiesofcollapsedobjectstotheinitialspectrumoflineardensityuctuationsfromwhichtheygrew.Theseanalyticaldescriptionsmustthenbetestedagainsttheresultsofthenumericalsimulations.Iftheywork,theyprovideinsightintothenumericalresults,andprovidethebasisforsim-pliedcalculationsandmodellingwhichcancoveramuch   2  C.LaceyandS.Cole  widerrangeofparameterspacethanisfeasiblewiththenumericalsimulationsalone.TheanalyticalapproachwaspioneeredbyPress& Schechter(1974)(hereafterPS),whoderived,ratherheuris-tically,anexpressionforthemassspectrumofcollapsed,virializedobjectsresulting,viadissipationlessgravitationalclusteringofinitiallycoldmatter,frominitialdensityuctu-ationsobeyingGaussianstatistics.Thebasisofthemethod istoderiveathresholdvalueofthelinearoverdensityforcollapseofsphericalperturbations,andthencalculatethefractionofmassinthelineardensityeldthatisabovethisthresholdwhensmoothedonvariousscales.ThePSmassfunctionformulahassincebeenwidelyappliedtoavarietyofproblems,includinggravitationallensingbydark halos(Narayan&White1988),abundanceofclustersand theirinuenceonthecosmicmicrowavebackgroundviatheSunyaev-Zel'dovicheect(Cole&Kaiser1988),andgalaxyformation(Cole&Kaiser1989;White&Frenk1991).An alternativederivationofthePSmassfunctionwaspresented byBond  etal. (1991)(hereafterBCEK),who,byconsider-ingtherandomwalkoflinearoverdensity(ataxedloca-tion)asafunctionofsmoothingscale,obtainedarigoroussolutiontotheproblemofabove-thresholdregionslyingin-sideotherabove-thresholdregions(theso-called\cloud-in-cloud"problem).BCEKalsoshowedhowthederivedmassfunctiondependedonthelterusedtodenethespatialsmoothing,andthatthestandardPSformulainfactonlyre-sultsinthecaseof\sharp  k -space"ltering,whichisalsotheonlycaseforwhichexactanalyticalresultscanbeobtained.(RelatedapproximateresultswerealsoobtainedbyPeacock &Heavens(1990).)Inaddition,BCEKshowedhowtogobeyondacalculationofthemassfunctionatasingletime,toderivetheconditionalmassfunctionrelatingthehalomassesattwodierenttimes.Indepedently,Bower(1991)extendedtheoriginalmethodusedbyPressandSchechterandderivedanidenticalexpressionfortheconditionalmassfunction.ThisextensiontothePStheorywasthentaken upbyLacey&Cole(1993)(hereafterPaperI),whoused theconditionalmassfunctiontoderivearangeofresultsonthemergingofdarkhalos.Theseincludedtheinstan-taneousmergerrateasafunctionofthemassesofboth halosinvolved,andthedistributionofformationandsur-vivaltimesofhalosofagivenmassidentiedatagiven epoch,theformationtimebeingdenedastheearlierepoch whenthehalomasswasonlyhalfofthatattheidenti-cationepoch,andthesurvivaltimeaswhenthehalomasshasgrowntotwicethatattheidenticationepoch.Theex-pressionsforthesedependontheinitiallineaructuationsthroughtheirpowerspectrum,andonthebackgroundcos-mology(densityparameterandcosmologicalconstant).Preliminaryapplicationsoftheseresultsweremadetocon-strainingthevalueoffromthemergingofgalaxyclusters,andtoestimatingtherateofaccretionofsatellitesbythedisksofspiralgalaxies.Kaumann&White(1993)haveextendedtheutilityoftheanalyticalresultsofBCEKandBower(1991)bypre-sentingaMonteCarlomethodofgeneratingmergertrees,decribingtheformationhistoryofhalos,thatareconsistentwiththeanalyticalconditionalmassfunction.Thistech-niquehassubsequentlybeenutilizedinstudiesofgalaxyformationbyKaumann,White&Guiderdoni(1993)and Kaumann,Guiderdoni&White(1994).AnalternativeMonteCarloimplementationhasalsobeenusedinstudiesofgalaxyformationinCole etal. (1994).Analternativeanalyticalapproachistoassumethatobjectsformfrompeaksintheinitialdensityeld.Thishasbeenextensivelyusedtostudyclusteringofgalaxiesand clusters(Peacock&Heavens1985;Bardeen  etal. 1986),buthasbeenlessusedincalculatingmassfunctionsbecauseoftheproblemofdealingproperlywithpeakswhichlieinsideotherpeaks,andofidentifyingwhatmassobjectformsfrom agivensizepeak( e.g. Bond(1988)).Bond&Myers(1993a)haverecentlydevelopedanewmethodwhichcombinesas-pectsofthePress-Schechterandpeaksmethods,butthisrequiresonetogenerateandanalyzeMonteCarlorealiza-tionsofthelineardensityeld,andsoismuchmorecompli-catedtoapply,thoughstillsimplerthandoinganN-bodysimulation.OuraiminthispaperistotesttheanalyticalresultsformergingofdarkhalosderivedinPaperIagainstasetoflargeN-bodysimulations.Specically,wetesttheformulaeformergerprobabilitiesasafunctionofthemassesofthehalosinvolvedandofthedierenceincosmicepochs,andforthedistributionofformationepochsofhalos.WealsorevisitthequestionofhowwellthePSmassfunctionitselfworkscomparedtosimulations.Thelatterquestionhasbeeninves-tigatedpreviously,notablybyEfstathiou  etal. (1988)(here-afterEFWD)forasetofself-similarmodelswithpower-law initialpowerspectraand=1,andbyEfstathiou&Rees(1988),White etal. (1993)andBond&Myers(1993b)forColdDarkMatter(CDM)models,whoallfoundreason-ablygoodagreement.BCEKtestedtheconditionalmassfunctionformulaeagainstEFWD'ssimulations,andfound encouragingresults,butthesizeoftheirsimulations(32 3 particles)didnotallowverydetailedcomparisons.Inthispaper,weaddressthesequestionsusing128 3   2   10 6 par-ticleN-bodysimulationsofself-similarmodels,with=1andinitialpowerspectra P  ( k ) /  k n , n  =    2 ;   1 ; 0.With thislargenumberofparticles,itispossibletomakefairlydetailedtestsofthemergerformulae.Therehasbeensomediscussionofwhatisthebestltertousewiththestandard PSmassfunctionformula,andwhetherimprovedresultscanbeobtainedbychoosingathresholdlinearoverdensitydierentfromthatofthesimplesphericalcollapsemodel(Efstathiou&Rees1988;Carlberg&Couchman1989),and weinvestigatethisalso.Kaumann&White(1993)madeaqualitativecomparisonofsomeofthepropertiesofthemergerhistoriesfromtheirMonteCarlomethodwiththosefromaCDMsimulation,andarguedthattherewasreason-ableagreement.Animportantquestionthatariseswhencomparingan-alyticalpredictionsformassfunctions,mergerratesetc.ofdarkhaloswiththecorrespondingquantitiesinN-bodysim-ulations,ishowbesttoidentifythehalosinthesimula-tions.Giventhatthehalosfoundinsimulationsarenei-thercompletelyisolatednorexactlyinvirialequilibrium,thereseemsnouniquewaytodothis.Muchworkhasbeen basedonthepercolationor\friends-of-friends"algorithm (Davis etal. 1985)(DEFW),inwhichparticlesarelinked togetherwithotherparticlesintoagroupifthedistancetothenearestgroupmemberislessthanacertainfraction (usuallytakentobe0 : 2)ofthemeaninterparticlesepara-tion.However,othergroup-ndingschemeshavealsobeen used( e.g. Warren  etal. (1992),Bond&Myers(1993b)).It  Mergerrates  3  isimportanttoknowhowmuchthecomparisonwithanalyt-icalresultsdependsonthegroup-ndingschemeemployed,sowewillinvestigatetheeectofvaryingthelinkinglength inthefriends-of-friendsscheme,andalsouseanalternativemethodbasedonndingspheresofparticlesofacertain overdensity.Theplanofthepaperisasfollows:In  x 2wereview theanalyticalresultsonmergingderivedinPaperI.In  x 3wedescribetheN-bodysimulations,andin  x 4wedescribeourgroup-ndingschemes.In  x 5wecomparetheN-bodymassfunctionswiththePSformula.Thefollowingtwosec-tionstestthepredictionsformergingagainstthesimula-tions:mergerprobabilitiesasafunctionofmassin  x 6,and formationepochsin  x 7.Wepresentourconclusionsin  x 8. 2REVIEWOFANALYTICALMERGER RESULTS  2.1SpatialFilteringandRandomTrajectories Inthissection,wereviewtheanalyticalresultsonmerg-ingofdarkhalosderivedinPaperI,BCEKandBower(1991).Centraltotheapproachis spatialltering  ofthe lineardensityeld  .Theinitialconditionsforstructurefor-mationarespeciedasaGaussianrandomdensityeld   ( x )=   ( x  ) =   1havingsomepowerspectrum  P  ( k ),where  isthemeandensity.Thiseldissmoothedbyconvolvingitwithspherically-symmetriclters W  R  ( r )ofvariousradii R  ,havingFourierrepresentations c   W  R  ( k ): c   W  R  ( k )=  Z  W  R  ( j x  j )exp(   i k  : x  ) d 3 x  : (2.1)Thevarianceoftheeldaftersmoothingwithalterisrelatedtothepowerspectrumby   2 ( R  ) h  2 i R  =  Z  1  0 c   W  2 R  ( k ) P  ( k )4 k 2 dk (2.2)Welistbelowtheltersthatwewillbeusinginthispaper,andtheirFourierrepresentations.Wealsogivethe\naturalvolume" V  f thatisassociatedwithalterofradius R  f ,denedtobetheintegralof W  R  ( r ) =W  R  (0)overallspace. TopHat(TH): W  R  ( r )=    3 =   4 R  3T   r<R  T  0 r>R  T  (2.3) c   W  R  ( k )= 3( kR  T  ) 3 sin( kR  T  )   ( kR  T  )cos( kR  T  )](2.4) V  T  =(4 = 3) R  3T  (2.5) Gaussian(G): W  R  ( r )= 1(2   ) 3 = 2 R  3G  exp      r 2 2 R  2G    (2.6) c   W  R  ( k )=exp      k 2 2 R  2G    (2.7) V  G  =(2   ) 3 = 2 R  3G  (2.8) Sharp  k -space(SK): W  R  ( r )= 12   2 r 3 sin( r=R  S )   ( r=R  S )cos( r=R  S )](2.9) c   W  R  ( k )=    1 k<  1 =R  S 0 k>  1 =R  S (2.10) V  S =6   2 R  3S (2.11)Theltersarenormalizedaccordingtothecondition  R  1  0 W  R  ( r )4 r 2 dr =  c   W  R  ( k =0)=1.Thenaturalvolumeofalteristhus V  f =1 =W  R  ( r =0).(Inthecaseofthesharp  k -spacelter,thevolumeintegralsarealittleill-denedifdoneinrealspace,sincetheintegral R  r 0 W  R  ( r )4 r 2 dr ac-tuallyoscillatesaround1as r !1  .Ifdesired,thisminorproblemcanbecuredbymultiplying W  R  ( r )byexp(   r )beforedoingtheintegral,andtakingthelimit   !  0after-wards.)Thenaturalmassunderalteristhendenedas M  f =  V  f .Whenthedensityuctuationsaresmall( <<  1),theygrowaccordingtolinearperturbationtheory,  ( x  ;t ) /  D  ( t ),wherethelineargrowthfactor D  ( t )dependsontheback-groundcosmologicalmodel;for=1, D  ( t ) /  a ( t ) /  t 2 = 3 , a ( t )beingthecosmicexpansionfactor.(Weareassumingthatonlythegrowingmodeoflinearperturbationtheoryispresent.)Thenon-linearevolutioncanbecalculatedan-alyticallyforsphericalperturbations( e.g. Peebles(1980);PaperI);forauniformoverdensesphericaluctuation,thecollapsetimedependsonitsinitiallinearoverdensity.Itisconvenienttoworkintermsoftheinitialdensityeldex-trapolatedaccordingtolineartheorytosomexedreferenceepoch  t 0 ,forwhichwealsotake a ( t 0 )=1;fromnowon,thisiswhatwewillmeanby  ( x  ).Intermsofthisextrapolated   ,asphericalperturbationofmeanoverdensity  collapsesattime t if  =   c ( t ),where,for=1,wehavetheusualresult  c ( t )=   c0 =a ( t )=   c0 ( t 0 =t ) 2 = 3  c0 =3(12   ) 2 = 3 = 20   1 : 69(2.12)(Thegeneralizationofthisfor  <  1isderivedinPaperI;notethatthesecondlineofequation(2.1)ofthatpapercontainsatypographicalerror.)Theanalyticalformulaewenowpresentforthehalomassfunctions,conditionalmassfunctionsandlifetimesareexpressedintermsofthisthreshold   c ( t ),and    ( M  ),thevarianceofthesmoothedlineardensityeldasafunction ofthesmoothingmass. 2.2MassFunction  Thefractionofmassinhaloswithmass M  ,attime t ,perinterval dM  ,originallyderivedbyPress&Schechter(1974),is(PaperIequation(2.10)) df dM  ( M;t )=   c ( t )(2   ) 1 = 2   3 ( M  )  d  2 ( M  ) dM   exp       c ( t ) 2 2   ( M  ) 2  (2.13)   ( M  )isassumedtodeclinemonotonicallywithincreasing M  .(TomakethecorrespondencewiththeequationsinPa-perI,notethatwethereusedthenotation  S  =    2 ( M  ), !  =   c ( t ).)Thus,thecomovingnumberdensityofhalosofmass M  presentattime t ,per dM  ,is dn dM  ( M;t )=   4  C.LaceyandS.Cole    2     1 = 2 M  2  c ( t )   ( M  )  d ln   d ln  M   exp       c ( t ) 2 2   2 ( M  )  (2.14)where  isnowthemeandensityatthereferenceepoch  t 0 .Bydening      c ( t ) =  ( M  ),(2.13)canberewrittenas df d ln    =    2     1 = 2   exp(     2 = 2)(2.15)whichisindependentoftheformoftheuctuationspec-trum. 2.3ConditionalMassFunctionandMergerProbability  Theconditionalprobabilityforamasselementtobepartofahaloofmass M  1 attime t 1 ,giventhatitispartofalargerhaloofmass M  2 >M  1 atalatertime t 2 >t 1 ,isfoundbyconsideringtwodierentthresholds,  c ( t 1 )and   c ( t 2 ).Theresult,derivedsomewhatdierentlybyBCEKandBower(1991),fortheconditionalmassfractionperinterval dM  1 is(PaperIequation(2.15)) df dM  1 ( M  1 ;t 1 j M  2 ;t 2 )= (  c1    c2 )(2   ) 1 = 2 (   21     22 ) 3 = 2  d  21 dM  1  exp      (  c1    c2 ) 2 2(   21     22 )  (2.16)where   1 =    ( M  1 ),   2 =    ( M  2 ),  c1 =   c ( t 1 )and   c2 =   c ( t 2 ).Theonlyassumptionmadehereaboutthefunction   c ( t )isthatitmonotonicallydecreaseswithincreasing t .Thereverseconditionalprobability,for M  2 given  M  1 ,is(Pa-perIequation(2.16)) df dM  2 ( M  2 ;t 2 j M  1 ;t 1 )= 1(2   ) 1 = 2  c2 (  c1    c2 )  c1     21   22 (   21     22 )  3 = 2    d  22 dM  2  exp      (  c2   21    c1   22 ) 2 2   21   22 (   21     22 )  (2.17)Thisisobviouslythesameastheprobabilityforahaloofmass M  1 at t 1 tobeincorporatedintoahaloofmass M  2 > M  1 at t 2 >t 1 .Thus,ifweset M  2 =  M  1 +  M  and  t 2 =  t 1 +  t intheaboveformula,wegettheprobabilityforahalotogainmass  M  bymergingintime  t .Takingthelimit  t !  0thengivestheinstantaneousmergerrateasafunctionof M  1 and  M  (equation(2.18)ofPaperI). 2.4FormationTimes Supposeoneidentiesahaloofmass M  2 attime t 2 .Atan earliertime,onecanidentifytheprogenitorsofthishalo.Wedenetheformationtimeofthehaloidentiedatepoch  t 2 astheearliesttime t f <t 2 atwhichithasaprogenitorofmass M  1 atleasthalfof M  2 .Wendthecumulativeprobabilitydistributionfor t f (equation(2.26)ofPaperI) P  ( t f <t 1 j M  2 ;t 2 )=  Z  M  2 M  2 = 2 M  2 M  1 df dM  1 ( M  1 ;t 1 j M  2 ;t 2 ) dM  1 (2.18)Thedierentialprobabilitydistributionfor t f isthengiven by dpdt f ( t f j M  2 ;t 2 )=  Z  M  2 M  2 = 2 M  2 M  1  @ @t f h  df dM  1 ( M  1 ;t f j M  2 ;t 2 ) i dM  1 (2.19)WenotedinPaperIthattheexpressioncorrespondingtoequation(2.19)actuallyleadstoaslightmathematicalinconsistencyinsomecases:forpower-lawpowerspectra P  ( k ) /  k n with  n>  0,theprobabilitydensityfor t f goesslightlynegativeforsmall t 2   t f .InPaperI,wealsode-rivedformationtimedistributionsbasedonaMonteCarlomethod,whichdonothavetheproblemofnegativeproba-bilitydensity.Thesedistributionshavesimilarshapestotheanalyticalones,butwiththemeanshifted.Wewillseein  x 7thattheanalyticaldistributiongivesaremarkablygoodttotheN-bodyresults. 2.5Self-SimilarModels Theanalyticalresultspresentedabovemakenospecialas-sumptionsaboutthefunctions   ( M  )and   c ( t ),exceptthattheyaremonotonic.However,intestingtheseresultsagainstsimulations,wewillfocusonself-similarmodels,inwhich thedensityparameter=1,sothatthereisnochar-acteristictimeintheexpansionoftheuniverse,andin whichtheinitialdensityuctuationshaveascale-freespec-trum, P  ( k ) /  k n .Inthiscase,theevolutionofstruc-tureshouldbeself-similarintime.Thishassomeadvan-tages,tobediscussedin  x 3.Fromequation(2.2),weobtain    ( M  ) /  M    ( n +3) = 6 ,ingeneral.For   todeclinewithin-creasing M  ,werequire n>    3,whichisjustacondition forstructuretogrowhierarchically,withsmallobjectscol-lapsingrstandthenmergingtoformlargerobjects.Forthetophatlter,theintegralinequation(2.2)onlycon-vergesfor n<  1.WewillbeconsideringN-bodymodelswith  n  =    2 ;   1 ; 0.For=1,densityuctuationsgrowas D  ( t ) /  a ( t ) /  t 2 = 3 inlineartheory,sothatther.m.s.uctuationonco-movingscale k   1 isroughly p   4 k 3 P  ( k ) a ( t ).Thuswecan deneacharacteristicnon-linearwavenumber k ? ( t )by4 k 3 ? ( t ) P  ( k ? ( t )) a ( t ) 2 =1(2.20)Attime t ,uctuationsareoforderunityandarestartingtocollapseonacomovinglengthscale   k ? ( t )   1 .Inaself-similarmodel,thisshouldbetheonlycharacteristiclength-scaleforstructure.Inouranalyticalexpressionsformassfunctions,mergerprobabilitiesetc,itisconvenienttodenealter-dependentcharacteristicmassscale M  ? ( t )by   ( M  ? ( t ))=   c ( t )(2.21)where  c ( t )=   c0 =a ( t )for=1.Themass M  ? isrelatedtothelter-independentquantity k ? by M  ? ( t )=    f   c f ( n  )  c0   6 = ( n +3) k 3 ? ( t )(2.22)where  isthemeandensityatthereferenceepochwhen  a =1.   f relatesthevolumeofaltertoitsradiusthrough   Mergerrates  5  V  f     f R  3f (c.f. x 2.1),and  c f ( n  ),whichentersthroughtherelation(2.2)between    ( R  )and  P  ( k ),isdenedby c 2f ( n  )=  Z  1  0 c   W  2 R  =1 ( k ) k n +2 dk (2.23)Fortheltersweareusing, c f (   2)=1 : 373 ; 0 : 941 ; 1 : 000, c f (   1)=1 : 500 ; 0 : 707 ; 0 : 707and  c f (0)=2 : 170 ; 0 : 666 ; 0 : 577for TH;G;SK  ltersrespectively.Fromequations(2.20)and(2.22),thecharacteristicmassgrowsas M  ? ( t ) /  a 6 = ( n +3) .Themassfunction(2.13)canthenberewritten as df d ln  M  =    2     1 = 2   n  +36   M M  ?   ( n +3) = 6   exp      12   M M  ?   ( n +3) = 3  (2.24)inwhichformthemassandtimeonlyappearinthecom-bination  M=M  ? ( t ).Similarly,themergerprobability(2.17)canberewrittenasadistributionfor M  2 =M  1 dependingon  M  1 =M  ? ( a 1 )and  a 2 =a 1 ,andtheformationtimedistribution (2.19)canberewrittenasadistributionfor a f =a 2 dependingon  M  2 =M  ? ( a 2 ). 2.6Choiceoflteringandcollapsethreshold  TheBCEKderivationofthePSmassfunctionandoftheconditionalmassfunctionisbasedonsharp  k -spacel-tering.Onthehand,theoriginal,moreheuristic,deriva-tionbyPSthemselvesassumedtophatltering,asdid Bower'sderivationoftheconditionalmassfunction.MostapplicationsofthePSformulahavefollowedthelatterap-proachandusedthe   ( M  )relationfortophatltering(with  M  =(4 = 3) R  3 T  ),butsomehaveinsteadused    ( M  )forGaussianltering(with  M  =(2   ) 3 = 2 R  3 G  )( e.g. Efstathiou &Rees(1988)).Evenforagivenchoiceoflter,onecan obtaindierent   ( M  )relationssimplybychoosingadier-entmass-radiusrelationfromthe\natural"onediscussedin  x 2.1;afterall,itisnotobvioushowthemassofacollapsed objectisrelatedtotheproleofthelterusedtoidentifyit.BCEKsuggestedcalculatingtheltermassforagenerallterfrom  M  =(4 = 3) R  3 T  ,wherethe\eqivalenttophat"radius R  T  isdenedthroughtherelation    ( R  )=    TH  ( R  T  ),onthegroundsthatthecollapsethreshold   c ( t )isalsocal-culatedforatophatsphericalperturbation.Forpower-law powerspectra,thisrequiresmaking   f inequation(2.22)afunctionofspectralindexaswellasltertype,whileforageneral P  ( k )itmustbeafunctionof R  .Thisprocedureisequivalenttousingthe   ( M  )relationfortophatlteringinformulaelike(2.13)and(2.17),eveniftheseformulaearederivedforsharp  k -spaceltering.Inthispaper,weadoptanempiricalapproachtothechoiceoflterand  M  ( R  )relation.Wewillcomparethere-sultsofN-bodysimulationstotheformulaeusingtophat,Gaussianandsharp  k -spacelteringfor   ( R  ).Weassumeamass-radiusrelation  M  =    f R  3 ,with    f aconstantde-pendingontheltertypebutindependentofthepower-spectrum,butwillconsidervaluesof   f dierentfromthe\natural"onesgivenin  x 2.1.Arelatedissueconcernsthechoiceofcollapsethreshold,whichfor=1boilsdowntoachoicefor  c0 inequation (2.12).Whilethesphericalcollapsemodelgivesanunam-biguousanswer,onecantaketheviewthat,sincerealcol-lapseshavenon-tophatandnon-sphericaldensityproles,  c0 shouldberegardedasaphenomenologicalparameter,chosentogivethebestttoN-bodyresults( e.g. Bond &Myers(1993b)).SincethePSmassfunctionandotherformulaedependon    f and   c0 onlythrough  M  ? (equation 2.22),thereisadegeneracybetweenthesetwoparametersforanygivenspectralindex  n  ,butthisdegeneracyislifted assoonasoneconsidersresultsfordierent n  .Thechoiceof  c0 and    f willbeconsideredinrelationtothesimulationsin  x 5. 3N-BODYSIMULATIONS  Thesimulationswereperformedusingthehighresolution particle-particle-particle-mesh( P  3 M  )codeofEfstathiou  etal. (1985)(EDFW)with128 3   2   10 6 particles.Thelong-rangeforcewascomputedona256 3 mesh,whilethesofteningparameterfortheshort-rangeforcewaschosentobe  =0 : 2( L= 256),where L  isthesizeofthe(periodic)computationalbox.Thiscorrespondstotheinterparticleforcefallingtohalfofitspoint-massvalueataseparation  r   0 : 4    L= 3200.Initialpositionsandvelocitiesweregen-eratedbydisplacingparticlesfromauniform128 3 gridac-cordingtotheZel'dovichapproximation,assumingthelinearpowerspectrumandGaussianstatistics.Weranonesimu-lationforeachof n  =    2 ;   1 ; 0.For n  =    1and  n  =0,theinitialamplitudeofthepowerspectrumwaschosentoequalthewhitenoiselevelattheNyquistfrequencyoftheparticlegrid;for n  =    2,thischoicewasfoundtoleadtolargedeparturesfromself-similarbehaviourinthederived massfunctionsandrelatedquantities,sothissimulationwasinsteadstartedwhentheamplitudewassmallerbyafactor0.4.Weadoptedtheconventionofnormalizingtheexpansion factor a to1whenthevarianceofthelineartheorypowerspectruminatop-hatsphereofradius L= 32was1.Withthischoice,theinitialexpansionfactorswere a i =0 : 2 ; 0 : 15 ; 0 : 06respectivelyfor n  =    2 ;   1 ; 0.Theinitialr.m.s.1Ddisplace-mentsoftheparticleswereapproximately0 : 9 ; 0 : 5 ; 0 : 25in unitsoftheparticlegridspacing.Thetimeintegrationwasperformedusingthevariable p =  a  ,with    =2 = ( n  +3),andaconstantstepsize  p=p i =  14   (256 =L  ) 3 = 2   0 : 023   (EDFW).Foreachsimulationweoutputthepositionsandve-locitiesofalltheparticlesatmanyepochs.Thespacingoftheseoutputswaschosensothatthecharacteristicmass, M  ? ,increasedbyafactor p  2betweeneachsuccessiveout-put,whichcorrespondstoanincreaseofafactor2 ( n +3) = 12 intheexpansionfactor a .Thenalexpansionfactorsforthesimulationsweredeterminedbasicallybythecomputerresourcesavailable;forthelaterstages,theCPUtimewasdominatedbytheshort-rangeforcecalculationbyalargefactor.Thesimulationswerestoppedatexpansionfactors a =1 : 26 ; 2 : 00 ; 1 : 68respectivelyfor n  =    2 ;   1 ; 0.Thisgaveusbetween20and24usefuloutputsfromeachsimulation.Self-similarmodelshavetwoadvantagesfromthepointofviewoftestingouranalyticalpredictionsagainstsimula-tions.(i)Wecancheckwhetherthesimulationsobeytheself-similarscalingwhichphysicallytheyshould.Inparticular,thisallowsustotestwhetherthesimulationswerestarted atasmallenoughexpansionfactor.Wecanalsodelineate
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