On the reliability of merger-trees and the mass gr

arXiv:astro-ph/0508531v124Aug2005Onthereliabilityofmerger-treesandthemassgrowthhistoriesofdarkmatterhaloesN.Hiotelis1stExperimentalLyceumofAthens,Ipitou15,Plaka,10557,Athens,Greece,e-mail:[e-mail]1,21Bo˘gazi¸ciUniversity,PhysicsDepartment,80815Bebek,Istanbul,Turkey2DipartimentodiMatematica,Universit`aStatalediBergamo,viadeiCaniana,2,24127,Bergamo,Italye-mail:[e-mail]ertreesrealizationstostudytheformationofdarkmatterhaloes.Theconstructionofmerger-treesisbasedonthreedifferentpicturesaboutthefor-mationofstructuresintheUniverse.Thesepicturesinclude:thesphericalcollapse(SC),theellipsoidalcollapse(EC)andthenon-radialcollapse(NR).Thereliabil-ityofmerger-treeshasbeenexaminedcomparingtheirpredictionsrelatedtothedistributionofthenumberofprogenitors,aswellasthedistributionofformationtimes,withthepredictionsofanalyticalrelations.Thecomparisonyieldsaverysat-isfactoryagreement.Subsequently,themassgrowthhistories(MGH)ofhaloeshavebeenstudiedandtheirformationscalefactorshavebeenderived.Thisderivationhasbeenbasedontwodifferentdefinitionsthatare:(a)thescalefactorwhenthehaloreacheshalfitspresentdaymassand(b)thescalefactorwhenthemassgrowthratefallsbelowsomespecificvalue.Formationscalefactorsfollowapproximatelypowerlawsofmass.IthasalsobeenshownthatMGHsareingoodagreementwithmodelsproposedintheliteraturethatarebasedontheresultsofN-bodysimulations.Theagreementisfoundtobeexcellentforsmallhaloesbut,attheearlyepochsoftheformationoflargehaloes,MGHsseemtobesteeperthanthosepredictedbythemodelsbasedonN-bodysimulations.ThisrapidgrowthofmassofheavyhaloesislikelytoberelatedtoasteepercentraldensityprofileindicatedbytheresultsofsomeN-bodysimulations.Keywords:galaxies:halos–formation–structure,methods:numerical–analytical,cosmology:darkmatterPACS:98.62.Gq,98.62.A,95.35.+dPreprintsubmittedto5February20081IntroductionItislikelythatstructuresintheUniversegrowfromsmallinitiallyGaussiandensityperturbationsthatprogressivelydetachfromthegeneralexpansion,reachamaximumradiusandthencollapsetoformboundobjects.Largerhaloesareformedhierarchicallybymergersbetweensmallerones.Twodifferentkindsofmethodsarewidelyusedforthestudyofthestruc-tureformation.ThefirstoneisN-bodysimulationsthatareabletofollowtheevolutionofalargenumberofparticlesundertheinfluenceofthemu-tualgravityfrominitialconditionstothepresentepoch.Thesecondoneissemi-analyticalmethods.Amongthese,Press-Schechter(PS)approachanditsextensions(EPS)areofgreatinterest,sincetheyallowustocomputemassfunctions(Press&Schechter1974;Bondetal1991),toapproximatemerginghistories(Lacey&Cole1993,LC93hereafter,Bower1991,Sheth&Lemson1999b)andtoestimatethespatialclusteringofdarkmatterhaloes(Mo&White1996;Catelanetal1998,Sheth&Lemson1999a).Inthispaper,wepresentmerger-treesbasedonMonteCarlorealizations.Thisapproachcangivesignificantinformationregardingtheprocessoftheforma-tionofhaloes.WefocusonthemassgrowthhistoriesofhaloesandwecomparethesewiththepredictionsofN-bodysimulations.Thispaperisorganizedasfollows:InSect.2basicequationsaresummarized.InSect.3thealgorithmfortheconstructionofmerger-treesaswellastestsregardingthereliabilityofthisalgorithmarepresented.Mass-growthhisto-riesarepresentedinSect.4,whiletheresultsaresummarizedanddiscussedinSect.5.2Basicequationsandmerger-treesrealizationsInanexpandinguniverse,aregioncollapsesattimet,ifitsoverdensityatthattimeexceedssomethreshold.Thelinearextrapolationofthisthresholduptothepresenttimeiscalledabarrier,B.Alikelyformofthisbarrierisasfollows:B(S,t)=qαS∗[1+β(S/αS∗)γ](1)1Presentaddress:Roikou17-19,NeosKosmos,Athens,11743Greece2InEq.(1)α,βandγareconstants,S∗≡S∗(t)≡δ2c(t),whereδc(t)isthelinearextrapolationuptothepresentdayoftheinitialoverdensityofasphericallysymmetricregion,thatcollapsedattimet.Additionally,S≡σ2(M),whereσ2(M)isthepresentdaymassdispersiononcomovingscalecontainingmassM.Sdependsontheassumedpowerspectrum.Thesphericalcollapsemodel(SC)hasabarrierthatdoesnotdependonthemass(eg.LC93).Forthismodelthevaluesoftheparametersareα=1andβ=0.Theellipsoidalcollapsemodel(EC)(Sheth&Tormen1999,ST99hereafter)hasabarrierthatdependsonthemass(movingbarrier).Thevaluesoftheparametersareα=0.707,β=0.485,γ=0.615andareadoptedeitherfromthedynamicsofellipsoidalcollapseorfromfitstotheresultsofN-bodysimulations.Additionally,thenon-radial(NR)modelofDelPopolo&Gambera(1998)-thattakesintoaccountthetidalinteractionwithneighbors-correspondstoα=0.707,β=0.375andγ=0.585.Sheth&Tormen(2002)connectedtheformofthebarrierwiththeformofthemultiplicityfunction.Theyshowthatgivenamasselement-thatisapartofahaloofmassM0attimet0-theprobabilitythatatearliertimetthismasselementwasapartofasmallerhalowithmassMisgivenbytheequation:f(S,t/S0,t0)dS=1√2π|T(S,t/S0,t0)|(ΔS)3/2exp−(ΔB)22ΔS#dS(2)whereΔS=S−S0andΔB=B(S,t)−B(S0,t0)withS=S(M),S0=S(M0).ThefunctionTisgivenby:T(S,t/S0,t0)=B(S,t)−B(S0,t0)+5Xn=1[S0−S]nn!∂n∂SnB(S,t).(3)SettingS0=0,andB(S0,t0)=0inEq.3,wecanpredicttheunconditionalmassprobabilityf(S,t),thatistheprobabilitythatamasselementisapartofahaloofmassM,attimet.Theq