An excursion set model for the distribution of dar

arXiv:astro-ph/9805319v126May1998Mon.Not.R.Astron.Soc.000,000–000(0000)Printed1February2008(MNLATEXstylefilev1.4)AnexcursionsetmodelforthedistributionofdarkmatteranddarkmatterhaloesRaviK.ShethMax-PlanckInstitutf¨urAstrophysik,Karl-Schwarzschild-Str.1,85740Garching,GermanyEmail:[e-mail]1998April;inoriginalform1997SeptemberABSTRACTAmodelofthegravitationallyevolveddarkmatterdistribution,intheEulerianspace,isdeveloped.Itisasimpleextensionoftheexcursionsetmodelthatiscommonlyusedtoestimatethemassfunctionofcollapseddarkmatterhaloes.Inadditiontodescribingtheevolutionofthedarkmatteritself,themodelallowsonetodescribetheevolutionoftheEulerianspacedistributionofthehaloes.Itcanalsobeusedtodescribedensityprofiles,onscaleslargerthanthevirialradius,ofthesehaloes,andtoquantifythewayinwhichmatterflowsinandoutofEuleriancells.WhentheinitialLagrangianspacedistributioniswhitenoiseGaussian,themodelsuggeststhattheInverseGaussiandistributionshouldprovideareasonablygoodapproximationtotheevolvedEuleriandensityfield,inagreementwithnumericalsimulations.ApplicationofthismodeltoclusteringfrommoregeneralGaussianinitialconditionsisdiscussedattheend.Keywords:methods:analytical–galaxies:clusters:general–galaxies:formation–cosmology:theory–darkmatter.1INTRODUCTIONThehypothesisthat,incomovingcoordinates,initiallydenserregionscontractmorerapidlythanlessdenseregions,andthatsufficientlyunderdenseregionsexpand,issimple,reasonable,andpowerful.Asaconsequenceofthisexpan-sionandcontraction,thedensitydistributionintheinitialLagrangianspacewillbedifferentfromthatintheevolved,Eulerianspace.Supposethat,astheuniverseevolves,thenumberofexpandingandcontractingregionsisconserved—onlytheircomovingsizechanges—andthemasswithineachsuchregionisalsoconserved.Ifwehaveamodelforthewayinwhichtheevolutionofthesizeofaregiondependsonitsinitialsizeanddensity,andwealsohaveamodelfortheinitialnumberofregionsasafunctionofinitialsizeanddensity,thenwecancomputethedistributionofsizesanddensitiesatsomelatertime.Forexample,supposethattheinitialLagrangiandensitydistributionisaGaussianrandomfield,andthattheevolutionofregionsisgivenbythespher-icalcollapsemodel.Thenitshouldbepossibletoconstructamodelforp(M0|R,z),wherep(M0|R,z)isthefractionofregionsofsizeRthat,atz,containmassM0.Thequantityp(M0|R,z)isoftencalledtheEulerianprobabilitydistribu-tionfunction.InthePress-Schechter(1974)approach,atanygiventime,allmatterisintheformofcollapsedobjects,usuallycalledhaloes,andthedistributionofhalomassesevolveswithtime.Atanytime,thematterwithinarandomlyplacedcellRisdividedamongmanycollapsedhaloes.Thus,p(M0|R,z)dependsbothonthehalomassfunction,andonthespatialdistributionofthehaloes.Bondetal.(1991)showedhowtoestimatetheevolutionofthehalomassfunc-tioniftheinitialLagrangiandistributionisGaussian(alsoseeLacey&Cole1993).Theydidnotshowhowtoestimatethespatialdistributionofthesehaloes,butMo&White(1996)showedhowthismightbeaccomplished.Thispaperdevelopsamodelthatcombines,self-consistently,theBondetal.(1991)excursionsetapproachwiththeMo&White(1996)modelfortheEulerianspacehalodistribution.Themodeldevelopedhereallowsonetosimultaneouslydescribeboththedistributionofthedarkmatter,i.e.p(M0|R,z),andthatofthehaloes.See,e.g.,Mo&White(1996)forwhysuchamodelisuseful.Section2describesthemodel.Itshowswhythedis-tributionoffirstcrossingsofabarrierwhoseheightisnotconstant,byBrownianmotionrandomwalks,isuse-ful.Theshapeofthebarrierassociatedwiththespheri-calcollapsemodelisgiveninSection2.2.Therelationbe-tweenthefirstcrossingdistributionandtheEuleriandis-tributionp(M0|R,z)isdiscussedinSection2.3.Section2.4showsthat,inthecontextofthemodeldevelopedhere,thehalomassfunctionisrelatedtothesmallcellsizelimit(i.e.R→0)ofp(M0|R,z).ItdiscussestheBondetal.(1991)excursionsetresultsinthiscontext,andthenshowshowthemodelcanbeusedtodescribethespatialdistributionc0000RAS2R.K.Shethofthehaloesaswell.TheresultsofMo&White(1996)arediscussedinSection2.5.Section2.6showsthattheassoci-atedtwobarrierproblemcanbeusedtoprovideinformationabouttheevolveddensityprofile,andalsoaboutthewayinwhichmatterflowsinandoutofEuleriancells.Thefirstcrossingdistributionassociatedwiththesphericalcollapsebarriermustbeobtainednumerically.Therefore,toillustratetheusefulnessofourapproach,Sec-tion3showstheresultsofassumingthattheinitialdistri-butioniswhite-noiseGaussian,andthatthebarriershapeissimplerthanthatassociatedwiththesphericalcollapsemodel.InSection3,thebarrierisassumedtobelinearforanumberofreasons.Firstly,thislinearbarriercanbeun-derstoodasarisingfromasimplevariantofthesphericalcollapsemodel(Section3.1).Secondly,thebarriercrossingdistributioncanbecomputedanalytically(e.g.Schr¨odinger1915).ThedetailsofthederivationarepresentedinAp-pendixA.Thirdly,theEulerianprobabilitydistributionas-sociatedwiththefirstcrossingsofthislinearbarrierisIn-verseGaussian(Section3.3),andtheInverseGaussianpro-videsagoodfittotheEuleriandistributionmeasuredinnumericalsimulationsofclusteringfromwhitenoiseinitialconditions(Section3.6).Finally,studiesofclusteringfromPoissoninitialconditionsalso