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arXiv:astro-ph/0009306v221Aug2002MultiresolutionAnalysisofSubstructureinDarkMatterHalosMichaelD.Seymour1andLawrenceM.Widrow2DepartmentofPhysics,Queen’sUniversity,Kingston,Ontario,CanadaK7L3N6ABSTRACTMultiresolutionanalysisisappliedtotheproblemofhaloidentificationincosmologi-calN-bodysimulations.Theproceduremakesuseofadiscretewavelettransformknownasthealgorithme`atrousandsegmentationanalysis.Ithastheabilitytofindsubhalosinthedenseregionsofaparenthaloandcandiscernthemultiplelevelsofsubstructureexpectedinthehierarchicalclusteringscenario.Asanillustration,a500,000particledarkmatterhaloisanalyzedandover600subhalosarefound.Statisticalpropertiesofthesubhalopopulationarediscussed.Subjectheadings:1.IntroductionThehierarchicalclusteringhypothesisprovidesanattractiveparadigmfortheformationofstructureinauniversedominatedbycolddarkmatter.Small-scaleobjectsformfirstandmergetoyieldsystemsofincreasingsize.Thishighlynon-linearprocesshasbeenstudiedextensivelyusingN-bodysimulationswithparticularattentionpaidtothesurvivalofsubhalosonceamergereventhasoccurred.Earlyresultssuggestedthatsubstructures(i.e.,subhaloswithinhalos)areerasedefficiently(White1976;Frenketal.1988).Thisso-calledovermergingproblemplaguedinvestiga-tionsofgalaxyclusterformationsinceitleadtotheconclusionthattheconstituentgalaxiesdonotsurvive.However,recenthighresolutionsimulations(Ghignaetal.1998,Klypinetal.1999)togetherwithanalyticwork(Moore,Katz,&Lake1996)demonstratedthattheovermergingprob-lemwasdueentirelytothepoormassandspatialresolutionofearlysimulations.Indeed,Mooreetal.(1999)foundthatongalacticscales,simulatedhalosmayhavetoomuchsubstructure.Theirhighresolutionsimulationofa1012M⊙(i.e.,Milky-Waysized)halorevealedover500M&108M⊙subhalos,afactorof50greaterthanthenumberofvisiblesatellitesintheMilkyWay.AnessentialelementintheanalysisofcosmologicalN-bodysimulationsistheidentificationofphysicalstructures,namelyhalosandsubhalos.Therearenowanumberofalgorithmsavailabletodothissuchasfriends-of-friends(FOF;Davisetal.1985),DENMAX(Bertschinger&Gelb1seymour@astro.queensu.ca2widrow@astro.queensu.ca–2–1991;Gelb&Bertschinger1994),andSKID(Governatoetal.1997).TheFOFalgorithmidentifiesstructuresbylinkingallparticlepairsseparatedbylessthanauser-supplieddistanceknownasthelinkingparameter.Foraparticularchoiceoflinkingparameter,thealgorithmproducesauniquelistofstructures.DENMAX,andthecloselyrelatedalgorthmSKID,arebasedoncontoursurfacesofthethree-dimensionaldensityfield.Eachlocalmaximumofthedensityfieldisassumedtocorrespondtothecenterofahaloorsubhalo:Allparticlesinteriortothelastclosedcontoursurroundingagivenmaximumareassignedtothecorrespondinghalo.Twochallenges,broughttotheforebythedramaticimprovementsinthemassandforceresolutionofpresent-daysimulations,nowconfrontthesemethods:(1)identificationofsubhaloswithinthedenseregionsofaparenthalo;and(2)analysisofmultiplelevelsofsubstructure.Agivensimulationparticlemaybeamemberofasmallclumpthatis,inturn,gravitationallyboundtoalarger(sub)halo.Likewise,thesubhalomaybegravitationallyboundtoagalacticorcluster-sizedhalo.Suchaparticleismostaccuratelydescribedasbeingamemberofthreedistinctstructures.Thus,anyschemewhichassignsagivenparticletoatmostonestructurecannothopetocapturethehierarchicalnatureofdarkmatterhalos.EarlyincarnationsoftheFOFalgorithmreliedonasinglelinkinglength.Ifthelinkinglengthissettobetoolarge,subhalosintheinnerpartsoflargehalosaremissed.Withasmalllinkinglength,ontheotherhand,thealgorithmpicksoutsubstructurebutlosesinformationonlargescales.Theseproblemscanbeavoidedifoneusesa“hierarchical”versionofFOF(Klypinetal.1999)whereinthealgorithmisrunseveraltimeswithdifferentlinkinglengths.DENMAXandSKIDarebestsuitedtofindingsubstructuresincetheylocateallmaximainthedensityfield.Thehierarchicalnatureofclusteringcanbestudiedbyapplyingsmoothingfilterstothedensityfieldandrerunningthealgorithm.Inthispaper,wedescribeamultiresolutionanalysis(MRA)thathandles,inanaturalway,themultiplelevelsstructurefoundincosmologicalN-bodysimulations.MRAreferstoageneralclassoftoolsthatprovideasimplehierarchicalrepresentationofasignal,thesignal,inourcase,beingthedensityfieldinasimulation.Ateachresolution,theanalysispicksoutthedetailsofthesignalatacharacteristicscale.Thus,MRAcanbethoughtofasa“mathematicalmicroscope”:Coarseresolution(lowmagnification)probeslarge-scalestructuresinthesignalwhilefineresolution(highmagnification)probessmall-scalestructures.WeemployaspecificMRAthatisbasedonthewavelettransformknownasthealgorithme`atrous.Awavelettransformallowsonetoanalyzeasignalsimultaneouslyinscaleandposition.Thetrans-formaccomplishesthistaskbyconvolvingthesignalwithaspecialtypeofwindowfunctionknownasawavelet.Waveletsmusthavecompactsupportandintegratetozero.Thewavelettransformthereforeprobeslocalpropertiesofthedensityfieldandisinsensitivetoameanbackground.Thewaveletsusedinaparticularimplementationofthetransformarechosentobetranslationsanddilationsofasingleprototypeknownasthemotherwavelet.Alowresolutionanalysisofasignalisachievedbyusinglarge-scaleversionsofthemotherwavelet:highresolutionisachieved