The distribution function of dark matter in massiv

arXiv:0802.0429v2[astro-ph]16May2008Mon.Not.R.Astron.Soc.000,000–000(0000)Printed16May2008(MNLATEXstylefilev2.2)ThedistributionfunctionofdarkmatterinmassivehaloesRadoslawWojtak,1EwaL.Lokas,1GaryA.Mamon,2,3StefanGottl¨ober,4AnatolyKlypin5andYehudaHoffman61NicolausCopernicusAstronomicalCenter,Bartycka18,00-716Warsaw,Poland2Institutd’AstrophysiquedeParis(UMR7095:CNRSandUniversit´ePierre&MarieCurie),98bisBdArago,F-75014Paris,France3GEPI(UMR8111:CNRSandUniversit´eDenisDiderot),ObservatoiredeParis,F-92195Meudon,France4AstrophysikalischesInstitutPotsdam,AnderSternwarte16,14482Potsdam,Germany5DepartmentofAstronomy,NewMexicoStateUniversity,Box30001,Departament4500,LasCruces,NM880003,USA6RacahInstituteofPhysics,HebrewUniversity,Jerusalem91904,Israel16May2008ABSTRACTWestudythedistributionfunction(DF)ofdarkmatterparticlesinhaloesofmassrange1014–1015M⊙.InthenumericalpartofthisworkwemeasuretheDFforasampleofrelaxedhaloesformedinthesimulationofastandardΛCDMmodel.TheDFisexpressedasafunctionofenergyEandtheabsolutevalueoftheangularmomentumL,aformsuitableforcomparisonwiththeoreticalmodels.Byproperscalingweobtaintheresultsthatdonotdependonthevirialmassofthehaloes.WedemonstratethattheDFcanbeseparatedintoenergyandangularmomentumcomponentsandproposeaphenomenologicalmodeloftheDFintheformfE(E)[1+L2/(2L20)]−β∞+β0L−2β0.Thisformulationinvolvesthreeparametersdescribingtheanisotropyprofileintermsofitsasymptoticvalues(β0andβ∞)andthescaleoftransitionbetweenthem(L0).TheenergypartfE(E)isobtainedviainversionoftheintegralforspatialdensity.Weprovideastraightforwardnumericalschemeforthisprocedureaswellasasimpleanalyticalapproximationforatypicalhaloformedinthesimulation.TheDFmodelisextensivelycomparedwiththesimulations:usingthemodelparametersobtainedfromfittingtheanisotropyprofile,werecovertheDFfromthesimulationaswellastheprofilesofthedispersionandkurtosisofradialandtangentialvelocities.Finally,weshowthatourDFmodelreproducesthepower-lawbehaviourofphasespacedensityQ=ρ(r)/σ3(r).Keywords:galaxies:clusters:general–galaxies:kinematicsanddynamics–cos-mology:darkmatter1INTRODUCTIONThedistributionfunction(DF)providesthemostgeneralandcompletewayofstatisticaldescriptionofdarkmatter(DM)haloes.Itcarriesmaximuminformationonthespatialandvelocitydistributionsofparticlesinsuchobjects.OurknowledgeontheDFisstillbeingimproved,mostlyduetonumericalexperiments.Inthelastfewyearscosmologicalsimulationshaverevealedincreasinglydetailedfeaturesofphase-spacestructureofDMhaloes.ThesenumericalresultsprovideusefulconstraintsontheoreticalmodelsoftheDF.Onepropertyofinterestinthisfieldistheanisotropyofthevelocitydispersiontensor.Ithasbeendemonstratedthattheouterpartsofthehaloesexhibitmoreradiallyanisotropictrajectoriesthanthehalocentre(seee.g.Col´ın,Klypin&Kravtsov2000;Fukushige&Makino2001;Wojtaketal.2005;Mamon&Lokas2005;Cuestaetal.2007).Thisfea-ture,besidesthewell-studieddensityprofile,hasbeencon-sideredasthemainpointofreferenceintheattemptsatconstructionofareliablemodeloftheDF.Sofar,afewapproachestothisproblemhavebeenpro-posed.Cuddeford(1991)generalizedtheOsipkov-Merrittmodel(Osipkov1979;Merritt1985)totheDFwhichgener-atesanarbitraryanisotropyinthehalocentreandbecomesfullyradialatinfinity.Althoughananalyticalinversionforthesemodelsexists,theanisotropyprofilecannotberec-onciledwiththenumericalresults:therisefromcentraltoouteranisotropyistoosharpandtheouterorbitsaretooradial(seeMamon&Lokas2005).An&Evans(2006a)no-ticedthatanon-trivialprofileoftheanisotropycanbeob-tainedfromasumofDFswithaconstantanisotropyforwhichananalyticalinversionisknown(Cuddeford1991;c0000RAS2R.Wojtaketal.Kochanek1996;Wilkinson&Evans1999).However,there-sultinganisotropyprofilesaredecreasingfunctionsofra-diusanddonotagreewiththosemeasuredincosmologi-calsimulations.RecentlyaveryelegantmethodhasbeenpresentedbyBaes&vanHese(2007).Theauthorsintro-ducedageneralansatzfortheanisotropyprofileandthen,foragivenpotential-densitypair,derivedtheDFasaseriesofsomespecialfunctions.Thisapproachworkswellundertheconditionthatthepotentialcanbeexpressedasanele-mentaryfunctionofthecorrespondingdensity.Thisrequire-ment,however,isnotsatisfiedbymanymodels,includingtheNFWdensityprofile(Navarro,Frenk&White1997)whichiscommonlyusedasagoodapproximationoftheuniversaldensityprofileofDMhaloes.TheDFinferredfromthesimulationgivesapossibil-itytotestdirectlytheanalyticalmodels.AccordingtotheJeanstheoremanysphericallysymmetricsysteminthestateofequilibriumshouldpossessaDFwhichisafunctiononlyofenergyandtheabsolutevalueofangularmomentum.Thistheoreticalpostulatewastakenintoaccountinthecompu-tationcarriedoutbyVoglis(1994)andNatarajan,Hjorth&vanKampen(1997).InthefirstcasetheDFwasobtainedforasinglerelaxedhalowhichformedfromcosmologicallycon-sistentinitialconditions.ItwasshownthatthereweretwomaincontributionstotheDF,thehalopopulationandthecorepopulationofparticles.Bothwereeffectivelydescribedbytwoindependentphenomenologicalfits.Natarajanetal.(1997)determinedtheDFforasampleofcluster-sizehaloesformedincosmologicalsimulations.Theirselectionofob-jectsincludedthosewithsubstructures