Thermodynamics and dynamics of a 1-D gravitational

arXiv:astro-ph/0601390v227Sep20061Thedynamicsoflarge-scalestructureformationintheuniversebygravitationalinstabilitystillpresentsmanyopenissuesandismostlystudiedthroughnumericalsim-ulations.Thismotivatesthestudyofsimplermodelswhichcanbeinvestigatedbyanalyticalmeansinordertounder-standthemainprocessesatwork.Thus,wedescribehereaone-dimensionalself-gravitatingsystemderivedfromthecosmologicalcontext,whichleadstoaneffectiveexternalpotentialinadditiontothestandardgravitationalself-interaction.Asafirststepweconsidersmalltimessothattheexpansioncanbeneglected.Thenwepresentathermodynamicalanalysisofthissystemaswellasthestabilitypropertiesoftheassociatedhydrodynamicalandcollisionlesssystems.Weconsiderthemeanfieldlimit(i.e.continuumlimit)toperformananalyticalstudy.Wefindasecond-orderphasetransitionatTc1fromanhomogeneousequilibriumathightemperaturetoaclusteredphase(withadensitypeakatoneoftheboundariesofthesystem)atlowtemperature.Therealsoexistsaninfiniteseriesofun-stableequilibriawhichappearatlowertemperaturesTcn,reflectingthescale-freenatureofthegravitationalinter-actionandtheusualJeansinstability.Wefindthat,asforthesimilarHMF(Hamiltonianmeanfield)model,allthreemicro-canonical,canonicalandgrand-canonicalen-semblesagreewitheachother,aswellaswiththestabil-itypropertiesassociatedwithahydrodynamicalapproach.Ontheotherhand,thecollisionlessdynamicsgovernedbytheVlasovequationyieldsthesameresultsexceptthatatlowTtheequilibriumassociatedwithtwodensitypeaks(oneateachboundary)becomesstable.Keywords.gravitation;cosmology:theory–large-scalestructureofUniverseAstronomy&Astrophysicsmanuscriptno.4472February4,2008(DOI:willbeinsertedbyhandlater)Thermodynamicsanddynamicsofa1-DgravitationalsystemP.ValageasServicedePhysiqueTh´eorique,CENSaclay,91191Gif-sur-Yvette,FranceReceived4November2005/Accepted11January2006Abstract.1.IntroductionInstandardcosmologicalscenariosthelargescalestruc-turesweobserveinthepresentuniverse(suchasclusters,filamentsorvoids)haveformedthroughtheamplifica-tionbygravitationalinstabilityofsmallprimordialper-turbations,seePeebles(1980).Moreover,theamplitudeofthesedensityfluctuationsincreasesatsmallscalesasintheCDMmodel(Peebles1982)whichleadstoahi-erarchicalprocesswheresmallerscalesbecomenon-linearfirst.Then,atlargescalesoratearlytimesonecanuseaperturbativeapproachtostudytheevolutionofini-tialfluctuations(Fry1984;Goroffetal.1986;Bernardeau1992;Valageas2001).Next,theweaklynon-linearregimemaybeinvestigatedthroughtheZeldovichapproxima-tion(Shandarin&Zeldovich1989)ortheadhesionmodel(Gurbatovetal.1989).However,thehighlynon-linearregime(whichcorrespondstocollapsedstructuressuchasclustersofgalaxies)hasmostlyremainedoutofreachofsystematicapproaches.Thus,onemayusethePress-Schechterapproximation(Press&Schechter1974)ortheexcursion-setformalismofBondetal.(1991)toobtainthestatisticsofjust-virializedhalosorthesaddle-pointapproachofValageas(2002)forrarevoids.Theseapprox-imationsfocusonspecificobjectswhichmaybe“recog-nized”fromindividualfeaturesinthelineardensityfielditselfanddonotfollowthesystemasawholethroughitsnon-linearevolution.AsystematicmethodtodosowasrecentlydevelopedinValageas(2004)butitsapplicationtothecollisionlessdynamicshasnotbeenperformedyetforthehighlynon-linearregime.Therefore,thenon-linearregimeofcosmologicalstructureformationismostlystud-iedthroughnumericalsimulations.Inordertosimplifythisdifficultproblemonecaninves-tigateone-dimensional(1-D)systemswhichareeasiertostudyboththroughnumericalandanalyticalapproaches.Infact,1-Dsystemssuchasthesystemofparallelmasssheets(Camm1950)havebeenstudiedforalongtimetoexploretheevolutionofisolatedN−bodysystemswhichonlyinteractthroughclassicalgravity.Inparticular,theyhavebeenusedtoinvestigaterelaxationprocessesandtotestLynden-Bell’spredictionforthefinalstateofviolentrelaxation(Lynden-Bell1967).Such1-Dsystemscanal-readyexhibitcomplexbehaviours.Forinstance,Luwel&Severne(1985)foundthatcollisionaleffectsarenotsufficienttorelaxthestationarywaterbagconfigurationtowardsthermodynamicalequilibriumwhereasRouet&Feix(1999)showedthatholesinphase-spaceintheini-tialdistributionfunctioncanpersistoverlongtimesandpreventefficientrelaxation.Ontheotherhand,thesystemcanbreakupintosmallerclustersasinHohl&Feix(1967)ordevelopafractalstructureasinKoyama&Konishi(2001).Inthispaper,weconsiderthe1-Dgravitationalsys-temobtainedbystudying1-Ddensityfluctuationsina3-Dcosmologicalbackground.ThismodelhasalreadybeenstudiedmostlythroughnumericalsimulationinAurell&Fanelli(2001),Aurelletal.(2001)andFanelli&Aurell(2002),bothwithandwithoutcosmologicalexpansion.Herewerestrictourselvestotime-scalesmuchsmallerthantheHubbletimesothattheexpansionoftheuni-versecanbeneglectedandwefocusonameanfieldanal-ysis(i.e.acontinuumlimit)whichisrelevantinthecos-mologicalcontext.Westudythethermodynamicsandsta-bilityofthissystem,similarlytothetheoreticalanalysisofChavanisetal.(2005)performedfortheHMFmodel(definedbyacosineinteraction)whichshowsasimilarbehaviour.Weintroducethismodelinsect.2,wheretheeffectofthecosmologicalbackgroundisseentoreducetoaneffectiveexternalpotentialVwhic