Evolution of Cosmological Density Distribution Function from(21)

We present a general framework to treat the evolution of one-point probability distribution function (PDF) for cosmic density $\delta$ and velocity-divergence fields $\theta$. In particular, we derive an evolution equation for the one-point PDFs and consid

sidertheEulerianjointPDFPE(δ,θ;t)(eq.[36])andtheLagrangianjointPDFPL(δ,t;δ′,t′)

(eq.[38]).Then,adoptingthesphericalandtheellipsoidalcollapsemodelsasrepresentativeLagrangianlocaldynamics,weexplicitlyevaluatetheEulerianPDFs,PE(δ)andPE(θ),as

wellasthejointPDFs.Theresultsfromtheellipsoidalcollapsemodelshowseveraldis-tinctproperties.WhilethePDFPE(δ)almostcoincideswiththeone-to-onemappingof

thesphericalcollapsemodel,thetailsofvelocity-divergencePDFPE(θ)largelydeviatefrom

thoseobtainedfromthesphericalmodel.Thesebehaviorshavealsobeencon rmedfromtheperturbativeanalysisofhigherordermoments.Ontheotherhand,evaluatingtheLa-grangianjointPDFoflocaldensity,PL(δ,t;δ′,t′),alargescatterintherelationbetweenthe

initialandtheevolveddensity eldswasfoundandtheirmeanrelationfailstomatchtheone-to-onemappingofsphericalcollapsemodel.Thisremarkablyreproducesthesamesitu-ationintheN-bodysimulation.Therefore,thelocalapproximationwithellipsoidalcollapsemodelprovidesasimpleandphysicallyreasonablemodelofone-pointstatistics,consistentwiththeleading-orderresultsofexactperturbationtheory.

Sincethepresentformalismdescribedinsection2isquitegeneral,theapproachdoesnotrestrictitsapplicabilitytothepressure-lesscosmological uid.Rather,onemayapplytothevarious uidsystemsinpresenceorabsenceofgravity.Asmentionedinsection

2.4,however,theapplicabilityorthevalidityoflocalapproximationofPDFs,inprinciple,sensitivelydependsonthechoiceofLagrangianlocaldynamics.Inthelastsection,simpleandintuitiveexampleswereexaminedfortheillustrativepurposes.TheresultsindicatethatthemultivariateLagrangiandynamicsratherthanthelocalmodelwithasinglevariablecandescribevariousstatisticalfeaturesof uidevolutionincludingthestochasticnature.

Perhaps,astraightforwardextensionofthepresenttreatmentistoincludethee ectofredshift-spacedistortionorprojectione ect,whichispracticallyimportantforpropercomparisonwithobservation.Beforeaddressingthisissue,however,rememberthemostprimarilyimportanceofthesmoothinge ect.Whiletheappropriateprescriptionfortop-hatsmoothing lterdoesexistinthelocalapproximationwiththesphericalcollapsemodel(e.g.,Bernardeau1994b;Protogeros&Scherrer1997;Fosalba&Gazta naga1998a),thesmoothinge ectontheapproximationusingellipsoidalcollapsemodelstillneedstobeinvestigated.Thisstepisinparticularacrucialtaskinordertoconstructamorephysicalprescriptionforone-pointstatisticsofcosmic eldsandtheanalysisisnowinprogress.Theresultswillbepresentedelsewhere(Ohta,Kayo&Taruya,inpreparation).

WearegratefultoY.Sutoforreadingthemanuscriptandcomments.I.KacknowledgesthesupportbyTakenaka-IkueikaiFellowship.Thisworkissupportedinpartbythegrand-in-aidforScienti cResearchofJapanSocietyforPromotionofScience(No.1470157).