Evolution of Cosmological Density Distribution Function from(17)

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

3.2.Results

IncomputingthePDFsfromtheabovelocalcollapsemodels,onemaypracticallyen-counterthecasewhenthelocaldensityin nitelydivergesat niteelapsedtimeforsomeregionsintheinitialparameterspace,whichhasnotbeentreatedinprevioussection.Toavoidtheunphysicaldivergences,wemustrestricttheintegralinthePDFstotheinitialpa-rameterspaceV(t),inwhichthelocaldensityδdoesnotdiverge.Indeed,thismodi cationslightlya ectsthenormalizationconditionforboththeLagrangianandtheEulerianPDFs,i.e., 1 L,E=1.AlthoughthisdoesnotalteranyqualitativefeaturesofPDFs,weconsider

somemodi cationstokeepthecorrectnormalizationandadoptingthisprocedureinap-pendixA,andtheresultsforone-pointandjointPDFsarepresentedbelow.Note,however,thattheperturbationcalculationdiscussedin3.2.1isfreefromtheseriousdivergencesandwithintheperturbativetreatment,onecanrigorouslydevelopthelocalapproximationforPDFs.

3.2.1.One-pointPDFs

Letusshowtheresultsoftheone-pointPDF.Figure2plotstheone-pointEulerianPDFsofthelocaldensity(top)andthevelocity-divergence(bottom)evaluatedatvariouslinear uctuationvaluesσl.Inbothpanels,thethicklinesrepresenttheresultsobtainedfromtheellipsoidalcollapsemodelwithlinearexternaltide,whilethethinlinesdenotethePDFsfromthesphericalcollapsemodels.IncomputingthePDFs,theLagrangianlocaldynamicsarenumericallysolvedwiththevariousinitialconditionspintheinitialparameterspaceV.Then,weightingbythePDFoftheinitialparameterPI(p),thePDFsareconstructed

bybinningtheevolvedresultsofthedensityδandthevelocity-divergenceθ,togetherwithappropriateconvolutionfactors(seeAppendixA).

Asexpected,theoverallbehaviorsofbothPDFsinFigure2arequalitativelysimilar,irrespectiveoftheLagrangianlocalmodels.Asincreasingthelinear uctuationvalueσl,whilethedensityPDFPE(δ)extendsoverthehigh-densityregionδ 1,thevelocity-

divergencePDFPE(θ)isnegativelyskewedanditextendsoverthenegativeregionθ 1.

Inlookingatthedi erencesineachlocalmodel,wereadilyobserveseveralremarkablefeatures.First,thedensityPDFscomputedfromboththesphericalandtheellipsoidalcollapsemodelsalmostagreewitheachother.At rstglance,thisseemstocontradictwithanaiveexpectationfromthelocaldynamicsinFigure1.However,onemightrathersuspectthattheagreementindensityPDFsisaccidental,duetothedistributionofinitialparametersPI(λi),whichis,atleast,consistentwithanaivepicturethatjointPDFPL(δ,t;δ′,t′)fromthe

ellipsoidalcollapsemodelexhibitsalargemountscatterandthemeanrelationbetweenδand