Evolution of Cosmological Density Distribution Function from(3)

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

1997).TheapproximationhasbeentestedagainstN-bodysimulationsandagoodagreementwasfoundeveninanon-linearregimeofthedensityperturbation.

Ontheotherhand,fromthenumericalstudy,Kayo,Taruya&Suto(2001)recentlyshowedthatthelog-normalPDFquantitativelyapproximatestheone-pointPDFP(δ)intheN-bodysimulationsemergingfromtheGaussianinitialcondition,irrespectiveoftheshapeofinitialpowerspectra.Thelog-normalPDFhasbeenlongknownasanempiricalmodelcharacterizingtheN-bodysimulations(e.g.,Kofmanetal.1994;Bernardeau&Kofman1995;Taylor&Watts2000)ortheobservedgalaxydistribution(e.g,Hamilton1985;Bouchetetal.1993;Kofmanetal.1994).Recently,agoodagreementwiththelog-normalmodelhasbeenreportedinthenumericalstudyofweaklensingstatistics(Taruyaetal.2002a),andanattempttoclarifytheoriginofthelog-normalbehaviorhasalsomade(Taruyaetal.2002b).Mathematically,thelog-normalPDFisobtainedfromaone-to-onelocalmappingbetweentheGaussianandthenon-lineardensity elds.Thus,thesuccessful ttotheN-bodysimulationmightbeinterpretedtoimplythattheevolutionoflocaldensitycanbewell-approximatedbytheone-to-onelocalmapping.Indeed,theanalyticPDFforthesphericalcollapseapproximationcanalsobeexpressedasaone-to-onelocalmappingviathesphericalcollapsemodel.

TheabovenaivepicturehasbeencriticallyexaminedbyKayo,Taruya&Suto(2001)evaluatingthejointPDFP(δ1,t1;δ2,t2),i.e.,jointprobabilityofthelocaldensity eldsδ1andδ2atthesamecomovingpositionbutatthedi erenttimest1andt2.Ithasbeenfoundthatalargeamountofscatterintherelationbetweenδ1andδ2showsupandtheirmeanrelationsigni cantlydeviatesfromthepredictionfromtheone-to-onelog-normalmapping.Althoughthismightnotbesurprising,thegoodagreementbetweenthelog-normalandthesimulatedPDFsbecomesmorecontrivedanddi culttobeexplainedinastraightforwardmanner.

De nitely,thefailureoftheone-to-onelocalmappingsomehowre ectsthenon-localityofthegravitationaldynamics.Thatis,theevolutionoflocaldensitycannotbedeterminedbytheinitialvalueofthelocaldensityonly.Rather,thelocaldensitycanbeexpressedasmulti-variatefunctionsoflocaldensityandotherlocalquantitiessuchasvelocity,gradientoflocaldensity,velocity-divergenceandsoon.Furthermore,initialvaluesoftheselocalquantitiesarerandomlydistributedoverthethree-dimensionalspace.Asaconsequence,evenifthedynamicsisdeterministic,therelationbetweentheevolvedandtheinitiallocaldensityinevitablybecomesstochastic.Inasense,thesituationisquitesimilartothenon-linearstochasticbiasingofgalaxies,i.e.,thestatisticaluncertaintybetweengalaxiesanddarkmatterarisingfromthehiddenvariables(e.g.,Dekel&Lahav1999;Taruya,Koyama&Soda1999;Taruya&Suto2000).Then,takingaccountofthisstochasticnature,the