Evolution of the Cosmological Density Distribution Function from the Local Collapse Model

We present a general framework to treat the evolution of one-pointprobability distribution function (PDF) for cosmic density $\delta$ andvelocity-divergence fields $\theta$. In particular, we derive an evolutionequation for the one-point PDFs and consider the stochastic nature associatedwith these quantities. Under the local approximation that the evolution ofcosmic fluid fields can be characterized by the Lagrangian local dynamics withfinite degrees of freedom, evolution equation for PDFs becomes a closed formand consistent formal solutions are constructed. Adopting this localapproximation, we explicitly evaluate the one-point PDFs $P(\delta)$ and$P(\theta)$ from the spherical and the ellipsoidal collapse models as therepresentative Lagrangian local dynamics. In a Gaussian initial condition,while the local density PDF from the ellipsoidal model almost coincides withthe that of the spherical model, differences between spherical and ellipsoidalcollapse model are found in the velocity-divergence PDF. Importantly, the jointPDF of local density, $P(\delta,t;\delta',t')$, evaluated at the sameLagrangian position but at the different times $t$ and $t'$ from theellipsoidal collapse model exhibits a large amount of scatter. The meanrelation between $\delta$ and $\delta'$ does fail to match the one-to-onemapping obtained from spherical collapse model. Moreover, the joint PDF$P(\delta;\theta)$ from the ellipsoidal collapse model shows a similarstochastic feature, both of which are indeed consistent with the recent resultfrom N-body simulations.Comment: 35 pages, 5 figures, accepted for publication in ApJ (2003