Evolution of Cosmological Density Distribution Function from the Local Collapse Model

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 consider the stochastic nature associated with these quantities. Under the local approximation that the evolution of cosmic fluid fields can be characterized by the Lagrangian local dynamics with finite degrees of freedom, evolution equation for PDFs becomes a closed form and consistent formal solutions are constructed. Adopting this local approximation, we explicitly evaluate the one-point PDFs $P(\delta)$ and $P(\theta)$ from the spherical and the ellipsoidal collapse models as the representative Lagrangian local dynamics. In a Gaussian initial condition, while the local density PDF from the ellipsoidal model almost coincides with the that of the spherical model, differences between spherical and ellipsoidal collapse model are found in the velocity-divergence PDF. Importantly, the joint PDF of local density, $P(\delta,t;\delta',t')$, evaluated at the same Lagrangian position but at the different times $t$ and $t'$ from the ellipsoidal collapse model exhibits a large amount of scatter. The mean relation between $\delta$ and $\delta'$ does fail to match the one-to-one mapping obtained from spherical collapse model. Moreover, the joint PDF $P(\delta;\theta)$ from the ellipsoidal collapse model shows a similar stochastic feature, both of which are indeed consistent with the recent result from N-body simulations.