Evolution of the cosmological density distribution function from the local collapse model

We present a general framework to treat the evolution of one-point probability distribution functions (PDFs) 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, the 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 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 that of the spherical model, differences between spherical and ellipsoidal collapse models are found in the velocity divergence PDF. These behaviors have also been confirmed from the perturbative analysis of higher order moments. 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 the 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. Hence, the local approximation with the ellipsoidal collapse model provides a simple but more physical model than the spherical collapse model of cosmological PDFs, consistent with the leading-order results of exact perturbation theory.