Cosmological Distribution Functions

Abstract

The evolution of probability distribution functions (PDFs) of continuous density, velocity and velocity derivatives ( deformation tensor) fields in the theory of cosmological gravitational instability are considered. We show that in the Newtonian theory the dynamical equations cannot be reduced to the closed set of Lagrangian equations. Since continuous fields from galaxy surveys need sufficiently large smoothing which exceeds the scale of nonlinearity, one can use the Zel'dovich approximation to describe the mildly non-linear matter evolution, which allows the closed set of Lagrangian equations. The closed kinetic equation for the joint PDF of cosmological continuous fields is derived in this approximation. The analytical theory of the cosmological PDFs with arbitrary (including Gaussian) initial statistics is developed, based on the solution on the kinetic equaiton. For Gaussian initial fluctuations, the PDFs are parametrized by only linear {\it rms} fluctuations $σ$ on given filtering scale. Density PDF $P(ρ, t)$ and PDF $M(λ_1, λ_2, λ_3; t)$ of eigenvalues of the deformation tensor field in the Eulerian space evolve very rapidly in non-linear regime. On the contrary, velocity PDF $Q(\vec v, t)$ remains invariant under non-linear evolution. For small $σ$ the Edgworth series is suggested to reconstruct