Cosmological velocity-density relation in the quasi-linear regime

Practical methods developed for extracting the mass-density fluctuation field of a cosmological system from the peculiar velocity field smoothed on scales of a few Mpc are presented. The methods are local and are based on quasi-linear approximations to the gravitational equations of the motion of a pressureless fluid. They address density fluctuations in the range between 0.7 and 4.5 and are tested against exact solutions in special configurations and against cosmological N-body simulations with Omega = 1 and with Omega = 2. Consideration is given to the exact solution of the continuity equation and to the exact solution of the dynamical Euler-Poisson equation, which turns out to coincide with the linear approximation. The density fluctuation field, based on Lagrangian Zel'dovich approximation, is derived in terms of the partial derivatives of the Eulerian velocity field. It is found that both the continuity equation and the Euler-Poisson equation are closely related to the true density, with a standard deviation of not more than 0.1.