Cosmological velocity-density relation in the quasi-linear regime

We develop practical methods for extracting the mass-density fluctuation field of a cosmological system, δ(x), from the peculiar velocity field smoothed on scales of a few Mpc. The methods are local. They are based on quasi-linear approximations to the gravitational equations of motion of a pressureless fluid and address density fluctuations in the range -0.7 ≤ δ ≤ 4.5. They are tested against exact solutions in special configurations and against cosmological N-body simulations with Ω = 1 and with Ω = 0.2. Two approximations are considered under the assumption of Zel'dovich displacements of particles with a universal time dependence: the exact solution of the continuity equation, δ, and the exact solution of the dynamical Euler-Poisson equation, δ_d, which turns out to coincide with the linear approximation, δ₀. A key new result is our derivation of the density fluctuation field, based on the Lagrangian Zel'dovich approximation, in terms of the partial derivatives of the Eulerian velocity field. We find that both δ_d and δ_c are tightly related to the true density, with a standard deviation ≤0.1. While δ_d systematically underestimates the true density, δc is an excellent approximation to δ, with an rms error ≲0.1. Alternatively, we find an empirical third-order approximation for δ(δ₀) with a similar rms error. Corrections involving second derivatives are less successful, with an rms error ∼0.3. The continuity density scheme is now being used in the POTENT analysis of the large-scale velocities. The tight relation between the true density and the linear approximation in the quasi-linear regime also suggests a method for the inverse problem of extracting the quasi-linear velocity from a given density field specified over a large region. This new inversion procedure can improve the prediction of the peculiar velocity field from galaxy redshift surveys such as the IRAS survey.