Evolution of one-point distributions from Gaussian initial fluctuations

We study the quasi-linear evolution of the one-point probability density functions (PDFs) of the smoothed density and velocity fields in a cosmological gravitating system beginning with Gaussian initial fluctuations. Our analytic results are based on the Zel'dovich approximation and laminar flow. A numerical analysis extends the results into the multistreaming regime using the smoothed fields of a CDM N-body simulation. We find that the PDF of velocity, both Lagrangian and Eulerian, remains Gaussian under the laminar Zel'dovich approximation, and it is almost indistinguishable from Gaussian in the simulations. The PDF of mass density deviates from a normal distribution early in the quasi-linear regime and it develops a shape remarkably similar to a lognormal distribution with one parameter, the rms density fluctuation σ. Applying these results to currently available data we find that the PDFs of the velocity and density fields, as recovered by the POTENT procedure from observed velocities assuming Ω = 1, or as deduced from a redshift survey of IRAS galaxies assuming that galaxies trace mass, are consistent with Gaussian initial fluctuations.