The quasi-linear evolution of the density field in models of gravitational instability

Two quasi-linear approximations, the frozen-flow approximation (FFA) and the frozen-potential approximation {FPA), have been proposed recently for studying the evolution of a collisionless, self-gravitating fluid. In FFA it is assumed that the velocity field remains unchanged from its value obtained from the linear theory, whereas in FPA the same approximation is made for the gravitational potential. In this paper, we compare these and the older Zel'dovich approximation by calculating the evolution of the density in perturbation theory. In particular, we compute the skewness, including the smoothing effects, and the kurtosis for FFA, FPA and the Zel'dovich approximation, and compare their relative accuracy.