LOCAL LAGRANGIAN APPROXIMATIONS FOR THE EVOLUTION OF THE DENSITY DISTRIBUTION FUNCTION IN LARGE-SCALE STRUCTURE

We examine local Lagrangian approximations for the gravitational evolu tion of the density distribution function. In these approximations, th e final density at a Lagrangian point q at a time t is taken to be a f unction only of t and the initial density at the same Lagrangian point . A general expression is given for the evolved density distribution f unction for such approximations, and we show that the vertex generatin g function for a local Lagrangian mapping applied to an initially Gaus sian density field bears a simple relation to the mapping itself. Usin g this result, we design a local Lagrangian mapping which reproduces n early exactly the hierarchical amplitudes given by perturbation theory for gravitational evolution. When extended to smoothed density fields and applied to Gaussian initial conditions, this mapping produces a f inal density distribution function in excellent agreement with full nu merical simulations of gravitational clustering. We also examine the a pplication of these local Lagrangian approximations to non-Gaussian in itial conditions.