Local Lagrangian approximations for the evolution of the density distribution function in large-scale structure

We examine local Lagrangian approximations for the gravitational evolution ofthe density distribution function. In these approximations, the final densityat a Lagrangian point q at a time t is taken to be a function only of t and ofthe initial density at the same Lagrangian point. A general expression is givenfor the evolved density distribution function for such approximations, and weshow that the vertex generating function for a local Lagrangian mapping appliedto an initially Gaussian density field bears a simple relation to the mappingitself. Using this result, we design a local Lagrangian mapping whichreproduces nearly exactly the hierarchical amplitudes given by perturbationtheory for gravitational evolution. When extended to smoothed density fieldsand applied to Gaussian initial conditions, this mapping produces a finaldensity distribution function in excellent agreement with full numericalsimulations of gravitational clustering. We also examine the application ofthese local Lagrangian approximations to non-Gaussian initial conditions.Comment: LaTeX, 22 pages, and 11 postscript figure