The performance of Lagrangian perturbation schemes at high resolution

Abstract

We present high--spatial resolution studies of the density field as predicted by Lagrangian perturbation approximations up to the third order. The first--order approximation is equivalent to the ``Zel'dovich approximation'' for the type of initial data analyzed. The study is performed for two simple models which allow studying of typical features of the clustering process in the early non--linear regime. We calculate the initial perturbation potentials as solutions of Poisson equations algebraically, and automate this calculation for a given initial random density field. The presented models may also be useful for other questions addressed to Lagrangian perturbation solutions and for the comparison of different approximation schemes. In an accompanying paper we investigate a detailed comparison with various N--body integrators using these models (Karakatsanis \& Buchert 1995). Results of the present paper include the following: 1. The collapse is accelerated significantly by the higher--order corrections confirming previous results by Moutarde \etal (1991); 2. the spatial structure of the density patterns predicted by the ``Zel'dovich approximation'' differs much from those predicted by the second-- and third--order Lagrangian approximations; 3. Second--order effects amount to internal substructures such as ``second generation'' --pancakes, --filaments and --clusters, as are also observed in N--body simulations; 4. The third--order effect gives rise to substructuring of the secondary mass--shells. The hierarchy of shell--crossing singularities that form features small high--density clumps at the intersections of caustics which we interprete as gravitational fragmentation.