Testing Higher--Order Lagrangian Perturbation Theory Against Numerical Simulations -- 2. Hierarchical Models

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of scales. The Lagrangian theory of gravitational instability of Friedmann--Lema\^\i tre cosmogonies investigated and solved up to the third order in the series of papers by Buchert (1989, 1992, 1993), Buchert \& Ehlers (1993), Buchert (1994), Ehlers \& Buchert (1994), is compared with numerical simulations. In this paper we study the dynamics of hierarchical models as a second step. In the first step (Buchert, Melott and Wei{\ss} 1994) we analyzed the performance of the Lagrangian schemes for pancake models, i.e., models which initially have a truncated power spectrum. We here explore whether the results found for pancake models carry over to hierarchical models which are evolved deeply into the non--linear regime............We find that for spectra with negative power--index the second--order scheme performs considerably better than TZA in terms of statistics which probe the dynamics, and slightly better in terms of low--order statistics like the power--spectrum. In cases with much small--scale power the gain from the higher--order schemes is small, but still measurable. However, in contrast to the results found for pancake models, where the higher--order schemes get worse than TZA at late non--linear stages and on small scales, we here find that the second--order model is as robust as TZA, retaining the improvement at later stages and on smaller scales. In view of these