A pr 1 99 4 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–Lemaˆıtre cosmogonies investigated and solved up to the third order in the series of papers by 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 Weiß 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 smooth the initial data by using a variety of filter types and filter scales in order to determine the optimal performance of the analytical models, as has been done for the " Zel'dovich– approximation " – hereafter TZA – (as a subclass of the irrotational Lagrangian first-order solution) in previous work (Melott et al. 1994a). We study cross–correlation statistics employed in previous work for power–law spectra having indices in the range (−3, +1). 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 results we expect that the second– order truncated Lagrangian model is especially useful for the modelling of standard dark matter models such as Hot–, Cold–, and Mixed–Dark–Matter.