TESTING HIGHER-ORDER LAGRANGIAN PERTURBATION-THEORY AGAINST NUMERICALSIMULATIONS .2. HIERARCHICAL-MODELS

We present results showing an improvement of the accuracy of perturbat ion theory as applied to cosmological structure formation for a useful range of scales. The Lagrangian theory of gravitational instability o f Friedmann-Lemaitre cosmogonies is compared with numerical simulation s. In this paper we study the dynamics of hierarchical models as a sec ond step. In the first step (Buchert et al. 1994) we analyzed the perf ormance of the Lagrangian schemes for pancake models, the difference b eing that in the latter models the initial power spectrum is truncated . This work probed the quasi-linear and weakly non-linear regimes. We here explore whether the results found for pancake models carry over t o hierarchical models which are evolved deeply into the non-linear reg ime. We smooth the initial data by using a variety of filter types and filter scales in order to determine the optimal performance of the an alytical 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 cros s-correlation statistics employed in previous work for power-law spect ra having indices in the range (-3, +1). We find that for spectra with negative power-index the second-order scheme performs considerably be tter than TZA in terms of statistics which probe the dynamics, and sli ghtly better in terms of low-order statistics like the power-spectrum. In cases with much small-scale power the gain from the higher-order s chemes is small, but still measurable. However, in contrast to the res ults found for pancake models, where the higher-order schemes get wors e 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 improv ement at later stages and on smaller scales. In view of these results we expect that the second-order truncated Lagrangian model is especial ly useful for the modelling of standard dark matter models such as Hot -, Cold-, and Mixed-Dark-Matter.