Temporal optimization of Lagrangian perturbation schemes

The Lagrangian perturbation theory on Friedmann– Lemaı̂tre cosmologies is compared with numerical simulations (tree–, adaptive P3M– and PM codes). In previous work we have probed the large–scale performance of the Lagrangian perturbation solutions up to the third order by studying their cross–correlations with N–body simulations for various power spectra (Buchert et al. 1994, Melott et al. 1995, Weiß et al. 1996). Thereby, spatial optimization techniques were applied by (high–frequency–)filtering of the initial power spectra. In this work the novel method of temporal optimization [Shifted– Time–Approximation (STA) and Frozen–Time–Approximation (FTA)] is investigated and used. Themethod is designed to compensate the native property of Lagrangian perturbation solutions to delay the collapse of structures. The method can be treated analytically. Applying the STA and FTA prescriptions a significant improvement of the performance of Lagrangian perturbation schemes up to r.m.s density contrast of about 10 (as measured by cross–correlation, relative phase error and power– spectrum statistics) is observed. Using this tool we investigate a local study of special clustering models of dark matter as candidates for typical elements of the large–scale structure in the Universe, and so also focus on the performance of the perturbation solutions on smaller scales at high–spatial resolution. The models analyzed were presented in (Buchert et al. 1997) and allow studying typical features of the clustering process in the non–linear regime. The spatial and temporal limits of applicability of the solutions at second and third order are determined and compared with the first–order solution, which is equivalent to the “Zel’dovich approximation” (Zel’dovich 1970, 1973) for the type of initial data analyzed.