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Astron. Astrophys. 318, 1-10 (1997)

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The performance of Lagrangian perturbation schemes at high resolution

Thomas Buchert 1, Georgios Karakatsanis 2, 3, Robert Klaffl 2 and Peter Schiller 2

1 Theoretische Physik, Ludwig-Maximilians-Universität, Theresienstr. 37, D-80333 München, Germany
2 Max-Planck-Institut für Astrophysik, Postfach 1523, D-85740 Garching, Germany
3 National Observatory of Athens, Lofos Nimfon, Thesio, 18115 Athens, Greece

Received 27 October 1995 / Accepted 18 March 1996


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 et al. 1996).

Results of the present paper include the following: 1. the collapse is accelerated significantly by the higher-order corrections confirming previous results by Moutarde et al. (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.

Key words: gravitation – instabilities – methods: analytical – cosmology: large-scale structure of universe

Send offprint requests to: T. Buchert

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998