Lagrangian theory of gravitational instability of Friedman-Lemaitre cosmologies -- second-order approach: an improved model for non-linear clustering

A large class of solutions for second-order irrotational perturbations is derived in the framework of the Lagrangian theory of gravitational instability of a homogeneous and isotropic universe investigated in earlier papers. The solutions are evaluated in detail for perturbations in a flat background universe. The form of the solutions is designed for use in studies of the formation of large-scale structure from generic initial conditions. Some general remarks on the properties of the solutions are made. The result is illustrated by a special case and discussed. In particular, it is found that sheetlike structures stay compact after shell-crossing (as in the competing `adhesion model'), and that the collapse of first objects occurs earlier (as expected from numerical simulations) in the second-order approach. Both these properties compensate shortcomings of the `Zel'dovich approximation'. In contrast to the `adhesion model', the nth-order Lagrangian perturbation solutions also describe internal structures of self-gravitating pancakes (=2n + 1 stream systems) in terms of the nth orbit crossings within pancakes.