Third‐Order Perturbative Approach to Gravitational Instability: Evolution of Isolated Structures and Environmental Effects

Abstract

In an Ω0 = 1 universe, within the classical Eulerian theory of gravitational instability, the redshift evolution of a peculiar velocity field in a region with arbitrary initial density contrast is derived, for the first time, in real space and third-order perturbation theory. A vector proportional to the gravitational acceleration can also be expanded in terms of the redshift and the initial density contrast. The results are applied to isolated (spherically symmetric) superclusters and voids. Using reasonable models and the exact solution, we tested the accuracy of three (linear, second order, and third order) approaches. A numerical example showed that the relative error of the third-order solutions (average density contrast and peculiar velocity) is less than 5% when 0 < δ 1. In another example, a relative error was derived (at -1 < δ < 0) of less than 10% (average density contrast) to 2% (peculiar velocity). On the other hand, second-order environmental dynamical terms (supercluster-supercluster, supercluster-void, and void-void complexes) have been also obtained. In the complexes (which contain two large-scale structures with spherical symmetry at recombination), the global peculiar flow can be described as a natural (but not trivial) superposition of two effective peculiar flows. Given a member of a complex, its effective peculiar velocity field is the sum of a spherically symmetric radial field (which is equal to the peculiar velocity field obtained from an isolated evolution) and an environmental (due to the interaction with the companion) field. In general, the external tides can be comparable to the internal ones. The imprint of the environment fields in the mean radial effective peculiar flows is also studied.