Nonlinear evolution of cosmological inhomogeneities

Abstract

The nonlinear evolution of a cosmologically significant fluid is studied up to shell crossing. The magnetic part of the Weyl tensor, the pressure and the vorticity vanish. A suitable spatial grid is chosen. The relativistic Ellis equations are particularized on the world lines defined by the nodes of the grid and, then, the resulting equations are numerically solved. The integrations are performed in suitable Lagrangian inertial coordinates, in which the differential equations become ordinary. After the integration, a method to change from Lagrangian to Eulerian coordinates is applied. This approach has been outlined with the essential aim of studying the evolution of large scale cosmological structures. In this case, no important relativistic effects are expected, but a relativistic approach based on the Ellis formalism appears to be suitable due to two main reasons: it facilitates a rigorous choice of the initial conditions according to the gauge invariant Bardeen formalism, and it is not more involved than some nonrelativistic schemes. In order to test the method, its results are compared with the Tolman-Bondi solution in the spherically symmetric case. The comparison is very encouraging. In the limit of vanishing numerical errors, our method approaches an exact 3-dimension solution to the problem of structure evolution in the mildly nonlinear regime; hence, this solution improves on the Zel'dovich one, which is exact only in the 1-dimension case. Both solutions apply up to shell crossing. The extension of the proposed approach beyond caustic formation deserves attention.