Caustics in Self-gravitating N-body systems and Cosmological Large Scale Structures

Extended galaxy surveys revealed that at a large scale the Universe consists of matter concentrations in the form of galactic clusters, filaments and vast regions devoid of galaxies. In this paper we demonstrate the generation of caustics in self-gravitating N-body system and the formation of cosmological large-scale structure due to the gravitational focusing. The gravitational caustics are space regions where the density of matter is higher than the average density in the surrounding Universe. The gravitational caustics can represent galaxies, galactic clusters and filaments and the regions between caustics, the voids. In our approach the dynamics of a self-gravitating N-body system is formulated in terms of a geodesic flow on a curved Riemannian manifold of dimension 3N equipped by the Maupertuis's metric. The regions of negative sectional curvatures are responsible for the exponential instability of geodesic trajectories, while the regions of positive sectional curvatures are responsible for the gravitational focusing of geodesics and generation of caustics. The stability of geodesic trajectories is analysed by means of the Jacobi equation and the gravitational focusing of geodesics by means of the Raychaundhuri equation. By solving these equations we estimated the characteristic relaxation time scales and the time scale of generation of gravitational caustics.