Gravitational Structure Formation

We discuss the formation of the first structures in gravitational N-body simulations. The role of two-body interaction is found to be a crucial element and an analogy with the dynamics of the Coulomb lattice, well-studied in solid state physics, is discussed. The standard models of the formation of large scale structure of the universe is based on the gravitational growth of small initial density fluctuations in a homogeneous and isotropic medium (e.g. Peebles 1980). For example, in the so-called Cold Dark Matter (CDM) model, particles interact only gravitationally and they are cold, i.e. with very small initial velocity dispersion. This situation allows one to model this system with a collision-less Boltzmann equation and, for sufficiently large scales, pressure-less fluid equations. Then it is possible to solve in a perturbative way, for small density fluctuations, these fluid equations (see e.g. Peebles 1980). However this treatment is inapplicable in the strong nonlinear regime. Then, the most widely used tool to study gravitational clustering in the various regimes is by means of N-body simulations (NBS) which are based on the computation of the dynamics of self-gravitating particles in expanding universe. These simulations can be performed by considering an infinite periodic system, i.e. a finite system with periodic boundary condition. Despite the simplicity of the system, in which dynamics are Newtonian at all but the smallest scales, the analytic understanding of this crucial problem is limited to the regime of very small fluctuations where a linear analysis can be performed. In the cosmological case, the problem can be approximated to Newtonian but the equation of motions are modified because of the expanding background (Peebles, 1980). As discussed below, we find instructive to consider some simplified cases where the expansion is not included and then study the differences introduced by space expansion. An important point should be stressed: for numerical reasons, the cosmological density field must be discretized into “macro-particles” interacting gravitationally which are tens of order of magnitudes heavier than the (elementary) CDM particles due to computer limitations. This procedure introduces