A new approach to gravitational clustering: a path-integral formalism and large-N expansions

Abstract

We show that the formation of large-scale structures through gravitational instability in the expanding universe can be fully described through a path-integral formalism. We derive the action S[f] which gives the statistical weight associated with any phase-space distribution function f(x,p,t). This action S describes both the average over the Gaussian initial conditions and the Vlasov-Poisson dynamics. Next, applying a standard method borrowed from field theory we generalize our problem to an N-field system and we look for an expansion over powers of 1/N. We describe three such methods and we derive the corresponding equations of motion at the lowest non-trivial order for the case of gravitational clustering. This yields a set of non-linear equations for the mean $\fb$ and the two-point correlation G of the phase-space distribution f, as well as for the response function R. These systematic schemes match the usual perturbative expansion on quasi-linear scales but should also be able to handle the non-linear regime. Our approach can also be extended to non-Gaussian initial conditions and may serve as a basis for other tools borrowed from field theory.