Eulerian Perturbation Theory in Non-Flat Universes: Second Order Approximation

The problem of solving perturbatively the equations describing the evolution of self-gravitating collisionless matter in an expanding universe considerably simplifies when directly formulated in terms of the gravitational and velocity potentials: the problem can be solved exactly, rather than approximately, even for cosmological models with arbitrary density parameter Omega. The Eulerian approach we present here allows us to calculate the higher-order moments of the initially Gaussian density and velocity fields: in particular, we compute the gravitationally induced skewness of the density and velocity-divergence fields for any value of Omega, confirming the extremely weak Omega-dependence of the skewness previously obtained via Lagrangian perturbation theory. Our results show that the separability assumption of higher-order Eulerian perturbative solutions is restricted to the Einstein-de Sitter case only.