The gravity-induced quasi-Gaussian correlation hierarchy

We calculate the correlation hierarchy that first emerges from a Gaussian field, with a given initial spectrum, owing to the nonlinearities of the dynamics in an expanding universe. The mean matter p-point correlation functions are shown to be given by a set of characteristic numbers S_p, so that 〈δ^p〉_c = S_p〈δ²〉^p-1 at the lowest order in 〈δ²〉, where 〈...〉_c stands for the connected part of the expectation values. These parameters, calculated assuming no cosmological constant, enlarge at every order the results obtained by Peebles and Fry for S₃ and S₄. They turn out to be completely independent of the initial spectrum, and almost independent of Ω. The (Poisson sampling) resulting void probability function, P₀, is presented. We find that (ln P₀)/nV ∼ -(nVσ²)^-3/7 for large (nVσ²), where n and σ² are respectively the density and the variance of the field in the volume V. The density distribution is significantly different from a Gaussian distribution and is found to be asymmetrical, always with a positive density, and with an extended large tail for the large overdensities. Extrapolations toward high nonlinear regimes are also discussed.