Cosmological perturbation theory and the spherical collapse model - I. Gaussian initial conditions

We present a simple and intuitive approximation for solving the perturbatio n theory (PT) of small cosmic fluctuations. We consider only the sphericall y symmetric or monopole contribution to the PT integrals, which yields the exact result for tree-graphs (i.e. at leading order). We find that the non- linear evolution in Lagrangian space is then given by a simple local transf ormation over the initial conditions, although it is not local in Euler spa ce. This transformation is found to be described by the spherical collapse (SC) dynamics, as it is the exact solution in the shearless (and therefore local) approximation in Lagrangian space. Taking advantage of this property , it is straightforward to derive the one-point cumulants, xi(J), for both the unsmoothed and smoothed density fields to arbitrary order in the pertur bative regime. To leading-order this reproduces, and provides us with a sim ple explanation for, the exact results obtained by Bernardeau. We then show that the SC model leads to accurate estimates for the next corrective term s when compared with the results derived in the exact perturbation theory m aking use of the loop calculations. The agreement is within a few per cent for the hierarchical ratios S-J = xi(J)/xi(2)(J-1). We compare our analytic results with N-body simulations, which turn out to be in very good agreeme nt up to scales where sigma approximate to 1. A similar treatment is presen ted to estimate higher order corrections in the Zel'dovich approximation. T hese results represent a powerful and readily usable tool to produce analyt ical predictions that describe the gravitational clustering of large-scale structure in the weakly non-linear regime.