Cosmological Perturbation Theory and the Spherical Collapse model; 1, Gaussian initial conditions

We present a simple and intuitive approximation for solving perturbation theory (PT) of small cosmic fluctuations. We consider only the spherically symmetric or monopole contribution to the PT integrals, which yields the exact result for tree-graphs. We find that the non-linear evolution is then given by a local transformation over the initial conditions. In the case of gravitational evolution of density fluctuations, this transformation is found to be 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 perturbative regime. To leading order this reproduces, and provides with a simple explanation for, the exact results obtained by Bernardeau (1992, 1994). We then show that the SC model leads to accurate estimates for the next corrective terms when compared to the results derived in the exact PT making use of the loop calculations (Scoccimarro & Frieman 1996). 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 to N-body simulations, which turn out to be in very good agreement up to scales where $\sigma \approx 1$. A similar treatment is presented to estimate higher order corrections in the Zel'dovich approximation. These results represent a powerful and easy-to-use tool to produce analytical predictions for the cosmological fields in the weakly non-linear regime. A generalisation of the current approach to non-Gussian initial conditions and non-flat FRW universes will be provided in two accompanying papers (Papers II and III)