Cosmological perturbation theory and the spherical collapse model - III. The velocity divergence field and the Omega dependence

Cosmological perturbation theory (PT) is a useful tool to study the cumulan ts of the density and velocity fields in the large-scale structure of the U niverse. In Papers I and II of this series we saw that the spherical collap se (SC) model provides the exact solution to PT at tree-level and gives a g ood approximation to the loop corrections (next-to-leading orders), indicat ing negligible tidal effects. Here, we derive predictions for the (smoothed ) cumulants of the velocity divergence field theta drop del.v for an irrota tional fluid in the SC model. By comparing these with the exact analytic re sults of Scoccimarro & Frieman, it is shown that, at least for the unsmooth ed case, the loop corrections to the cumulants of theta are dominated by ti dal effects. However, most of the tidal contribution seems to cancel out wh en computing the hierarchical ratios, T-J = <theta(J)>/<theta(2)>(J-1). We also extend the work presented in Papers I and II to give predictions for t he cumulants of the density and velocity divergence fields in non-flat spac es. In particular, we show the equivalence between the spherically symmetri c solution to the equations of motion in the SC model (given in terms of th e density) and that of the Lagrangian PT approach (given in terms of the di splacement field). It is shown that the Omega dependence is very weak for b oth cosmic fields even at one loop (a 10 per cent effect at most), except f or the overall factor f(Omega) that couples to the velocity divergence.