An Improved Semianalytical Spherical Collapse Model for Nonlinear Density Evolution

Abstract

We derive a semianalytical extension of the spherical collapse model of structure formation that takes account of the effects of deviations from spherical symmetry and shell crossing, which are important in the nonlinear regime. Our model is designed so that it predicts a relation between the peculiar velocity and density contrast that agrees with the results of N-body simulations in the region where such a comparison can sensibly be made. Prior to turnaround, when the unmodified spherical collapse model is expected to be a good approximation, the predictions of the two models coincide almost exactly. The effects of a late time dominating dark energy component are also taken into account. The improved spherical collapse model is a useful tool when one requires a good approximation not just to the evolution of the density contrast but also its trajectory. Moreover, the analytical fitting formulae presented is simple enough to be used anywhere where the standard spherical collapse might be used but with the advantage that it includes a realistic model of the effects of virialisation.

Spherical collapse model in time varying vacuum cosmologies

“…Following the notations of [54,73,74], we again provide in appendix C a fitting formula for ∆ vir (within a physical range of cosmological parameters: 0 ≤ γ ≤ 0.01). Notice that the Einstein-de Sitter value for ∆ vir is precisely 18π 2 , and was factorized in (C3) 4 .…”

Evaluating backreaction with the peak model of structure formation

“…In particular, there is no treatment of the stabilisation which ends the collapse. An effective treatment of the terms responsible for stabilisation was given in [130], and a improved model which covers the whole evolution from the linear regime to the stabilised phase was presented in [131]. One can also generalise into ellipsoidal collapse, and take shear and tidal effects into account [107,121,132].…”

“…One could integrate over a distribution of initial conditions to obtain the mean expansion rate corresponding to structures of a given density contrast, using the Buchert equations, as presented in [78,133]. While the approximation that the expansion rate depends only on the density contrast (used in [130,131]) is probably not valid for a single structure, it is true by construction for the expansion rate averaged over different structures.…”

“…For example, redoing the calculation above with the improved collapse model presented in [131] makes practically no difference to the results. The reason is that the volume in the overdense regions is so small that their impact is negligible.…”

Spherical collapse model and cluster formation beyond theΛcosmology: Indications for a clustered dark energy?

“…In any case, at very large redshifts the above density contrast tends to the Einstein-de Sitter value (∆ vir ∼ 18π 2 ), as it should. In this context, following the notations of [27,69,70], we provide an accurate fitting formula to ∆ vir (within a physical range of cosmological parameters)…”