Spherical collapse model with and without curvature

We investigate a spherical collapse model with and without the spatial curvature. We obtain the exact solutions of dynamical quantities such as the ratio of the scale factor to its value at the turnaround epoch and the ratio of the overdensity radius to its value at the turnaround time with general cosmological parameters. The exact solutions of the overdensity at the turnaround epoch for the different models are also obtained. Thus, we are able to obtain the nonlinear overdensity at any epoch for the given model. We obtain that the nonlinear overdensity of the Einstein de Sitter Universe is 18 pi^2(fr{1}{2pi} + fr{3}{4})^2 ~ 147 instead of the well known value 18 pi^2 ~ 178. In the open Universe, perturbations are virialized earlier than in flat Universe and thus clusters are denser at the virial epoch. Also the critical density threshold from the linear theory at the virialized epoch is obtained as fr{3}{20} (9pi+6)^{fr{2}{3}} ~ 1.58 instead of fr{3}{20} (12pi)^{fr{2}{3}} ~ 1.69. This value is same for the close and the open Universes. Because we obtain the analytic forms of dynamical quantities, we are able to estimate the abundances of both virialized and non-virialized clusters at any epoch. Also the temperature and luminosity functions are able to be computed at any epoch. Unfortunately, the current concordance model prefers the almost flat Universe and the above results might be restricted by the academic interests only. However, mathematical structure of the physical evolution equations of the curved space is identical with that of the dark energy with the equation of state w = -fr{1}{3}. Thus, we are able to extend these analytic solutions to the general dark energy model and they will provide the useful tools for probing the properties of dark energy.