  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.
