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 (EdS) universe at the virial epoch is 18π2(12π+34)2≃147 instead of the well-known value 18π2≃178. In the open universe, perturbations are virialized earlier than in the flat one and thus clusters are denser at the virial epoch. Also the critical density threshold of EdS universe from the linear theory at the virialized epoch is obtained as 320(9π+6)23≃1.58 instead of 320(12π)23≃1.69. This value is same for the close and the open universes. We find that the observed quantities at high redshifts are less sensitive between different models. Even though the low redshift cluster shows the stronger model dependence than high redshift one, the differences between models might be still too small to be distinguished by observations if the curvature is small. From these analytic forms of dynamical quantities, we are able to estimate the abundances of both virialized and non-virialized clusters and the temperature and luminosity functions at any epoch. The current concordance model prefers the almost flat universe and thus the above results might be restricted by the academic interests only. However, the mathematical structure of the evolution equations of physical quantities for the curved space is identical with that for the flat universe including the dark energy with the equation of state ωde=−13. Thus, we might be 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.