Spherical collapse model with non-clustering dark energy

We investigate a spherical overdensity model for the non-clustering dark energy (DE) with the constant equation of state, w in a flat universe. In this case, the exact solution for the evolution of the scale factor is obtained for general w. We also obtain the exact (when w = - 1/3) and the approximate (when w neq -1/3) solutions for the ratio of the overdensity radius to its value at the turnaround epoch (y) for general cosmological parameters. Also the exact and approximate solutions of the overdensity at the turnaround epoch (zeta) are obtained for general w. Thus, we are able to obtain the non-linear overdensity Delta = 1 + delta at any epoch for the given DE model. The non-linear overdensity at the virial epoch (Delta_{vir}) is obtained by using the virial theorem and the energy conservation. The non-linear overdensity of every DE model converges to that of the Einstein de Sitter universe ~ 147 when z_{vir}increases. We find that the observed quantities at high redshifts are insensitive to the different w models. The low-redshift cluster (z_{vir} ~ 0.04, i.e., z_{ta} ~ 0.7) shows the most model dependent feature and it should be a suitable object for testing DE models. Also as Omo increases, the model dependence of the observed quantities decreases. The error in the approximate solutions is at most 2% for a wide range of the parameter space. Even though the analytic forms of y and \zeta are obtained for the constant w, they can be generalized to the slowly varying w. Thus, these analytic forms of the scale factor, y, and zeta provide a very accurate and useful tool for measuring the properties of DE.