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.

for the different models are also obtained. Thus, we are able to obtain the nonlinear overdensity ∆ ≡ 1 + δ NL = ζ x We also show that there is the minimum z ta value with the given cosmological parameters for each model. We find that the observed quantities at high redshifts are less sensitive between different models. Thus, the high redshift cluster (z vir ∼ 0.5) is not a good object to probe the curvature of the Universe. The low redshift cluster (z vir ∼ 0.07 i.e. z ta ∼ 0.6) shows the stronger model dependence feature. However, it might be still too small to be distinguished. 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. Thus, these analytic forms of x, y, and ζ provide the accurate tools for probing the curvature of the Universe. 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 ω de = − 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.
Background evolution equations of physical quantities in a FRW Universe with the matter are given by where a is the scale factor, ρ m is the energy densities of the matter, ρ cr is the critical energy density, and k is chosen to be +1, 0, or −1 for spaces of constant positive, zero, or negative spatial curvature. In terms of the ratio of the matter density to the critical density Ω m , the above Friedmann equation (0.1) becomes which is valid for all times.
We consider a spherical perturbation in the matter density. ρ cluster is the matter density within the spherical overdensity radius R. The flatness condition is not held because of the perturbation in the matter (The curvature is not a constant inside the overdensity patch). Thus, we have the another equations governing the dynamics of the spherical perturbation [1] where ρ cluster is the energy density of the clustering matter. The radius of the overdensity R evolves slower than the scale factor a and reaches its maximum size R ta at the turnaround epoch z ta and then the system begins to collapse.
Cosmological parameters and the curvature of the Universe can be constrained from the growth of large scale structure and the abundance of rich clusters of galaxies. There have been numerous works related to this [2,3,4,5,6,7,8,9,10,11,12]. Most of them reach to the similar conclusions based on the conventional approximate solutions of the background scale factor and of the overdensity radius. It is natural to expect that the correct values for the virial radius and the nonlinear overdensity obtained from the exact solutions might be different from those obtained from the conventional approximate solutions. In order to investigate this, we need to obtain the exact solutions of physical quantities. Now we adopt the notations in Ref. [11] to investigate the evolutions of a and R x = a a ta , (0.7) where a ta is the scale factor of the background evolution at z ta . Then the equations (0.1) and (0.5) are rewritten as . Ω 0 m and Ω 0 k represent the present value of energy density contrasts of the matter and the curvature term, respectively. Equations (0.9) and (0.10) can be solved analytically and we will obtain them.
The analytic solution of Eq. (0.9) is given by where F is the hypergeometric function and we use the boundary condition x = 0 when τ = 0 (see Appendix for details). From this equation, the exact turnaround time τ ta is given by where we use the fact that x ta = 1, the relations Q ta = Ω 0 m Ω 0 m −1 (1 + z ta ) and τ = H ta √ Ω mta t. This exact analytic form of the turnaround time will be used to investigate the other quantities.
As expected, τ ta (t ta ) depends on Ω 0 m (i.e. Ω k ) and z ta as given in the Eq. (0.12). We show these properties of τ ta (t ta ) in on the values of Ω 0 m for the different choice of z ta models. The solid, dashed, and dotdashed lines (from top to bottom) correspond to z ta = 0.6, 1.2, and 2.0, respectively. Eq. (0.11) is the evolution of the background scale factor a and we can interpret it as the age of the Universe is a decreasing function of Ω 0 m . Larger Ω 0 m implies faster deceleration, which corresponds a more rapidly expanding universe early on. Also larger z ta means the earlier formation of the structure and thus gives the smaller t ta . We also show the Ω 0 m dependence of τ ta for the values of z ta in the right panel of Fig. 1. Because τ ta = H ta √ Ω mta t ta , τ ta becomes larger for the larger values of Ω 0 m . The evolution of y given in Eq. (0.10) described as the exact analytic solution (see Appendix)
As shown in the equation (0.15), ζ is inversely proportional to τ 2 ta . Thus, ζ decreases as Ω 0 m increases. Because ζ is the ratio of ρ cluster to ρ m at z = z ta , it means that the smaller overdensity takes longer time to turnaround and collapse. In Fig. 2, the solid, dashed, and dotdashed lines correspond to z ta = 0.6, 1.2, and 2.0, respectively. After we obtain the value of ζ, we are able to obtain the values of x and y at any τ without any ambiguity because the analytic solutions of x, y, and ζ given in Eqs. (0.11), (0.13), (0.14), and (0.15) are exact.
In order to better understand the above results, it is useful to investigate the virialized times for the different models. From the equation (0.14), we are able to obtain the collapsing time of each model normalized to the turnaround time for the EdS Universe is the potential energy associated with the spherical mass overdensity [1,4,5,14]. From the above equation we are able to obtain y vir = R vir Rta for any model After replacing y vir = 1 2 into Eq. (0.14), we obtain x vir is obtained from Eq. (0.11) with Eq. (0.19) where we use F [− 1 2ω de , 1 2 , 1 − 1 2ω de , 0] = 1 and τ ta = 2 3 . The commonly used assumption to obtain the nonlinear overdensity ∆ c in the EdS Universe is that τ c = 2τ ta = 4 3 which is the collapsing time for y c = 0 even though one uses y vir = 1 2 to obtain ∆ c . By using this assumption, x c = 2 However, we know the exact relation between x and y and thus we do not need to use the above assumption. If we insert y vir = 1 2 in the equation (0.22), then we obtain the correct τ vir = 1 + 2 3π < τ c . With this correct value of τ vir , we obtain the correct x vir = ( 3 2 + 1 π ) 2 3 by using the Eq. (0.21). The correct value of the nonlinear overdensity ∆ vir for the EdS universe is Thus, the minimum overdensity for the flat Universe is about 147 instead of 178. We show this in tables 1. As z ta increases ∆ c approaches to 147 instead of 178. Also for the closed universe ∆ vir can be smaller than this value. From the equation (0.7) we are able to obtain the virial epoch from the given z ta z vir = 1 + z ta ζ Ω mta t t ta  where we use √ ζ = π 2τta . The above result given in Eq. (0.30) is true for with and without curvature and thus it is valid for the close, flat, and open Universes. After we obtain the critical density threshold δ lin (z vir ) we are able to obtain δ lin at any epoch by using the relation where D g is the linear growth factor. There is the exact analytic form of D g for the dark energy with the equation of state ω de = − 1 3 [15]. Mathematically, this form of D g is identical with that of curved space and thus we can adopt this form of D g in these models.
From the analytic forms of dynamical quantities x, y, and ζ, it is straightforward 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 [16]. Thus, these analytic forms provide an accurate tool for probing the effect of the curvature on the clustering. As we mentioned, the mathematical structure of the physical evolution equations of the curved space is identical with that of the dark energy with the equation of state ω de = − 1 3 . Thus, these analytic solutions give the guideline for the extension of them to those of the general dark energy model and they will provide the useful tools for probing the properties of dark energy [17].

A Appendix
First, we derive the exact solution of τ as a function of x given in Eq. (0.11). After we replace the variables Z = x Qta and T = x ′ x , the integral in Eq. (0.11) becomes [18]  We also derive the exact analytic solution of τ as a function of y given in Eq. (0.10). This equation is solved as [19]