Although, the standard form of Press–Schechter mass function with $f(\nu)=\sqrt{{2}/{\pi }} \nu \mathrm{ e}^{-\frac{\nu }{2}}$ which discussed in Press & Schechter (1974) and Bond et al. (1991) can provide a good approximation of the predicted number density of haloes, it fails by predicting approximation too many low-mass haloes and too few high-mass ones (Sheth & Tormen 1999, 2002; Lima & Marassi 2004). Thus, in this study we apply another well-known fitting formula which first proposed in Sheth & Tormen (1999): 
(24)
\begin{eqnarray*}
f(\nu)=0.2709\sqrt{\dfrac{2}{\pi }}(1+1.1096\nu ^{0.6})\mathrm{ exp}(-\dfrac{0.707 \nu ^2}{2})\,\, .
\end{eqnarray*}
In a Gaussian density field, σ is given by 
(25)
\begin{eqnarray*}
\sigma ^2(R)=\dfrac{1}{2 \pi ^2}{\int _0}^\infty k^2 P(k) W^2(kR) \, {\rm d}k\,\, ,
\end{eqnarray*}
where R = (3M/4πρm0)1/3 is the radius of the spherical overdense region, W(kR) is the Fourier transform of a spherical top-hat profile with radius R and P(k) is the linear power spectrum of density fluctuations (Peebles 1993). To obtain the value of σ, we follow the procedure presented in Abramo et al. (2007a). Following on Ade et al. (2016), we use the normalization of matter power spectrum σ8 = 0.815 for ΛCDM cosmology. The number density of virialized haloes above a certain value of mass M at zc, the collapse redshift obtained by 
(26)
\begin{eqnarray*}
N(\: M,z)={\int _0}^\infty \dfrac{{\rm d}n(z)}{{\rm d}M^{\prime }}\, {\rm d}M^{\prime }\,\,. 
\end{eqnarray*}
The above limit of integration in equation (26) is $M=10^{18}\, \mathrm{ M}_{\rm \odot}\, \mathrm{ h}^{-1}$ which such gigantic structures could not in practice be observed. Now we can calculate the number density of virialized haloes in both homogeneous and clustered DE scenarios using equations (23) and (26). In this way the total mass of a halo is equal to the mass of pressureless matter perturbations. However, the virialization of dark matter perturbations in the non-linear regime cannot be independent from the properties of DE (Lahav et al. 1991; Maor & Lahav 2005; Creminelli et al. 2010; Basse, Bjlde & Wong 2011). Thus, in clustered DE scenarios, we should consider the contribution of perturbated DE components to the total mass of the haloes (Creminelli et al. 2010; Basse et al. 2011; Batista & Pace 2013). Based on the behaviour of wde(z), DE can reduce or enhance the total mass of the virialized halo. One can obtain ϵ(z), the ratio of DE mass to be taken into account with respect to the mass of dark matter, from: 
(27)
\begin{eqnarray*}
\epsilon (z)=\dfrac{m_{\rm DE}}{m_{\rm DM}}\,\, ,
\end{eqnarray*}
where the value of mDE depends on what we consider as the mass of DE component. When one only considers the contribution of the perturbations of DE, the mDE takes the form 
(28)
\begin{eqnarray*}
{m_{\rm DE}}^{\mathrm{ Perturbed}}=4 \pi \bar{\rho }_{\rm DE}{\int _0}^{R_{\rm vir}} \, {\rm d}R R^2 \delta _{\rm DE}(1+3{c_{\rm eff}}^2)\,\,. 
\end{eqnarray*}
In the other hand, if we assume both DE contributions of perturbation and background level, the total mass of DE in virialized haloes takes this new form
(29)
\begin{eqnarray*}
{m_{\rm DE}}^{\mathrm{ Total}}=4 \pi \bar{\rho }_{\rm DE}{\int _0}^{R_{\rm vir}} {\rm d}R R^2 [(1+3 w_{\rm DE})+ \delta _{\rm DE}(1+3{c_{\rm eff}}^2)]. 
\end{eqnarray*}
The quantities inside a spherical collapsing region in the framework of the top-hat profile, evolve only with cosmic time. Thus from equation (28) one can find 
(30)
\begin{eqnarray*}
\epsilon (z)=\dfrac{\Omega _{\rm DE}}{\Omega _{\rm DM}}\dfrac{\delta _{\rm DE}}{1+\delta _{\rm DM}}\,\, 
\end{eqnarray*}
and from equation (29) we can obtain 
(31)
\begin{eqnarray*}
\epsilon (z)=\dfrac{\Omega _{\rm DE}}{\Omega _{\rm DM}}\dfrac{1+3 w_{\rm DE}+\delta _{\rm DE}}{1+\delta _{\rm DM}}\,\, .
\end{eqnarray*}
The mass of dark matter also is obtained from (see also Batista & Pace 2013): 
(32)
\begin{eqnarray*}
{m_{\rm DM}}=4 \pi \bar{\rho }_{\rm DM}{\int _0}^{R_{\rm vir}} \, {\rm d}R R^2 (1+ \delta _{\rm DM})\,\,. 
\end{eqnarray*}
In Fig. 5 we plot the evolution of ϵ(z) using equation (30) as the definition of DE mass. We observe that, at high redshift, where the role of DE is less important, ϵ for both of parametrizations becomes negligible. This parameter has a greater value in the case of parametrization (2).
Citation: Creminelli et al. 2010
Citation function: Uses
Citation: Creminelli et al. 2010
Citation function: Uses