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Cluster number counts in quintessence models

Even though the abundance and evolution of clusters have been used to study the cosmological parameters including the properties of dark energy owing to their pure dependence on the geometry of the Universe and the power spectrum, it is necessary to pay particular attention to the effects of dark energy on the analysis. We obtain the explicit dark energy dependent {\it rms} linear mass fluctuation $σ_8$ which is consistent with the CMB normalization with less than $2$ % errors for general constant dark energy equation of state, $\oQ$. Thus, we do not have any degeneracy between $σ_8$ and the matter energy density contrast $\Omo$. When we use the correct value of the critical density threshold $δ_{c} = 1.58$ obtained recently \cite{09090826, 09100126} into the cluster number density $n$ calculation in the Press-Schechter (PS) formalism, $n$ increases as compared to the one obtained by using $δ_{c} = 1.69$ by about $60$, $80$, and $110$ % at $z = 0$, $0.5$, and $1$, respectively. Thus, PS formalism predicts the cluster number consistent with both simulation and observed data at the high mass region. We also introduce the improved coefficients of Sheth-Tormen (ST) formalism, which is consistent with the recently suggested mass function \cite{10052239}. We found that changing $\oQ$ by $Δ\oQ = -0.1$ from $\oQ = -1.0$ causes the changing of the comoving numbers of high mass clusters of $M = 10^{16} h^{-1} M_{\odot}$ by about $20$ and $40$ % at $z = 0$ and $1$, respectively.