5 universe contained in gravitationally bound structures (such as galaxy clusters) with matter fluctuations greater than a critical value δ$_{c}$ , which is the linearly extrapolated density threshold above which structures collapse (Eke, Cole & Frenk 1996). Assuming that the density con- trast is normally distributed with zero mean and variance σ $^{2}$( M, z ) we have: Note, that the prime denotes derivatives with respect to the scale factor. Useful expressions of the growth factor can be found for the ΛCDM cosmology in Peebles (1993), for dark energy models with w = const in Silveira, & Waga (1994) for dark energy models with a time vary- ing equation of state in Linder & Cahn (2007) and for the scalar tensor models in Gannouji & Polarski (2008). In this work the growth factor evolution for the current cosmological model is derived by solving numerically eq. (26). Note, that the growth factors are normalized to unity at the present time. where In this kind of studies it is common to parametrize the rms mass fluctuation amplitude at 8 h − $^{1}$Mpc which can be expressed as a function of redshift as σ ( M, z ) = σ$_{8}$ ( z ) = D ( z ) σ$_{8}$ . The current cosmological models are normalized by the analysis of the WMAP 5 years data σ$_{8}$ = 0 . 80 (Komatsu et al. 2009). The integration of eq.(28) provides the fraction of the universe, on some specific mass scale, that has already collapsed producing cosmic structures (galaxy clusters) at redshift z and is given by Richstone et al. (1992): P ( δ, z ) = 1 √$_{2}$$_{πσ}$$_{(}$$_{M, z}$$_{)}$ exp [$_{−}$ δ 2 2 σ $^{2}$( M, z ) ] . (28) X ( a ) = Ω$_{m}$ $^{1}$− Ω$_{m}$ e − 3 ∫ 1 a w ( y ) d ln y = Ω$_{m}$ a − 3 δH 2 . (27) P ( z ) = ∫ ∞ δ$_{c}$ P ( δ, z ) dδ = 1 2 [ $^{1}$− erf ( δ$_{c}$ √$_{2}$$_{σ}$$_{8}$$_{(}$$_{z}$$_{)}$ )] . (29) In this section we briefly investigate the cluster forma- tion processes by generalizing the basic equations which govern the behavior of the matter perturbations within the framework of the current dark energy models. Also we compare our predictions with those found for the tra- ditional Λ cosmology. This can help us understand better the theoretical expectations of the dark energy models as well as the variants from the usual Λ cosmology. Notice, that for the model of modified gravity (BRG) we use δ$_{c}$ ≃ 1 . 47 (see Schafer & Koyama 2008), for the EDE model we use δ$_{c}$ ≃ 1 . 4 (see Bartelmann, Doran & Wetterich 2006). For the rest of the dark energy models, due to the fact that w ≃ − 1 close to the present epoch, we utilize a δ$_{c}$ approximation which is given by Weinberg & Kamionkowski (2003 see their eq.18). 5.1. The formation of galaxy clusters FIG. 1: The predicted fractional rate of cluster formation as a function of redshift for the current cosmological models ( σ 8 = 0 . 80). The points represent the following cosmological models: (a) BRG (open stars), (b) LRDE (open squares), (c) TCM (open triangles), (d) EDE (open circles) and (e) PNGB (solid triangles). The lines represent: (a) CPL model (long dashed), (b) HRDE model (dot line) and VCG model (dashed line). Obviously the above generic form of eq.(29) depends on the choice of the background cosmology. The next step is to normalize the probability to give the number of clusters which have already collapsed by the epoch z (cumulative distribution), divided by the number of clus- ters which have collapsed at the present epoch ( z = 0), F ( z ) = P ( z ) /P (0). In figure 1 we present, in a logarith- mic scale, the behavior of normalized cluster formation rate as a function of redshift for the various dark energy models. In general, prior to z ∼ 0 . 2 the cluster formation has terminated due to the fact that the matter fluctua- tion field, D ( z ), effectively freezes. For the traditional Λ cosmology we find the known behavior in which galaxy clusters appear to be formed at high redshifts z ∼ 2 (Basilakos 2003; Basilakos, Sanchez & Perivolaropoulos 2009 and references therein). From figure 1 it becomes also clear, that the vast majority of the dark energy mod- els seem to have a cluster formation rate which is close to that predicted by the usual Λ cosmology (see solid line in figure 1). However, for the BRG and EDE cosmologi- cal scenarios we find that galaxy clusters appear to form earlier ( z ∼ 3) than in the CPL, LRDE, HRDE, TCM, PNGB and VCG dark energy models. The concept of estimating the fractional rate of clus- ter formation has been proposed by different authors (eg., Weinberg 1987; Richstone, Loeb & Turner 1992). In par- ticular, these authors introduced a methodology which computes the rate at which mass joins virialized struc- tures, which grow from small initial perturbations in the universe. The basic tool is the Press & Schechter (1974) formalism which considers the fraction of mass in the