4 The basic assumption here is that at early epochs the amount of dark energy is not negligible (Doran, Stern & Thommes 2006 and references therein). In this model, the total dark energy component is given by: Recently, Contopoulos & Basilakos (2007) proposed a novel idea which is based on the following consideration (hereafter TCM): if the cosmic magnetic field is gener- ated in sources (such as galaxy clusters) whose overall dimensions remain unchanged during the expansion of the Universe, the stretching of this field by the cosmic expansion generates a tension (negative pressure) that could possibly explain a small fraction of the dark en- $^{ergy (}$∼ 5 − 10%). In this flat cosmological scenario the normalized Hubble flow becomes: where Ω$_{e}$ is the early dark energy density and w$_{0}$ is the equation of state parameter at the present epoch. Notice, that the EDE model was designed to simultaneously (a) mimic the effects of the late dark energy and (b) provide a decelerated expansion of the universe at high redshifts. The normalized Hubble parameter is written as: where Ω$_{B}$ is the density parameter for the cosmic mag- netic fields and Ω$_{Λ}$ = 1 − Ω$_{m}$ − Ω$_{B}$. The equation of state parameter which is related to magnetic tension is (see eq.3) while using eq.(3), we can obtain the equation of state parameter as a function of the scale factor. From the joint likelihood analysis we find that Ω$_{e}$ = 0 . 05 ± 0 . 04 and w$_{0}$ = − 1 . 14 +0 . 18 − 0 10 (for the prior of Ω$_{m}$ = 0 . 28) with χ 2 $_{tot}$(Ω$_{e}$ , w$_{0}$ ) /dof ≃ 308 . 7 / 307. Ω$_{X}$( a ) = $^{1}$− Ω$_{m}$ − Ω$_{e}$$^{(1}$− a − 3 w 0 ) − Ω$_{m}$ a 3 w 0 + Ω$_{e}$$^{(1}$− a − 3 w 0 ) , (23) E $^{2}$( a ) = Ω$_{m}$ a − 3 $^{1}$− Ω$_{X}$( a ) , (24) E $^{2}$( a ) = Ω$_{m}$ $_{a}$− 3 + Ω$_{Λ}$ + Ω$_{B}$ $_{a}$− 3+ n , (18) w ( a ) = − 3Ω$_{Λ}$ + n Ω$_{B}$ a − 3+ n 3(Ω$_{Λ}$ + Ω$_{B}$ a − 3+ $^{n}$) . (19) 4.6. The tension of cosmological magnetic fields - TCM model 4.9. The Variable Chaplygin Gas - VCG model 4.7. The Pseudo-Nambu Goldstone boson - PNGB model To this end, we sample Ω$_{B}$ ∈ [0 , 0 . 3] and n ∈ [0 , 10] and we find that for the prior of Ω$_{m}$ = 0 . 28 the best fit values are: Ω$_{B}$ = 0 . 10 ± 0 . 10 and n = 3 . 60 +4 . 5 − 2 6 with χ 2 $_{tot}$(Ω$_{B}$ , n ) /dof ≃ 308 . 9 / 307. Let us consider now a completely different model namely the variable Chaplygin gas (hereafter VCG) which corresponds to a Born-Infeld tachyon action (Bento, Bertolami & Sen 2004). Recently, an interesting family of Chaplygin gas models was found to be consis- tent with the current observational data (Dev, Alcaniz & Jain 2003). In the framework of a spatially flat geome- try, it can be shown that the normalized Hubble function takes the following formula: In this cosmological model the dark energy equation of state parameter is expressed with the aid the poten- tial V ( φ ) ∝ [1 + cos ( φ/F )] (Abrahamse et al 2008 and references therein): where w$_{0}$ ∈ $^{[}$−2 $^{,}$− 0 . 4] and F ∈ [0 , 8]. In case of a ≪ 1 we get w ( a ) −→ − 1. Based on this parametrization the normalized Hubble function is given by where Ω$_{b}$ ≃ 0 . 021 h − 2 is the density parameter for the baryonic matter (see Kirkman et al. 2003) and B$_{s}$ ∈ [0 . 01 , 0 . 51] and n ∈ $^{[}$−4 , 4]. The effective matter density parameter is: Ω eff m = Ω$_{b}$$^{+(1}$−Ω$_{b}$$_{)}$√$_{B}$$_{s}$$_{. We find that the}$ best fit parameters are B$_{s}$ = 0 . $^{05}$±0 . 02 and n = 1 . 58 +0 . 35 − 0 43 (Ω eff m ≃ 0 . 26) with χ 2 $_{tot}$( B$_{s}$, n ) /dof ≃ 314 . 7 / 307. In this context, the corresponding dark energy density is Notice, that the likelihood function peaks at w$_{0}$ = − 1 . $^{04}$± 0 . 17 and F = 5 . 9 ± 3 . 2 with χ 2 $_{tot}$( w$_{0}$, F ) /dof ≃ 317 / 307. The evolution equation of the growth factor for models where the dark energy fluid has a vanishing anisotropic stress and the matter fluid is not coupled to other matter species is given by (Linder & Jenkins 2003) 5. EVOLUTION OF MATTER PERTURBATIONS E $^{2}$( a ) = Ω$_{b}$ a − $^{$^{3}$+(1}$− Ω$_{b}$) √ B$_{s}$a − 6 $^{+ (1}$− B$_{s}$ ) a − n , (25) w ( a ) = − 1 + (1 + w$_{0}$ ) a F , (20) E $^{2}$( a ) = Ω$_{m}$ a − 3 $^{+ (1}$− Ω$_{m}$) ρ$_{X}$ ( a ) . (21) ρ$_{X}$ ( a ) = exp [ 3(1 + w$_{0}$ ) F $^{(1}$− a $^{F}$) ] . (22) D ′′ + 3 2 [ $^{1}$− w ( a ) 1 + X ( a ) ] D ′ a − 3 2 X ( a ) 1 + X ( a ) D a 2 = 0 , (26) 4.8. The early dark energy - EDE model Another cosmological scenario that we include in our paper is the early dark energy model (hereafter EDE).