content_image/390347.jpg content_image/390348.jpg content_image/390362.jpg content_image/390363.jpg content_image/390364.jpg content_image/390345.jpg content_image/390361.jpg content_image/390365.jpg content_image/390346.jpg
## 4.6. The tension of cosmological magnetic fields - TCM model 

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$_{B}$ 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) 

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. 

## 4.7. The Pseudo-Nambu Goldstone boson - PNGB model 

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 

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. 

## 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). 

<img src='content_image/390347.jpg'>

<img src='content_image/390348.jpg'>

<img src='content_image/390362.jpg'>

<img src='content_image/390363.jpg'>

<img src='content_image/390364.jpg'>

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: 

<img src='content_image/390345.jpg'>

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: 

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. 

## 4.9. The Variable Chaplygin Gas - VCG model 

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: 

<img src='content_image/390361.jpg'>

where Ω$_{b}$ ≃ 0 . 021 h − 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. 

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) 

<img src='content_image/390365.jpg'>

## 5. EVOLUTION OF MATTER PERTURBATIONS 

<img src='content_image/390346.jpg'>

