A Parameterized Variable Dark Energy Model: Structure Formation and Observational Constraints

In this paper we investigate a simple parameterization scheme of the quintessence model given by Wetterich (2004). The crucial parameter of this model is the bending parameter $b$, which is related to the amount of dark energy in the early universe. Using the linear perturbation and the spherical infall approximations, we investigate the evolution of matter density perturbations in the variable dark energy model, and obtain an analytical expression for the growth index $f$. We show that increasing $b$ leads to less growth of the density contrast $\delta$, and also decreases the growth index. Giving a fitting formula for the growth index at the present time we verify that the approximation relation $f\simeq\Omega_m^{\alpha}$ also holds in this model. To compare predictions of the model with observations, we use the Supernovae type Ia (SNIa) Gold Sample and the parameters of the large scale structure determined by the 2-degree Field Galaxy Redshift Survey (2dFGRS). The best fit values for the model parameters by marginalizing on the remained ones, are $\Omega_m=0.21_{-0.06}^{+0.07}$, $w_0=-2.05_{-2.05}^{+0.65}$ and $b=4.05^{+7.05}_{-2.25}$ at $1\sigma$ confidence level. As a final test we calculate the age of universe for different choices of the free parameters in this model and compare it with the age of old stars and some high redshift objects. Then we show that the predictions of this variable dark energy model are consistent with the age observation of old star and can solve the "age crisis" problem.