Accuracy of the growth index in the presence of dark energy perturbations

We present the analytical solutions for the evolution of matter density perturbations, for a model with a constant dark energy equation of state $w$ but when the effects of the dark energy perturbations are properly taken into account. We consider two cases, the first when the sound speed of the perturbations is zero $c_s^2=0$ and the general case $0<c_s^2 \leq 1$. In the first case our solution is exact, while in the second case we found an approximate solution which works to better than $0.3\%$ accuracy for $k>10 H_0$ or equivalently $k/h>0.0033 \textrm{Mpc}^{-1}$. We also estimate the corrections to the growth index $\gamma(z)$, commonly used to parametrize the growth-rate. We find that these corrections due to the DE perturbations affect the growth index $\gamma$ at the $3\%$ level. We also compare our new expressions for the growth index with other expressions already present in the literature and we find that the latter are less accurate than the ones we propose here. Therefore, our analytical calculations are necessary as the theoretical predictions for the fundamental parameters to be constrained by the upcoming surveys need to be as accurate as possible, especially since we are entering in the precise cosmology era where parameters will be measured to the percent level.