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.
