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 (DE) 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 $0<c_s^2\le 1$. 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 $k>10H_0$ or equivalently $k/h>0.0033{\mathrm{Mpc}}^{-1}$. 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.

Figure 1

The percent difference of the effect of the DE perturbations of $\gamma$ as part of the total and as a function of the equation of state $w$ for $c_s^2=0$. In practice we plot $100*\frac{{\gamma }_{\mathrm{DE}}}{{\gamma }_m}$, with the latter given by the zero order part of Eq. (40).

Figure 2

A comparison of all the different expressions for $\gamma$ for $(c_s^2,w,{\mathrm{\Omega }}_{{\mathrm{m}}_0})=(0,-0.8,0.3)$. The solid black line corresponds to the expression for the analytic solution of Eq. (36), the dashed black line to the numerical solution from CAMB, the dotted line to the wCDM model with no DE perturbations, the dot-dashed black line to the semianalytic expression of Eq. (29), while the red and dashed-red lines correspond to the zero-order terms for expansions for $\gamma$ without and with DE perturbations, respectively.

Figure 3

Left: Comparison of the analytical solution for $f\sigma 8(z)$ (dashed line) vs the numerical solution of the full system of differential equations for the DE perturbations by CAMB (solid black line), the case with no DE perturbations (dotted line) and the ${\mathrm{\Omega }}_{\mathrm{m}}(a{)}^{\gamma }$ parametrization with $\gamma$ given by Eq. (29) (dot-dashed line) for $c_s^2=0$, $k=200H_0$ and $(w,{\mathrm{\Omega }}_{{\mathrm{m}}_0},{\sigma }_{8,0})=(-0.8,0.3,0.8)$. The gray points correspond to mock $f\sigma 8$ data based on the specifications of an LSST-like survey [22], as it was done in Ref. [23]. Right: Comparison of the analytical solution (black dashed line) vs the numerical solution (solid black line) of the ODE of Eq. (1) for $c_s^2=1$ and $k=10H_0$.

Figure 4

The percent difference between the analytical solution and the numerical one from CAMB, for various values of the sound speed $c_s^2$ and the scale $k$.