Growth of matter perturbations in clustered holographic dark energy cosmologies

We investigate the growth of matter fluctuations in holographic dark energy cosmologies. First we use an overall statistical analysis involving the latest observational data in order to place constraints on the cosmological parameters. Then we test the range of validity of the holographic dark energy models at the perturbation level and its variants from the concordance $\mathrm{\Lambda }$ cosmology. Specifically, we provide a new analytical approach in order to derive, for the first time, the growth index of matter perturbations. Considering a homogeneous holographic dark energy we find that the growth index is $\gamma \approx \frac{4}{7}$ which is somewhat larger ($\sim 4.8\mathrm{\%}$) than that of the usual $\mathrm{\Lambda }$ cosmology, $\gamma^{(\Lambda)}\approx \frac{6}{11}$. Finally, if we allow clustering in the holographic dark energy models then the asymptotic value of the growth index is given in terms of the effective sound speed $c_{\mathrm{e}\mathrm{f}\mathrm{f}}^2$, namely $\gamma \approx \frac{3(1-c_{\rm eff}^2)}{7}$.