Growth index of matter perturbations in running vacuum models

We derive for the first time the growth index of matter perturbations of the FLRW flat cosmological models in which the vacuum energy depends on redshift. A particularly well motivated model of this type is the so-called quantum field vacuum, in which apart from a leading constant term $\Lambda_0$ there is also a $H^{2}$-dependence in the functional form of vacuum, namely $\Lambda(H)=\Lambda_{0}+3\nu (H^{2}-H^{2}_{0})$. Since $|\nu|\ll1$ this form endows the vacuum energy of a mild dynamics which affects the evolution of the main cosmological observables at the background and perturbation levels. Specifically, at the perturbation level we find that the growth index of the running vacuum cosmological model is $\gamma_{\Lambda_{H}} \approx \frac{6+3\nu}{11-12\nu}$ and thus it nicely extends analytically the result of the $\Lambda$CDM model, $\gamma_{\Lambda}\approx 6/11$.