We derive for the first time the growth index of matter perturbations of the Friedmann-Lemaître-Robertson-Walker (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 ${\mathrm{\Lambda }}_0$, there is also an $H^2$ dependence in the functional form of the vacuum, namely, $\mathrm{\Lambda }(H)={\mathrm{\Lambda }}_0+3\nu (H^2-H_0^2)$. Since $|\nu |\ll 1$, 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 ${\gamma }_{{\mathrm{\Lambda }}_H}\approx \frac{6+3\nu }{11-12\nu }$, and thus it nicely extends analytically the result of the $\mathrm{\Lambda }\mathrm{CDM}$ model, ${\gamma }_{\mathrm{\Lambda }}\approx 6/11$.

• Received 25 September 2015

DOI:https://doi.org/10.1103/PhysRevD.92.123501