Testing the growth rate in homogeneous and inhomogeneous interacting vacuum models

In this work we consider a class of interacting vacuum corresponding to a generalised Chaplygin gas (gCg) cosmology. In particular we analyse two different scenarios at perturbation level for the same background interaction characterised by the parameter $\alpha$: (i) matter that follows geodesics, corresponding to homogeneous vacuum, and (ii) a covariant ansatz for vacuum density perturbations. In the latter case, we show that the vacuum perturbations are very tiny as compared to matter perturbations on sub-horizon scales. In spite of that, depending on the value of the Chaplygin gas parameter $\alpha$, vacuum perturbations suppress or enhance the matter growth rate as compared to the case (i). We use Cosmic Microwave Background (CMB), type Ia supernovae (SNe) and Redshift Space Distortion (RSD) measurements to test the observational viability of the model. We found that the mean value of our joint analysis clearly favours a positive interaction, i.e., an energy flux from dark matter to dark energy, with $\alpha \approx 0.143$ in both cases, while the cosmological standard model, recovered for $\alpha$=0, is ruled out by 3$\sigma$ confidence level. Noteworthy, the positive value of interaction can alleviate both the $H_0$ and $S_8$ tension for the dataset considered here.