Density perturbations for running vacuum: a successful approach to structure formation and to the $\sigma_8$-tension

Recent studies suggest that dynamical dark energy (DDE) provides a better fit to the rising affluence of modern cosmological observations than the concordance model ($\Lambda$CDM) with a rigid cosmological constant, $\Lambda$. Such is the case with the running vacuum models (RVMs) and to some extent also with a simple XCDM parametrization. Apart from the cosmic microwave background (CMB) anisotropies, the most crucial datasets potentially carrying the DDE signature are: i) baryonic acoustic oscillations (BAO), and ii) direct large scale structure (LSS) formation data (i.e. the observations on $f(z)\sigma_8(z)$ at different redshifts). As it turns out, analyses mainly focusing on CMB and with insufficient BAO+LSS input, or those just making use of gravitational weak-lensing data for the description of structure formation, generally fail to capture the DDE signature, whereas the few existing studies using a rich set of CMB+BAO+LSS data (see in particular Sol\`a, G\'omez-Valent & de Cruz P\'erez 2015,2017; and Zhao et al. 2017) do converge to the remarkable conclusion that DDE might well be encoded in the current cosmological observations. Being the issue so pressing, here we explain both analytically and numerically the origin of the possible hints of DDE in the context of RVMs, which arise at a significance level of $3-4\sigma$. By performing a detailed study on the matter and vacuum perturbations within the RVMs, and comparing with the XCDM, we show why the running vacuum fully relaxes the existing $\sigma_8$-tension and accounts for the LSS formation data much better than the concordance model.