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.
