Relaxing the $\sigma_8$-tension through running vacuum in the Universe
  It has recently been shown that the class of running vacuum models (RVMs) has
the capacity to fit the overall cosmological observations better than the
concordance $\Lambda$CDM model, therefore supporting the possibility of
dynamical dark energy (DE). Apart from the cosmic microwave background (CMB)
anisotropies, the most crucial datasets involved are: i) baryonic acoustic
oscillations (BAO), and ii) direct large scale structure (LSS) formation data.
Analyses mainly focusing on CMB and with insufficient BAO+LSS input generally
fail to capture the dynamical DE signature, whereas the few existing studies
accounting for the wealth of known 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 dynamical DE might well be encoded in the
current cosmological observations at a $3-4\sigma$ c.l. A decisive factor is
the persistent $\sigma_8$-tension between the $\Lambda$CDM and the data.
Because the issue is obviously pressing, we devote this work to explain how and
why running vacuum in the expanding universe successfully relaxes the existing
$\sigma_8$-tension and describes the LSS formation data significantly better
than the $\Lambda$CDM.
