Running vacuum against the $H_0$ and $\sigma_8$ tensions

The cosmological term, $\Lambda$, was introduced $104$ years ago by Einstein in his gravitational field equations. Whether $\Lambda$ is a rigid quantity or a dynamical variable in cosmology has been a matter of debate for many years, especially after the introduction of the general notion of dark energy (DE). $\Lambda$ is associated to the vacuum energy density, $\rho_{\rm vac}$, and one may expect that it evolves slowly with the cosmological expansion. Herein we present a devoted study testing this possibility using the promising class of running vacuum models (RVM's). We use a large string $SNIa+BAO+H(z)+LSS+CMB$ of modern cosmological data, in which for the first time the CMB part involves the full Planck 2018 likelihood for these models. We test the dependence of the results on the threshold redshift $z_*$ at which the vacuum dynamics is activated in the recent past and find positive signals up to $\sim4.0\sigma$ for $z_*\simeq 1$. The RVM's prove very competitive against the standard $\Lambda$CDM model and give a handle for solving the $\sigma_8$ tension and alleviating the $H_0$ one.