Vacuum dynamics in the Universe versus a rigid $\Lambda=$const
  In this year, in which we celebrate 100 years of the cosmological term,
$\Lambda$, in Einstein's gravitational field equations, we are still facing the
crucial question whether $\Lambda$ is truly a fundamental constant or a mildly
evolving dynamical variable. After many theoretical attempts to understand the
meaning of $\Lambda$, and in view of the enhanced accuracy of the cosmological
observations, it seems now mandatory that this issue should be first settled
empirically before further theoretical speculations on its ultimate nature. In
this work, we summarize the situation of some of these studies. Devoted
analyses made recently show that the $\Lambda=$const. hypothesis, despite being
the simplest, may well not be the most favored one. The overall fit to the
cosmological observables $SNIa+BAO+H(z)+LSS+CMB$ singles out the class RVM of
the "running" vacuum models, in which $\Lambda=\Lambda(H)$ is an affine
power-law function of the Hubble rate. It turns out that the performance of the
RVM as compared to the "concordance" $\Lambda$CDM model (with $\Lambda=$const.)
is much better. The evidence in support of the RVM may reach $\sim 4\sigma$
c.l., and is bolstered with Akaike and Bayesian criteria providing strong
evidence in favor of the RVM option. We also address the implications of this
framework on the tension between the CMB and local measurements of the current
Hubble parameter.
