Cosmological constant vis-a-vis dynamical vacuum: bold challenging the   $\Lambda$CDM
  Next year we will celebrate 100 years of the cosmological term, $\Lambda$, in Einstein's gravitational field equations, also 50 years since the cosmological constant problem was first formulated by Zeldovich, and almost about two decades of the observational evidence that a non-vanishing, positive, $\Lambda$-term could be the simplest phenomenological explanation for the observed acceleration of the Universe. This mixed state of affairs already shows that we do no currently understand the theoretical nature of $\Lambda$. In particular, we are still facing the crucial question whether $\Lambda$ is truly a fundamental constant or a mildly evolving dynamical variable. At this point the matter should be settled once more empirically and, amazingly enough, the wealth of observational data at our disposal can presently shed true light on it. In this short review I summarize the situation of some of these studies. It turns out that the $\Lambda=$const. hypothesis, despite being the simplest, may well not be the most favored one when we put it in hard-fought competition with specific dynamical models of the vacuum energy. Recently it has been shown that the overall fit to the cosmological observables $SNIa+BAO+H(z)+LSS+BBN+CMB$ do favor the class of "running" vacuum models (RVM's) -- in which $\Lambda=\Lambda(H)$ is a function of the Hubble rate -- against the "concordance" $\Lambda$CDM model. The support is at an unprecedented level of $\sim4\sigma$ and is backed up with Akaike and Bayesian criteria leading to compelling evidence in favor of the RVM option and other related dynamical vacuum models. I also address the implications of this framework on the possible time evolution of the fundamental constants of Nature. 
Joan Sola
astro-ph.CO gr-qc hep-ph hep-th