Dynamical dark energy versus $\Lambda=$const. in light of observations

After about two decades of the first observational papers confirming the accelerated expansion of the universe, we are still facing the question whether the cause of it is a rigid cosmological constant $\Lambda$-term or a mildly evolving dynamical dark energy (DDE). While studies focusing mainly on CMB measurements do not perceive signs of physics beyond the $\Lambda$CDM, in this work we show that if we take a large string $SNIa+BAO+H(z)+LSS+CMB$ of modern cosmological observations, in which not only the CMB but also a rich sample of large scale structure formation data are included, one can extract $\sim 3.3\sigma$ signs of DDE using a simple XCDM parameterization. These signs can be enhanced up to near $3.8\sigma$ in the context of the running vacuum model (RVM), in which the vacuum energy density is in interaction with dark matter. Recently the RVM has been shown to provide an efficient and economical solution to the $\sigma_8$-tension, which is one of the intriguing phenomenological problems that has not been possible to solve within the $\Lambda$CDM so far. This fact contributes to strengthen the possibility that dynamical vacuum energy, or in general DDE, could be presently favored by the observations.