Running vacuum in Brans & Dicke theory: A possible cure for the σ8 and H0 tensions

Extensions of the gravitational framework of Brans-Dicke (BD) are studied by considering two different scenarios: i) `BD-Lambda CDM', in which a rigid cosmological constant, Lambda, is included, thus constituting a BD version of the vanilla concordance Lambda CDM model (the current standard model of cosmology with flat three-dimensional geometry), and ii) `BD-RVM', a generalization of i) in which the vacuum energy density (VED), rho(vac), is a running quantity evolving with the square of the Hubble rate: delta rho(vac)(H)proportional to nu m(Pl)(2)(H-2-H-0(2)) (with |nu|<< 1 and H-0 being the present value of the Hubble rate). This dynamical scenario is motivated by recent studies of quantum field theory (QFT) in curved spacetime, which lead to the running vacuum model (RVM). In both cases, rigid or running p(vac), the GR limit can be recovered smoothly. We solve the background as well as the perturbation equations for each cosmological model and test their performance against the modern wealth of cosmological data, namely a compilation of the latest SNIa+H(z)+BAO+LSS+CMB observations. We utilize the AIC and DIC statistical information criteria in order to determine if they can fit better the observations than the concordance model. The two BD extensions are tested by considering three different datasets. According to the AIC and DIC criteria, both BD extensions (i) and (ii) are competitive, but the second one (the BD-RVM scenario) is particularly favored when it is compared with the vanilla model. This fact may indicate that the current observations favor a mild dynamical evolution of the Newtonian coupling G(N) as well as of the VED. While further studies will be necessary, the results presented here suggest that the Brans & Dicke theory with running vacuum could have the potential to alleviate the two tensions at the same time.