Composite running vacuum in the Universe: implications on the cosmological tensions

The possibility that the vacuum energy density (VED) could be time dependent in the expanding Universe is intuitively more reasonable than just a rigid cosmological constant for the entire cosmic history. The framework of the running vacuum model (RVM) is a salient example derived from QFT in curved spacetime, wherein the VED appears as a power series of the Hubble rate, $H(t)$, and its derivatives. The RVM contributes to alleviate the cosmological tensions and at a more fundamental level it also helps to smooth out certain hardcore aspects of the cosmological constant problem. Composite dark energy (DE) extensions of the RVM are possible, in which the DE is a mixed fluid made out of running vacuum and an entity X called ``phantom matter'' which, I should stress, is radically different from phantom DE, since the former produces positive pressure like ordinary matter (therein its name). The prototype is the old $\Lambda$XCDM model from 2006. Recently a simplified version of the latter, called the $w$XCDM, has proven capable to yield a truly outstanding fit to the cosmological data with dramatic implications for mitigating the cosmological tensions. The analysis is sensitive to the kind of BAO data used, transversal (2D) or anisotropic (3D). For both BAO types the overall fit quality is substantially better than for the standard $\Lambda$CDM model, and the growth tension becomes alleviated. Nevertheless, only for 2D BAO the $H_0$-tension can be completely cut down. Most notably, this scenario favours (at $3.3\sigma$ c.l.) quintessence-like behavior around our time, as recently reported by the DESI Collaboration. In what follows I will summarize in very simple phenomenological terms the essentials of the RVM idea and its composite extensions, as well as the remarkable practical implications ensuing from such scenarios.