Implications of Dynamical Dark Energy in the expansion of the Universe and the Structure Formation

Abstract: In this PhD thesis a wide variety of cosmological models beyond the $\Lambda$CDM are studied in detail. Great emphasis is put on the running vacuum models (RVM's), which can be motivated in the context of Quantum Field Theory in curved spacetime. They consider the possibility of a smoothly evolving vacuum energy density that inherits its time-dependence from cosmological variables, such as the Hubble rate and its time derivative. The analysis presented, however, is not limited only to the dynamical vacuum models but considers also models of dark energy such as the Peebles & Ratra scalar field model or simple parameterizations like the XCDM or the CPL. Their theoretical predictions are tested against the wealth of cosmological data and a fitting procedure is carried out in order to obtain constraints over the free parameters that characterize the models under study. Finally, an extended discussion about the $\sigma_8$ and $H_0$ tensions is provided and a possible solution for both of them is described within the framework of Brans and Dicke gravity with a cosmological constant, which mimics the RVM with a mild time-evolving gravitational coupling $G$. The main conclusion of this study is that appreciable signals in favour of a time-evolving dark energy density are found in the current data.