Abstract: A universe with a Cosmological Constant has been the reigning paradigm in the last decades. However, the Cosmological Principle opens a window to the possibility of having a dynamical Vacuum Energy density, ρvac(t). In this thesis, we have studied the renormalization procedure of the vacuum energy with contributions from scalars and fermions in the context of quantum field theory (QFT). The quantum scaling with the renormalization point manifests as cosmic evolution, expressed as a series of powers of the Hubble parameter, ρvac(H). At low energies, it consists of an additive term and a small dynamical component ∼ νH2 (|ν| ≪ 1), characteristic of the Running Vacuum Models. Higher powers may be relevant in the early universe and could naturally drive a possible mechanism for inflation.

We elucidate new features of the running vacuum energy with many interesting consequences from the phenomenological point of view. In particular, we explore their possible implications for the current σ8 and H0 tensions of modern cosmology by studying models of Modified Gravity and Dynamical Vacuum Energy Density.