Cosmic expansion and structure formation in running vacuum cosmologies
  We investigate the dynamics of the FLRW flat cosmological models in which the
vacuum energy varies with redshift. A particularly well motivated model of this
type is the so-called quantum field vacuum, in which both kind of terms $H^{2}$
and constant appear in the effective dark energy density affecting the
evolution of the main cosmological functions at the background and perturbation
levels. Specifically, it turns out that the functional form of the quantum
vacuum endows the vacuum energy of a mild dynamical evolution which could be
observed nowadays and appears as dynamical dark energy. Interestingly, the
low-energy behaviour is very close to the usual $\Lambda$CDM model, but it is
by no means identical. Finally, within the framework of the quantum field
vacuum we generalize the large scale structure properties, namely growth of
matter perturbations, cluster number counts and spherical collapse model.
