The $\bar{\Lambda}CDM$ cosmology: from inflation to dark energy through running $\Lambda$

Perhaps the deepest mystery of our accelerating Universe in expansion is the existence of a tiny and rigid cosmological constant, $\Lambda$. Its size is many orders of magnitude below the expected one in the standard model of particle physics. However, an expanding Universe is not expected to have a static vacuum energy density. We should rather observe a mildly dynamical behavior $\delta\Lambda(t)\sim R\sim H^2(t)$ with the expansion rate $H$. At the same time, it is natural to think that the huge value of the primeval vacuum energy (presumably connected to some grand unified theory) was responsible for the initial inflationary phase. In the traditional inflaton models such phase is inserted by hand in the early epoch of the cosmic evolution, and it is assumed to match the concordance $\Lambda$CDM regime during the radiation epoch. Here, instead, we consider a class of dynamical vacuum models which incorporate into a single vacuum structure $\bar{\Lambda}(H)$ the rapid stage of inflation, followed by the radiation and cold matter epochs, until achieving our dark energy Universe. The early behavior of the model bares resemblance with Starobinsky's inflation and ptovides a solution to the large entropy problem. It is compatible with the latest cosmological data on Hubble expansion and structure formation, and presents distinctive observational features that can be tested in the near future.