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Perhaps the deepest mystery of our accelerating Universe in expansion is theexistence of a tiny and rigid cosmological constant, $\Lambda$. Its size ismany orders of magnitude below the expected one in the standard model ofparticle physics. However, an expanding Universe is not expected to have astatic vacuum energy density. We should rather observe a mildly dynamicalbehavior $\delta\Lambda(t)\sim R\sim H^2(t)$ with the expansion rate $H$. Atthe same time, it is natural to think that the huge value of the primevalvacuum energy (presumably connected to some grand unified theory) wasresponsible for the initial inflationary phase. In the traditional inflatonmodels such phase is inserted by hand in the early epoch of the cosmicevolution, and it is assumed to match the concordance $\Lambda$CDM regimeduring the radiation epoch. Here, instead, we consider a class of dynamicalvacuum models which incorporate into a single vacuum structure$\bar{\Lambda}(H)$ the rapid stage of inflation, followed by the radiation andcold matter epochs, until achieving our dark energy Universe. The earlybehavior of the model bares resemblance with Starobinsky's inflation andptovides a solution to the large entropy problem. It is compatible with thelatest cosmological data on Hubble expansion and structure formation, andpresents distinctive observational features that can be tested in the nearfuture.

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