Running Vacuum in the expanding Universe: a unified QFT paradigm for Inflation and Dark Energy

The concordance [Formula: see text]CDM model, based on a rigid [Formula: see text]-term for the entire cosmic history, has been in crisis for a long time. In our expanding Universe, an evolving [Formula: see text] with the expansion is intuitively much more reasonable, especially if a fundamental theoretical reason can be given for it. In the running vacuum model (RVM) framework, based on quantum field theory (QFT) in curved spacetime, quantum effects induce a vacuum energy density (VED) [Formula: see text] which is a function of the Hubble rate [Formula: see text] and its time derivatives, [Formula: see text]. Currently, [Formula: see text] evolves very slowly with the expansion, [Formula: see text], and this fact provides a possible fundamental origin of dark energy (DE), conceived as dynamical vacuum energy. The RVM framework leads to a “renormalized [Formula: see text]CDM”, an upgraded version of the concordance model which can help cure the cosmological tensions. In it, [Formula: see text] is also evolving, but much slower (logarithmically with [Formula: see text]): [Formula: see text]. In the very early universe, on the other hand, the vacuum fluctuations of the quantized matter fields induce higher (even) powers, e.g. [Formula: see text], capable of triggering fast inflation in a very short period, in which [Formula: see text] is very large and approximately constant. This is the mechanism of ‘RVM-inflation’. It does not require an ‘inflaton’ field since inflation is brought about by pure QFT effects on the dynamical background. It differs from Starobinsky’s inflation, where [Formula: see text] is never constant. Furthermore, the dynamics of [Formula: see text] and [Formula: see text] can also have implications on the frequently discussed possibility that the fundamental ‘constants’ of Nature can be mildly evolving with the cosmic expansion. Putting things together, a unified QFT framework of dark energy and inflation ensues as a realistic theory for the description of the universe as a whole on fundamental grounds. In it, dynamical VED is predicted and is much welcomed, since it fits in with current DESI measurements, preferring dynamical DE over a rigid [Formula: see text] term.