Towards a unified quantum field theory of dark energy and inflation: unstable de Sitter vacuum and running vacuum

Inflation is a necessary cosmic mechanism to cure basic inconsistencies of the standard model of cosmology. These problems are usually `fixed' by postulating the existence of a scalar field (called the ``inflaton''). However, other less ad hoc options are possible. In the running vacuum model (RVM) framework, the vacuum energy density (VED) $ρ_{\rm vac}$ is a function of the Hubble rate $H$ and its time derivatives. In this context, the VED is dynamical (there is no rigid cosmological constant $Λ$). In the FLRW epoch, $ρ_{\rm vac}$ evolves very slowly with expansion. In contrast, in the very early universe the vacuum fluctuations induce higher powers $H^N$ capable of unleashing fast inflation in a short period in which $H\simeq$ const. We call this mechanism `RVM-inflation'. It does not require an inflaton field since inflation is brought about by pure quantum field theory (QFT) effects on the dynamical background. It is different from Starobinsky's inflation, in which $H$ is never constant. In this work, we study a closely related scenario: the decay of the exact de Sitter vacuum into FLRW spacetime in its radiation epoch and the subsequent impact on the current universe, and compare with the RVM. We find that in both cases inflation is driven by $H^4$ powers together with subleading contributions of order $H^2$ that ease a graceful-exit transition into the radiation-dominated epoch, where the FLRW regime starts and ultimately develops a mildly evolving VED in the late universe: $δρ_{\rm vac}\sim {\cal O}(m_{\rm Pl} ^2 H^2)$. The proposal presented here aims at a unified QFT approach to inflation and dark energy (conceived as dynamical vacuum energy) with potentially measurable phenomenological consequences in the present universe, and constitutes a first step toward establishing its full theoretical and phenomenological consistency.