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

TL;DR

The paper establishes a semiclassical QFT framework in curved spacetime in which inflation and dark energy emerge from a dynamical vacuum energy density ρ_vac that runs with the Hubble rate. Using off-shell adiabatic renormalization, it derives renormalized vacuum EMTs and shows that the early Universe can experience H^4-driven inflation, while subleading H^2 terms drive a smooth transition to the radiation-dominated era and a mildly evolving ρ_vac at late times. It compares two inflationary realizations—running vacuum models and unstable de Sitter decay—finding strikingly similar late-time VED running, though their fundamental vacuum EoS differs (−1 in de Sitter, dynamical in RVM). The framework naturally links inflation to dynamical DE as a fossil remnant of primeval vacuum energy, potentially addressing cosmological tensions without introducing ad hoc inflatons or quintessence fields, and predicting observable signatures in the current cosmic expansion.

Abstract

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) is a function of the Hubble rate $H$ and its time derivatives: $ρ_{\rm vac}=ρ_{\rm vac}(H, \dot{H},\ddot{H},\dots)$. 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, as befits the observed $Λ\simeq$const. behavior. 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 outcome is an unified QFT approach to inflation and dark energy (conceived as dynamical vacuum energy) with potentially measurable phenomenological consequences in the present universe which can help to cure the cosmological tensions.

Figures (8)

• Figure 1: Dependence of the VED, $\rho_\mathrm{vac}$, normalized to the Planck density in the inflationary period for the unstable Sitter case as a function of $m/H$. We plot specifically Eq. \ref{['eq:VED_Ren_final2']}. The various contributions from the terms of $\mathcal{O}(H^4)$ and $\mathcal{O}(H^2)$ which add up to the total $\rho_\mathrm{vac}$ (red curve) are also displayed. We can take into account the contribution of more scalar fields with the same non-minimal coupling and mass by including a multiplicity factor $N_0$. For this numerical example, we take $N_0=1000$ with $\xi=0.168$ (for all scalar fields) and $m=0.001m_\mathrm{Pl}$ (see Sec.\ref{['sec:Inflation']} for the numerical analysis and notation).
• Figure 2: Hubble rate $H_I$ from Eq. \ref{['eq:HIA']} and temperature $T_I$ of radiation immediately after inflation, Eq. \ref{['eq:TempdeSitter']}, for different multiplicities of the fields, $N_0$, and for $\xi$ (assumed common to all of them) in the allowed region for inflation, as discussed in the text. It is seen that for large enough $N_0$ the condition \ref{['eq:subplankian']} is fulfilled all over the permitted range, and $T_I<m_{\rm Pl}$.
• Figure 3: De Sitter decay scenario: energy densities for vacuum and radiation for $\xi>1/6$, as a function of the scale factor normalized to the transition point, $\hat{a}=a/a_*$. We compare the numerical solution to Eq. \ref{['eq:diffHdeSitter']} (solid lines) with the (RVM-like) analytical approximation (dashed lines) given in \ref{['eq:infaprox2']}-\ref{['eq:infaprox3']}. As in previous figures, we use $m=0.001m_\mathrm{Pl}$ for the scalar field mass.
• Figure 4: Same as Fig. \ref{['fig:ved_greater']} but taking $\xi<1/6$.
• Figure 5: Energy density ($\rho_{\rm vac}$) and pressure ($P_{\rm vac}$) of the unstable de Sitter vacuum for the same cases considered in Figs. \ref{['fig:ved_greater']} and \ref{['fig:ved_lower']}. Because $\rho_{\rm vac}$ and $P_{\rm vac}$ in the last region take much higher values the features observed on the plots on the left cannot be appreciated in those on the right.