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

Abstract: The concordance $Λ$CDM model, based on a rigid $Λ$-term for the entire cosmic history, has been in crisis for a long time. In our expanding Universe, an evolving $Λ$ with the expansion is intuitively much more reasonable. In the running vacuum model (RVM) framework, based on quantum field theory (QFT) in curved spacetime, quantum effects induce a vacuum energy density (VED) $ρ<em>{\rm vac}=Λ/(8πG)$ which is a function of the Hubble rate $H$ and its time derivatives, $$ρ</em>{\rm vac}=ρ<em>{\rm vac}(H, \dot{H},\ddot{H},\dots)$$. Currently, $ρ</em>{\rm vac}$ evolves very slowly with the expansion, $$δρ<em>{\rm vac}\sim {\cal O}(m</em>{Pl} ² H<sup>2)$$, and this fact provides a possible fundamental origin of dark energy (DE), conceived as dynamical vacuum energy. In the RVM, Newton's $G$ is also evolving, but much more slowly (logarithmically with $H$): $G=G(\ln H)$. In the very early universe, the vacuum fluctuations induce higher (even) powers, e.g. $\sim H<sup>4$, capable of triggering fast inflation in a very short period, in which $H$ is very large and approximately constant. This is the mechanism of RVM-inflation'. It does not require aninflaton' field since inflation is brought about by pure QFT effects on the dynamical background. It differs from Starobinsky's inflation, where $H$ is never constant. Furthermore, the dynamics of $ρ_{\rm vac}(H)$ and $G(H)$ 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 $Λ$ term.

• The paper introduces a first-principles QFT framework where the vacuum energy density runs with the Hubble parameter, unifying inflation and dark energy without a separate inflaton field.
• It employs off-shell renormalization with the adiabatic regularization prescription in FLRW backgrounds to eliminate quartic divergences and drive cosmic transitions naturally.
• The model yields improved fits for H0 and σ8 tensions while predicting mild dynamical dark energy and potential cosmological variations in fundamental constants.

Quantum Field Theory, Running Vacuum, and a Unified Cosmological Paradigm

Introduction and Motivation

The paper "Running Vacuum in the expanding Universe: a unified QFT paradigm for Inflation and Dark Energy" (2606.05352) synthesizes extensive theoretical advances in quantum field theory in curved spacetime (QFTCS) to construct a renormalization program for vacuum energy in cosmology. The principal aim is to provide a first-principles framework that addresses both early-universe inflation and late-time dark energy (DE), thereby offering a unified dynamical paradigm that supersedes the ad hoc separation of the cosmological constant (Λ) and inflation typically implemented in standard cosmology. The model, called the Running Vacuum Model (RVM), is sharply contrasted with the conventional ΛCDM, which assumes a static Λ throughout cosmic evolution.

The work asserts that the vacuum energy density (VED), $\rho_\mathrm{vac}$, is a function of the Hubble parameter $H$ and its derivatives, emerging naturally from QFT renormalization in FLRW backgrounds via the adiabatic regularization prescription (ARP) generalized to an off-shell renormalization scale. This perspective crucially escapes both the old and new cosmological constant problems, avoids fine-tuning, and can be phenomenologically tuned to reconcile the persistent $H_0$ and $\sigma_8$ tensions in late-time cosmological data.

QFT Analysis of the Running Vacuum

The central technical achievement of the paper is a rigorous, covariant off-shell renormalization of the vacuum energy-momentum tensor (EMT) for generic quantized fields non-minimally coupled to the FLRW background. Unlike most conventional treatments relying on minimal subtraction schemes in Minkowski spacetime—thereby lacking cosmological relevance and carrying arbitrary finite counterterms—the RVM approach operates directly in cosmological backgrounds, employing ARP and subtraction at an arbitrary physical scale $M$ (later identified with $H$).

The vacuum EMT is composed of the classical vacuum term (bare $\rho_\Lambda$) and the ZPE of quantum fluctuations. The renormalized VED acquires the following explicit structure (neglecting higher-derivative and non-renormalizable contributions): $$\rho_\mathrm{vac}(H) \approx \rho_\Lambda + \sum_i \frac{c_i}{(4\pi)^2} \left\{ H^2 m_i^2 + {\cal O}(H^4) \right\}$$ where $m_i$ are the physical field masses and $c_i$ are coefficients dependent on spin, coupling, and multiplicity. Crucially, quartic divergences $H$0 are eliminated in the renormalization difference between epochs, as demanded by general covariance and enforced by the off-shell ARP. The running emerges as a consequence of tracing the finite evolution between $H$1 and $H$2, avoiding unnaturally large vacuum masses.

Figure 1: Transition from inflation to the radiation epoch. On the left, the numerical solution of the VED (solid line) versus the approximate analytical solution; on the right, the radiation energy is also shown, both with $H$3.

Inflation as Vacuum Dynamics: No Inflaton Required

One of the most forceful claims is that in the early universe, the running VED is dominated by the $H$4 term, naturally generating a period of quasi-de Sitter expansion with no need for a distinct inflaton scalar or engineered potential. This is a direct quantum effect from GUT-scale fields, with non-minimal couplings $H$5 required for positive inflation-driving coefficients.

$H$6

This mechanism enables a "graceful exit": as $H$7 decreases, the $H$8 term becomes dominant, automatically terminating inflation and commencing a radiation-dominated epoch. This dynamical transition, and the smooth decay of primordial VED into standard-model radiation, are confirmed both analytically and numerically.

Figure 2: Equation of state parameter of the running vacuum in the very early universe as a function of normalized scale factor, $H$9. The EoS interpolates between $H_0$0 during inflation to $H_0$1 during radiation domination.

Late-time Cosmology, Fitting Data, and Cosmic Tensions

For $H_0$2 (all known masses), expanding the renormalized VED yields a leading linear running: $H_0$3 with $H_0$4 a calculable, softly suppressed parameter. The theory thus predicts a mild dynamical dark energy component. Observationally, fitting SNIa, BAO, LSS, and CMB data with RVM yields $H_0$5 in the range $H_0$6, well below limits from primordial nucleosynthesis and pulsar timing.

As highlighted by the paper, the RVM delivers improved fits to the $H_0$7 and $H_0$8 tensions, which plague the traditional ΛCDM—this is both statistically significant and robust to dataset choices.

Figure 3: Easing the $H_0$9 and $\sigma_8$0 tensions in the (σ₈, H₀) and ($\sigma_8$1, $\sigma_8$2) planes. Contours at $\sigma_8$3, $\sigma_8$4, and $\sigma_8$5 confidence, with $\sigma_8$6, showing superior concordance for the RVM.

Equation-of-State and Theoretical Consistency

The RVM yields predictions for the vacuum EoS parameter $\sigma_8$7 that may deviate slightly from $\sigma_8$8 in the late universe, depending on $\sigma_8$9. At high $M$0, the EoS tracks the dominant component—a chameleonic feature distinguishing the RVM from simple scalar field models.

Figure 4: The running vacuum EoS as a function of redshift. RVM tracks quintessence or phantom-like DE for $M$1 or $M$2 depending on the sign of $M$3.

QFT Phenomena and Variation of Fundamental Constants

The paper discusses far-reaching implications: if all couplings and masses run with $M$4, the "constants" of physics may drift cosmologically, with RVM predicting explicit correlations between changes in $M$5, $M$6, and particle mass ratios—predictions connectable to atomic clocks and astrophysical bounds.

Resolution of Cosmological Problems

The smooth, non-fine-tuned evolution of the VED in RVM solves the entropy and horizon problems causally—no particle horizon forms, entropy production is linked directly to vacuum decay during inflation, and the present-day entropy budget of the observable universe is recovered quantitatively.

Theoretical Robustness and Future Outlook

The effective action formalism recapitulates all results directly from the heat kernel and DeWitt-Schwinger expansion, cross-validating the mode function approach. The dynamical running dictated by renormalization is thus structurally enforced, and deviations from their form would imply either a failure of QFTCS or breakage of covariance.

Potential future directions include coupling the RVM with string-based or anomaly-driven corrections, probing the possibility of observable cosmic drift in lab-based measurements, and developing non-perturbative frameworks to extend the results toward quantum gravity.

Conclusion

The Running Vacuum Model, formulated from first-principles QFT renormalization in curved cosmological backgrounds, provides a technically natural, dynamically consistent mechanism for both inflation and dark energy without recourse to fine-tuned parameters or hypothetical inflaton fields. Its softly evolving vacuum energy density, free from quartic mass hazards and robustly linked to the cosmic expansion, aligns with a wide set of empirical data and resolves outstanding cosmological anomalies. The link to the possible variation of physical 'constants' opens up intriguing connections across fundamental physics, cosmology, and experimental testability. The RVM thus stands as a theoretically mature, unifying paradigm for cosmic acceleration in all eras.