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arxiv: 2606.05352 · v1 · pith:WUXRRDK3new · submitted 2026-06-03 · 🌀 gr-qc · astro-ph.CO· hep-ph· hep-th

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

Joan Sol\`a Peracaula This is my paper

Pith reviewed 2026-06-28 04:57 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-phhep-th
keywords running vacuum modelquantum field theorycurved spacetimeinflationdark energydynamical vacuumHubble parameter
0
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The pith

Quantum effects from QFT in curved spacetime induce an evolving vacuum energy that drives both inflation and dark energy in a unified framework.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes that the vacuum energy density in the expanding universe is not constant but runs with the Hubble rate and its derivatives according to quantum field theory in curved spacetime. This running allows higher powers like H to the fourth to cause rapid inflation early on without an inflaton field, while lower order terms provide the slow evolution needed for dark energy today. Newton's gravitational constant also evolves slowly with the logarithm of H. The model replaces the rigid cosmological constant with a dynamical vacuum energy that emerges naturally from QFT. It suggests this framework can describe the entire cosmic history on fundamental grounds and aligns with observations favoring dynamical dark energy.

Core claim

In the running vacuum model, quantum fluctuations induce a vacuum energy density ρ_vac depending on H and its time derivatives, with ~H^4 terms triggering inflation at early times and ~H^2 terms driving dark energy at late times, while G evolves as G(ln H), providing a unified QFT paradigm without additional fields.

What carries the argument

The running vacuum energy density ρ_vac(H, Ḣ, Ḧ, ...) induced by QFT in curved spacetime, which evolves with the expansion rate.

If this is right

  • Dynamical vacuum energy replaces the fixed Lambda term and fits DESI data preferring evolving dark energy.
  • Inflation arises purely from vacuum fluctuations at high H without requiring an inflaton scalar field.
  • Newton's constant G evolves logarithmically with H, implying mild time variation of fundamental constants.
  • The model unifies early and late universe acceleration in one QFT-based description.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the RVM holds, it could resolve current cosmological tensions by allowing dynamical dark energy.
  • Similar running effects might appear in other contexts involving QFT on dynamical backgrounds.
  • Future measurements of varying constants or precise DE evolution could test the predicted forms of ρ_vac.

Load-bearing premise

Quantum field theory calculations in curved spacetime directly yield vacuum energy densities with powers of the Hubble rate large enough to dominate the expansion at the required epochs.

What would settle it

Detection of a strictly constant vacuum energy density throughout cosmic history or no evidence for any evolution in G would contradict the model's predictions.

Figures

Figures reproduced from arXiv: 2606.05352 by Joan Sol\`a Peracaula.

Figure 1
Figure 1. Figure 1: Transition from inflation to the radiation epoch. On the left it is shown the numerical [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Equation of state parameter of the running vacuum in the very early universe, as a [PITH_FULL_IMAGE:figures/full_fig_p045_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Unstable de Sitter vacuum decay: shown are the energy densities for vacuum and [PITH_FULL_IMAGE:figures/full_fig_p049_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The running vacuum EoS (128) as a function of the redshift [34–36]. It mimics [PITH_FULL_IMAGE:figures/full_fig_p052_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Easing the H0 and σ8-tensions. Contours at 1σ, 2σ and 3σ c.l. in the (σ8-H0) and (S8- H0) planes, where S8 ≡ σ8 p Ω0 m/0.3. The considered models are RVM, ΛCDM and the generic DE parameterization XCDM (or wCDM) for the data sets SnIa+BAO+H(z)+LSS+CMB used in [74]. Only the RVM alleviates the H0 tension and at the same time it reduces the S8 one. 5.3 Running of Newton’s G with the cosmic expansion From the … view at source ↗
read the original abstract

The concordance $\Lambda$CDM model, based on a rigid $\Lambda$-term for the entire cosmic history, has been in crisis for a long time. In our expanding Universe, an evolving $\Lambda$ 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) $\rho_{\rm vac}=\Lambda/(8\pi G)$ which is a function of the Hubble rate $H$ and its time derivatives, $\rho_{\rm vac}=\rho_{\rm vac}(H, \dot{H},\ddot{H},\dots)$. Currently, $\rho_{\rm vac}$ evolves very slowly with the expansion, $\delta\rho_{\rm vac}\sim {\cal O}(m_{Pl} ^2 H^2)$, 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. $\sim H^4$, 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 an `inflaton' 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 $\rho_{\rm vac}(H)$ 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 $\Lambda$ term.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that within the running vacuum model (RVM) based on QFT in curved spacetime, the vacuum energy density takes the form ρ_vac = ρ_vac(H, Ḣ, Ḧ, …) with higher even powers such as H^4 inducing inflation in the early universe without an inflaton field, while the term O(m_Pl² H²) provides the dynamical dark energy at late times. Newton's constant is also running as G = G(ln H). This framework is presented as a unified QFT paradigm for inflation and dark energy that is consistent with DESI measurements favoring dynamical DE over a rigid Λ.

Significance. Should the QFT derivation of the specific functional form and coefficients hold without additional assumptions, this would represent a significant advance by providing a single mechanism rooted in quantum effects on curved spacetime for both the inflationary epoch and the current accelerated expansion. It would eliminate the need for an inflaton field and offer a dynamical origin for the vacuum energy, potentially resolving aspects of the cosmological constant problem. The logarithmic running of G adds implications for varying constants.

major comments (2)
  1. Abstract: The central claim that quantum effects induce ρ_vac with the specific powers H^4 (early universe) and O(m_Pl² H²) (late universe) dominating the dynamics at the required epochs is asserted but not derived in the provided text; the coefficients multiplying these terms are not shown to emerge parameter-free from a renormalization calculation in curved spacetime, which is load-bearing for the 'unified QFT paradigm' without additional mechanisms or tuning.
  2. Section on RVM-inflation: The statement that H is approximately constant during RVM-inflation due to the H^4 term, differing from Starobinsky inflation, lacks an explicit dynamical analysis or solution of the modified Friedmann equations showing the required slow-roll parameters or number of e-folds from the QFT-induced terms alone.
minor comments (2)
  1. The abstract mentions consistency with DESI measurements but does not include explicit data comparison or fit results.
  2. Notation for m_Pl and the precise definition of the functional dependence should be clarified with references to the underlying QFT computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: Abstract: The central claim that quantum effects induce ρ_vac with the specific powers H^4 (early universe) and O(m_Pl² H²) (late universe) dominating the dynamics at the required epochs is asserted but not derived in the provided text; the coefficients multiplying these terms are not shown to emerge parameter-free from a renormalization calculation in curved spacetime, which is load-bearing for the 'unified QFT paradigm' without additional mechanisms or tuning.

    Authors: The functional form ρ_vac=ρ_vac(H,Ḣ,Ḧ,…) with the indicated even powers follows from the standard renormalization of the vacuum energy in QFT on curved spacetime, as established in the RVM literature cited in the manuscript. The coefficients are fixed by the renormalization conditions rather than introduced by hand. To make the presentation fully self-contained, we will add a concise outline of the relevant renormalization steps (including how the H^4 and m_Pl²H² terms arise) in a new subsection of the revised manuscript. revision: yes

  2. Referee: Section on RVM-inflation: The statement that H is approximately constant during RVM-inflation due to the H^4 term, differing from Starobinsky inflation, lacks an explicit dynamical analysis or solution of the modified Friedmann equations showing the required slow-roll parameters or number of e-folds from the QFT-induced terms alone.

    Authors: We agree that an explicit solution of the modified Friedmann equations would strengthen the section. In the revised manuscript we will provide the dynamical analysis, solve the equations with the leading H^4 term, verify that H remains approximately constant, compute the slow-roll parameters, and estimate the number of e-folds generated by the QFT-induced vacuum energy alone. This will also make the contrast with Starobinsky inflation explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework presented as QFT-derived

full rationale

The abstract and description frame ρ_vac(H,Ḣ,...) and the H^4/H² powers as induced by QFT in curved spacetime on FLRW, with G(ln H) running as a separate slow effect. No equations are exhibited that define the target quantities in terms of themselves or rename fitted coefficients as predictions. The load-bearing step is the external QFT computation itself rather than an internal reduction or self-citation chain within this manuscript. This is the common case of a self-contained presentation against an external benchmark (QFT effective action), warranting score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; full derivations and coefficient determinations are unavailable, so the ledger is necessarily incomplete.

axioms (1)
  • domain assumption QFT in curved spacetime induces vacuum energy density as a function of the Hubble rate H and its time derivatives
    Stated as the foundational premise of the RVM in the abstract

pith-pipeline@v0.9.1-grok · 5948 in / 1371 out tokens · 27915 ms · 2026-06-28T04:57:14.864835+00:00 · methodology

discussion (0)

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Reference graph

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