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This paper argues that quantum vacuum fluctuations alone—no inflaton, no quintessence—can drive both primordial inflation and today's dark energy, with inflation powered by H^4 terms and the late-time vacuum evolving via a mild H^2 running

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2026-08-03 11:42 UTC pith:LKFEF27H

load-bearing objection The exact de Sitter EMT renormalization and its on-shell Bunch-Davies recovery are genuine technical achievements; the 'unified' dark-energy claim, however, rests on choosing M=H, which is a convention rather than a derivation. the 2 major comments →

arxiv 2601.05218 v3 pith:LKFEF27H submitted 2026-01-08 gr-qc astro-ph.COhep-phhep-th

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

Joan Sol\`a Peracaula , \`Alex Gonz\'alez-Fuentes , Cristian Moreno-Pulido This is my paper
classification gr-qc astro-ph.COhep-phhep-th PACS 04.62.+v98.80.-k95.36.+x
keywords running vacuum modeldynamical dark energyinflationde Sitter spacetimeoff-shell adiabatic renormalizationvacuum energy densityH^4 inflationcosmological constant problem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the observed expansion history—from primordial inflation to the present accelerated expansion—can be derived from pure quantum field theory on a curved spacetime, without inventing an inflaton, quintessence, or phantom fields. In this picture the vacuum energy density is not a rigid cosmological constant but a running quantity ρ_vac(H) whose form is fixed by renormalization. During inflation the leading quantum term is H^4, which keeps H nearly constant for a brief period and then decays smoothly into radiation; in the late universe the subleading H^2 term produces a slow drift of the vacuum energy of order δρ_vac ~ m_Pl^2 H^2 with a coefficient computable in QFT and small enough to pass cosmological data. If true, the paper claims, dark energy is a fossil remnant of inflation, and both epochs are connected in one framework that also softens the cosmological-constant problem and addresses the universe's entropy problem.

Core claim

The central discovery is that both scenarios—the running vacuum model and an unstable de Sitter vacuum, the latter treated here for the first time with off-shell adiabatic renormalization—yield the same unified structure for the vacuum energy density: a dominant H^4 term that triggers a brief quasi-de Sitter inflation, plus H^2 terms that mediate a smooth exit into the radiation epoch and later become the mildly evolving dark energy of today. Late in the universe, both converge to ρ_vac(H) = ρ_vac(H_0) + (3ν_eff/8π)m_Pl^2 (H^2 - H_0^2), with ν_eff a QFT-computable beta-function coefficient of order 10^-4 to 10^-3. The calculation reproduces the classic on-shell result, accounts for the confo

What carries the argument

The engine of the argument is the off-shell adiabatic renormalization prescription: the vacuum expectation value of the energy-momentum tensor is computed on-shell for the physical mass m, and the first four adiabatic orders are subtracted with the mass replaced by an arbitrary renormalization scale M. This leaves a finite vacuum energy that depends on M, and the physical running is obtained by identifying M with the Hubble rate H of the epoch, M = H. In de Sitter spacetime the field modes are exact Hankel functions of order ς = ς(m/H; ξ), where ξ is the non-minimal curvature coupling, and the same subtraction yields closed expressions for the vacuum energy and trace. The key structural resu

Load-bearing premise

The late-time running of the vacuum energy rests on choosing the Hubble rate H as the renormalization scale; change that choice and the claimed H^2 dark-energy evolution disappears.

What would settle it

Recompute the renormalized vacuum energy density in a curved background with the same off-shell subtraction but with the renormalization scale M held fixed at a constant (for instance, the particle mass m) rather than set to H. If the result contains no term proportional to m_Pl^2 (H^2 - H_0^2), the predicted late-time dark-energy running is a scheme artifact; a reader can check this calculation directly from Eqs. (31) and (32) of the paper by replacing M = H with M = const.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Inflation proceeds without an inflaton: the H^4 vacuum term drives a short period with H ≈ const., unlike R^2 inflation where Ḣ is constant, and the vacuum decays into radiation without a separate reheating phase.
  • The late-time vacuum energy is predicted to evolve as ρ_vac(H) = ρ_vac(H_0) + (3ν_eff/8π) m_Pl^2 (H^2 - H_0^2), with ν_eff in the range 10^-4–10^-3, matching the dynamical dark energy hinted at by observations and potentially relieving the Hubble and growth tensions.
  • No quintessence or phantom fields are needed: the quantum vacuum itself mimics these behaviors—the running vacuum through a redshift-dependent effective equation of state w(z), the de Sitter vacuum through the 'mirage' of a fixed w = -1 with evolving density.
  • The present-day entropy, S ~ 10^88, is generated causally during H^4-inflation through vacuum decay into radiation, linking the earliest and latest epochs of cosmic history in one consistent account.
  • The running of the vacuum energy is free of ~m^4 contributions; the beta function is ∝ m^2 H^2, so at the level of the running the fine-tuning that plagues the cosmological-constant problem is avoided.

Where Pith is reading between the lines

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

  • A decisive test of the scheme: perform the same off-shell subtraction with the renormalization scale M fixed at a constant (e.g., the particle mass) instead of M = H; if the H^2 term disappears, the claimed late-time running is an artifact of the scale choice and the unified dark-energy mechanism collapses.
  • Because the two scenarios coincide at low energies, late-time density measurements alone cannot distinguish the running vacuum from the decaying de Sitter vacuum; only a precise measurement of the dark-energy equation of state—w(z) deviating from -1 as in the RVM, versus a strict -1 for de Sitter—separates them.
  • The narrow allowed window for the non-minimal coupling in the de Sitter case (roughly 0.161 < ξ < 0.339 for a single field) turns precision cosmology into a probe of new physics near the GUT scale, since any field species with a coupling in that window contributes to inflation.
  • If the vacuum is genuinely running with ν_eff of order 10^-4 to 10^-3, independent late-time probes—such as the integrated Sachs-Wolfe effect or growth-rate measurements—should eventually reveal a small but systematic departure from ΛCDM at the percent level, a prediction sharper than the paper's own statistical fits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper proposes a semiclassical QFT framework in which the vacuum energy density of quantized matter fields, renormalized by an off-shell adiabatic prescription, drives both inflation and dark energy. The authors compute the renormalized vacuum energy-momentum tensor (EMT) for a non-minimally coupled scalar field in FLRW spacetime and in exact de Sitter spacetime, using Hankel-mode integrals with dimensional regularization. They claim that both the running vacuum model (RVM) and a decaying de Sitter vacuum yield an H^4-dominated inflationary phase with a subleading H^2 term that provides a graceful exit into radiation, and at late times a mildly evolving vacuum energy δρ_vac ∼ ν_eff m_Pl^2 H^2. The on-shell limit of their de Sitter computation reproduces the Bunch-Davies result. The paper argues that no inflaton, quintessence, or phantom fields are needed, and discusses phenomenological consequences including an effective quintessence/phantom behavior and a causal explanation of the cosmological entropy.

Significance. The manuscript contains a substantial and largely coherent calculation. The exact de Sitter mode-function computation, the use of dimensional-regularized Hankel integrals, and the explicit cancellation of UV divergences in the off-shell adiabatic subtraction are serious technical achievements. The recovery of the Bunch-Davies/Dowker-Critchley EMT at M=m (Eq. 106) is a genuine cross-check against an established result. If the M=H identification were physically justified, the paper would offer an attractive unified account of inflation and dynamical dark energy, with a calculable running coefficient and falsifiable late-time predictions. The discussion of entropy production is also a useful addition. However, the principal observable claim is conditional on an unproven scale identification that is not derived from first principles.

major comments (2)
  1. [Sec. 2.5, 4.1, 5.3 (Eqs. 31-32, 77-78, 80, 111, 129)] The late-time running VED, Eq. (37)/(140), and the H^4-inflation coefficient α, Eq. (111), are obtained by setting the renormalization scale M equal to H (and M_0=H_0). However, the paper does not derive this identification; Sec. 2.5 only says 'it has been suggested' with references to the authors' own program. In Eq. (31) the H^2 coefficient is (ξ-1/6)[M^2-m^2+m^2 ln(m^2/M^2)]/(16π^2), which vanishes for M=m. If M is fixed at any other scale, the coefficient and the H^4 terms change. Thus the central quantitative predictions are scheme-dependent. Please either derive M=H from a physical principle, demonstrate stability under a class of scale choices, or explicitly reframe the claims as scheme-dependent.
  2. [Sec. 5.3 (Eqs. 131-134)] The inflationary mechanism in the de Sitter scenario requires α>0, which for a single scalar field restricts ξ to the narrow interval (129), and to keep H_I sub-Planckian the paper requires N_0≳O(100) (Eq. 134). Since ξ, m, and N_0 are free parameters, the model does not yet provide a unique prediction for the inflation scale or for ν_eff. The paper should either tie this parameter region to a concrete GUT particle content or state more modestly that the framework demonstrates a mechanism rather than a complete unified theory.
minor comments (5)
  1. [Sec. 6.1] Typo: 'tbe current universe' should be 'the current universe'.
  2. [Sec. 5.5] Typo: 'is is' should be 'is'.
  3. [Eq. (82)] The notation O(H/m H^4) is confusing; use O((H/m)H^4).
  4. [Sec. 3.2] The prime notation is overloaded (Hubble derivative, mode derivative, Hankel function derivative); consider using a different symbol for Hankel derivatives to avoid possible confusion.
  5. [Fig. 1] The legend label 'O(m^2H^2), O(m^4 ln m^2/H^2)' seems to combine two different terms; clarify which curve corresponds to which contribution.

Circularity Check

2 steps flagged

Partial circularity (5): the principal late-time 'prediction' δρ_vac∝m_Pl²H² is generated by the M=H scale identification, imported as a suggested convention from the authors' own prior work, and vanishes at the independently validated on-shell scale; the H^4 inflation mechanism and the Bunch-Davies recovery are genuine independent QFT content.

specific steps
  1. ansatz smuggled in via citation [Sec. 2.5 (Eqs. 31-32); applied to de Sitter in Sec. 4.1 (Eqs. 77-78) and Sec. 6.1 (Eqs. 139-142)]
    "According to the standard practice in ordinary gauge theories, the choice of the scale should be made near the typical energy scale of the process. Here, the 'process' is, of course, the cosmic expansion, and it has been suggested to make the choice of M at the value of the Hubble rate H for each cosmic epoch under consideration...; see [43–46] for more details and [22] for a review. ... the on-shell value (M=m) of this expression is simply ρ_vac(m,H)=ρ_Λ(m), which is independent of H, as it should since the vacuum EMT has been subtracted at the point M."

    The abstract's central DE claim — 'ultimately develops a mildly evolving VED in the late universe: δρ_vac ~ O(m_Pl^2 H^2)' — is obtained in Sec. 2.5 as Eq. (32) by substituting the arbitrary renormalization scale M=H into the off-shell result ρ_vac(M,H), Eq. (31); the same substitution generates the de Sitter running in Sec. 4.1 (Eq. 78) and Sec. 6.1 (Eq. 140). The paper itself states that at M=m, ρ_vac(m,H)=ρ_Λ(m), i.e. the H²-running is absent. The M=H rule is justified only as 'it has been suggested', citing the authors' own [43–46] and review [22]. Fixing M at any other scale changes or removes the predicted δρ_vac ∝ H² term, and the one scale the paper independently validates (Bunch-Davies at M=m) is exactly where the running vanishes. The late-time DE prediction therefore reduces to

  2. other [Sec. 5.3 (Eqs. 111, 128-129); Sec. 2.6 (Eqs. 32, 38)]
    "if we set the on-shell condition M=m, the Bunch-Davies EMT given by (106) applies. In that case, the overall coefficient of H^4 would not be given by the full Eq. (111), since the linear term in ξ−1/6 (the only one contributing in the RVM case) would be missing. The corresponding (approximate) allowed range for ξ would then be 0.136<ξ<0.197, which is even narrower than (129)."

    The headline inflationary parameters — the H^4 coefficient α (Eq. 111), the allowed window 0.161<ξ<0.339 (Eq. 129), and hence H_I (Eq. 132) — are evaluated after the same M=H identification. The paper concedes that under the on-shell condition M=m, 'the overall coefficient of H^4 would not be given by the full Eq. (111)' and the ξ-range narrows to 0.136<ξ<0.197. Likewise the RVM H^4-inflation form (38) is read off from the M=H running, Eq. (32). So the quoted numbers are outputs of the self-cited M=H convention rather than scheme-independent QFT results. This is a partial reduction only: the H^4 mechanism itself is robust, since the trace-anomaly term H^4/(960π^2) survives even at M=m (Sec. 4.3).

full rationale

One genuinely load-bearing step imports the identification M=H from the authors' own program; the rest of the derivation is substantive. The new QFT computations — exact de Sitter mode functions, the off-shell renormalized EMT, the trace renormalization, and the recovery at M=m of the known Bunch-Davies result (Eq. 106, matching [87,89]) — are genuine calculations with external anchors (point-splitting and zeta-function/dimensional-regularization results obtained by other authors). That recovery strengthens the renormalization method, but it validates precisely the on-shell case in which the VED is H-independent; it does not validate the M=H running. The H^4 inflation mechanism is likewise robust in broad outline, since Eq. (77) contains H^4 and H^2 powers with M left free and the trace-anomaly piece survives the on-shell/conformal limit; the paper says the de Sitter result does not vanish for ξ=1/6, m=0. What is scheme-dependent is the quantitative content of most testable claims: the late-time law δρ_vac=(3ν_eff/8π)(H^2-H_0^2)m_Pl^2 (Eqs. 37, 140) is obtained by substituting M=H into ρ_vac(M,H) (Eqs. 31, 77), vanishes at M=m, and the only justification offered is 'it has been suggested... see [43–46]', i.e. the authors' prior work. The inflationary coefficient α and the ξ-window shift under a different scale choice, as the paper discloses. Conditioned on the M=H convention, ν_eff is honestly computed from QFT parameters (not fitted), and the resulting RVM form has been compared with cosmological data ([49–51] and earlier), which is external, falsifiable support for the convention's output rather than for its derivation. The entropy discussion is explicitly an order-of-magnitude consistency check tied to the measured T_γ0, not a prediction. Net: partial circularity — the DE upshot reduces to a self-cited ansatz, while the core de Sitter QFT and the H^4 mechanism retain independent content. Score 5.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 0 invented entities

The central claim rests on the semiclassical framework (quantized matter, classical background), the off-shell adiabatic subtraction (Eq. 20), the identification of the renormalization scale with the Hubble rate (M = H, Sec. 2.5), an initial de Sitter state whose origin is deferred to QG/string theory, and standard QFTCS assumptions (Bunch-Davies vacuum, WKB expansion). The model parameters ξ, m, N_0 are free; ν_eff is anchored by the group's own observational fits. No new entities (particles, forces, dimensions) are introduced beyond the running-vacuum framework itself.

free parameters (5)
  • ξ (non-minimal coupling) = Not fitted; constrained to 0.161 < ξ < 0.339 by the H^4-inflation positivity condition (Eq. 129); sign of ξ−1/6 sets qui
    Free parameter of the scalar action (Eq. 1). The allowed range is derived, not measured; the late-time DE behavior (sign of ν_eff) is controlled by this free input.
  • m (scalar field mass) = 0.001 m_Pl used in numerical examples
    Free; affects the size of ν_eff and the digamma-function behavior; any GUT-scale mass is allowed in principle.
  • N_0 (scalar field multiplicity) = 1000 in figures; ≳135 needed for sub-Planckian H_I
    Free; needed to keep the inflationary scale and temperature sub-Planckian (Eq. 131); typical GUT values are assumed.
  • ν_eff (late-time running coefficient) = |ν_eff| ≲ 10^-3 from the authors' own fits [49–51]
    Formally computable from (ξ, m) via Eq. (35), but numerically unconstrained without data; the phenomenological band comes from the same group's global fits, giving the DE 'prediction' its flexibility.
  • ρ_0^vac = Λ_obs/(8πG) = Observed cosmological constant
    Input normalization at the scale H_0 (Sec. 2.6). The model only predicts the running, not the overall scale — a stated limitation of its approach to the CCP.
axioms (8)
  • domain assumption Semiclassical limit: classical spacetime with quantized matter; graviton contributions neglected
    Sec. 2: 'the main assumption, which indeed serves as a golden rule of the entire approach'. Stated, argued to fail near the Planck scale, and relied upon throughout.
  • domain assumption Adiabatic/WKB expansion is valid and can be truncated at 4th order
    Secs. 2.2–2.3: needed for the subtraction term in the ARP (Eq. 20); an asymptotic series whose truncation is mandatory but uncontrolled.
  • ad hoc to paper Off-shell ARP (Eq. 20) is the physical renormalization prescription for the VED
    Eq. (20): the M-dependent subtraction defines the theory; different prescriptions would change the finite running.
  • ad hoc to paper Renormalization scale identified with the Hubble rate (M = H)
    Sec. 2.5: 'it has been suggested to make the choice of M at the value of the Hubble rate H', citing the group's own refs. [43–46]; converts the RG scale into cosmic dynamics and generates the ρ_vac(H) law.
  • standard math Bunch-Davies vacuum as initial state
    Sec. 3.1: standard QFTCS boundary condition deep inside the horizon.
  • domain assumption An initial de Sitter state exists at the start of cosmic history
    Secs. 3 and 5: origin deferred ('might ultimately be connected to QG or stringy RVM versions'); without this state there is no H^4 inflation.
  • domain assumption Vacuum decays into radiation with P_vac = −ρ_vac during inflation
    Sec. 5.2 (Eqs. 116–118): needed for the analytic solution (120)–(122); stated to hold up to Ḣ-terms; the microscopic decay is explicitly not derived ('We shall not enter the microscopic details of the transition').
  • domain assumption Adiabatic particle production: entropy per particle constant (σ̇ = 0) with β = (ρ+p)/n
    Sec. 6.3 (Eqs. 152–153): required for the entropy-production calculation.

pith-pipeline@v1.3.0-alltime-deepseek · 58604 in / 20265 out tokens · 202341 ms · 2026-08-03T11:42:52.354260+00:00 · methodology

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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) $\rho_{\rm vac}$ 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 $\Lambda$). In the FLRW epoch, $\rho_{\rm vac}$ 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: $\delta\rho_{\rm vac}\sim {\cal O}(m_{\rm Pl} ^2 H^2)$. 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.

Figures

Figures reproduced from arXiv: 2601.05218 by \`Alex Gonz\'alez-Fuentes, Cristian Moreno-Pulido, Joan Sol\`a Peracaula.

Figure 1
Figure 1. Figure 1: Dependence of the VED, ρvac, normalized to the Planck density in the inflationary period for the unstable Sitter case as a function of m/H. We plot specifically Eq. (83). The various contributions from the terms of O(H4 ) and O(H2 ) which add up to the total ρvac (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 in… view at source ↗
Figure 2
Figure 2. Figure 2: Hubble rate HI from Eq. (132) and temperature TI of radiation immediately after inflation, Eq. (137), for different multiplicities of the fields, N0, and for ξ (assumed common to all of them) in the allowed region for inflation, as discussed in the text. It is seen that for large enough N0 the condition (134) is fulfilled all over the permitted range, and TI < mPl. mentioned TI is of the order of the maxim… view at source ↗
Figure 3
Figure 3. Figure 3: De Sitter decay scenario: energy densities for vacuum and radiation for [PITH_FULL_IMAGE:figures/full_fig_p045_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as Fig. 3 but taking [PITH_FULL_IMAGE:figures/full_fig_p046_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Energy density (ρvac) and pressure (Pvac) of the unstable de Sitter vacuum for the same cases considered in Figs. 3 and 4. Because ρvac and Pvac 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. the post-inflationary epoch. In Fig.5, we show the vacuum density and pressure of the de Sitter decay model considering the regio… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the VED for the decaying de Sitter scenario in the current universe, [PITH_FULL_IMAGE:figures/full_fig_p050_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Equation of state parameter wvac in the recent universe within the RVM (Eq. (145)) for νeff > 0 (left plot) and νeff < 0 (right plot), corresponding to effective quintessence and phantom behaviors, respectively. For very low redshift z ≪ 1 the above expression for the EoS boils down to the simpler form wvac(z) ≃ −1 + νeff Ω 0 m Ω0 vac (1 + z) 3 . (146) As expected, it stays close to the classical value w c… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of temperature and entropy of the heat-bath of radiation as the inflationary [PITH_FULL_IMAGE:figures/full_fig_p056_8.png] view at source ↗

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