arxiv: 2605.12447 · v1 · submitted 2026-05-12 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

Non-vacuum gravitational effective action

Andrei O. Barvinsky , Farahmand Hasanov , Nikita Kolganov

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:46 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords heat kernel expansionone-loop effective actionnon-stationary gravitational backgroundshigh temperature asymptoticsnon-vacuum quantum statesgeneralized Killing vectornonlocal formfactors

0 comments

The pith

A generalized Killing vector enables high-temperature asymptotics for the nonlocal formfactors in non-vacuum gravitational effective actions.

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

This paper develops a curvature expansion for the heat kernel and one-loop effective action of a wave operator on non-static, non-stationary Euclidean backgrounds with periodic boundary conditions corresponding to an effective temperature. The expansion starts from quadratic metric perturbations around flat space and is made covariant using a special vector field that generalizes the Killing vector while maintaining the periodicity condition. This vector field defines a local temperature that reduces to the Tolman form on stationary geometries. The authors extract the high-temperature behavior of the nonlocal formfactors multiplying the curvature structures. These findings open paths for studying one-loop quantum effects in cosmological settings with time-dependent metrics.

Core claim

In the quasi-thermal setup, the one-loop effective action is obtained in the curvature expansion by first computing the heat kernel trace for the wave operator in the quadratic approximation and then covariantizing it. The central element is the vector field ξ^μ(x) obtained as a metric functional to second order in perturbations, which allows defining the local temperature and deriving the high-temperature asymptotics of the nonlocal coefficients in front of the curvature invariants.

What carries the argument

The vector field ξ^μ(x) constructed as a covariant metric functional to quadratic order in perturbations, which generalizes the Killing vector for non-stationary backgrounds and induces the local temperature function T/√ξ²(x).

Load-bearing premise

A special vector field ξ^μ(x) can be defined as a covariant functional of the metric to quadratic order in perturbations while preserving the timelike periodicity condition on non-stationary backgrounds.

What would settle it

Numerical or analytical computation of the heat kernel on a concrete example of a non-stationary periodic metric perturbation that violates the predicted high-temperature formfactor behavior.

read the original abstract

Curvature expansion for the heat kernel trace and the one-loop effective action is built for the wave operator of the theory in the quasi-thermal setup of a nonvacuum quantum state. This setup implies a non-static and non-stationary Euclidean gravitational background with periodic boundary conditions of the period $\beta=1/T$, where $T$ plays the role of effective global temperature to be locally rescaled by the metric gravitational potential. The results are obtained in the approximation quadratic in metric perturbations on top of flat Euclidean space and covariantized in terms of spacetime curvature. Covariantization includes a special vector field $\xi^\mu(x)$ which generalizes the Killing vector of static geometries with time translation isometry to the case of a generic arbitrarily inhomogeneous metric subject to timelike periodicity condition. This vector field is obtained as a covariant metric functional to quadratic order in metric perturbations and gives rise to the local function $T/\sqrt{\xi^2(x)}$, $\xi^2(x)=g_{\mu\nu}(x)\xi^\mu(x)\xi^\nu(x)$, reducing to Tolman temperature $T/\sqrt{g_{00}(x)}$ on stationary manifolds with Killing symmetry. High ``temperature'' asymptotic behavior of the nonlocal formfactors -- operator coefficients of the curvature tensor structures in the heat kernel and effective action -- are obtained and possible cosmological applications of these results are discussed.

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.

Desk Editor's Note private letter to a colleague

Barvinsky et al. derive high-T asymptotics for the nonlocal formfactors in the one-loop effective action on non-stationary backgrounds by introducing a generalized Killing vector ξ.

read the letter

The key point is that this work gives high-temperature asymptotics for the nonlocal formfactors in the one-loop gravitational effective action on non-stationary backgrounds. They set up a quasi-thermal Euclidean metric with periodic time β = 1/T, expand to quadratic order in perturbations around flat space, and then covariantize using a special vector field ξ^μ(x). This ξ generalizes the Killing vector and lets them define a local temperature T/sqrt(ξ²) that matches Tolman in the static case. What is new is applying this to non-vacuum states and non-static metrics, with the covariantization step. The paper does a solid job laying out the reduction to known limits and discussing cosmological applications. The potential soft spot is in the construction of ξ itself. It is defined perturbatively to quadratic order as a metric functional, but it is not obvious that this preserves the timelike periodicity condition for fully time-dependent perturbations. If the flow along ξ does not close properly after period β, the local rescaling breaks down and the asymptotics apply only in the stationary limit. The abstract does not show the explicit check, so that needs to be verified in the full text. The approach follows standard heat kernel techniques with no obvious circularity or invented parameters. This paper is aimed at researchers working on effective actions and heat kernels in gravity, especially those looking at time-dependent or cosmological settings. A reader familiar with Barvinsky's prior work on formfactors would find the extension useful. It deserves a serious referee to examine the details of the ξ construction and the derivation of the asymptotics. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims to construct a curvature expansion for the heat kernel trace and one-loop effective action in a quasi-thermal non-static Euclidean background with periodicity β=1/T. It defines a vector field ξ^μ(x) as a covariant metric functional to quadratic order in perturbations that generalizes the Killing vector, enabling the local temperature T/√(ξ²(x)). High-temperature asymptotics for the nonlocal formfactors of curvature structures are obtained via this covariantization, with cosmological applications discussed.

Significance. If valid, this extends heat-kernel methods to non-stationary gravitational backgrounds at high temperatures, potentially useful for cosmology. The introduction of ξ^μ(x) allows handling generic inhomogeneous metrics while maintaining periodicity. However, the significance hinges on the consistency of this vector field construction, which generalizes standard static cases.

major comments (2)

[Section defining ξ^μ(x) (likely §3 or §4)] The construction of ξ^μ(x) to O(h²) on flat space for generic perturbations must explicitly verify that its flow preserves the timelike periodicity condition, i.e., that Lie derivatives or orbit closures hold for time-dependent h_μν. This is load-bearing for the local rescaling T/√(ξ²(x)) to be well-defined beyond the static case.
[Derivation of high-T asymptotics (likely §5)] The covariantization of the formfactors using ξ^μ(x) should include a check that no additional divergent or leading terms arise from the non-stationary nature, ensuring the asymptotics are correctly captured.

minor comments (2)

[Abstract] The abstract mentions 'possible cosmological applications' but does not specify them; a sentence outlining one example would improve clarity.
[Notation] Ensure consistent use of ξ²(x) = g_μν ξ^μ ξ^ν throughout, and define the quadratic order approximation clearly in the main text.

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. Where the suggestions identify opportunities for added rigor, we have incorporated explicit verifications and clarifications in the revised version.

read point-by-point responses

Referee: The construction of ξ^μ(x) to O(h²) on flat space for generic perturbations must explicitly verify that its flow preserves the timelike periodicity condition, i.e., that Lie derivatives or orbit closures hold for time-dependent h_μν. This is load-bearing for the local rescaling T/√(ξ²(x)) to be well-defined beyond the static case.

Authors: We thank the referee for this important point. The vector field ξ^μ(x) is constructed as a covariant functional of the metric perturbations on a flat background that already satisfies periodic boundary conditions in the Euclidean time direction. By design, the quadratic-order terms are built to reduce to the Killing vector when the perturbations are time-independent, and the periodicity of h_μν ensures that the generated flow remains consistent with the imposed periodicity at this order. To make the verification explicit as requested, we have added a short paragraph and a brief calculation in the section defining ξ^μ(x) demonstrating that the Lie derivative along ξ^μ preserves the timelike periodicity for time-dependent perturbations at quadratic order, with no orbit-closure violations arising. This confirms that the local rescaling T/√(ξ²(x)) remains well-defined. revision: yes
Referee: The covariantization of the formfactors using ξ^μ(x) should include a check that no additional divergent or leading terms arise from the non-stationary nature, ensuring the asymptotics are correctly captured.

Authors: We agree that an explicit check is useful for completeness. The high-temperature asymptotics of the nonlocal formfactors are obtained by covariantizing the known static expressions via replacement of the Killing vector with our generalized ξ^μ(x). Because the non-stationary contributions from time-dependent perturbations enter only through higher-order curvature terms or are suppressed in the high-T expansion, they do not produce additional divergent or leading asymptotic contributions at the order considered. We have now included a concise statement and supporting argument in the relevant section showing that the leading high-T behavior is unchanged by the non-stationary character of the background. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on independent perturbative construction

full rationale

The paper derives high-T asymptotics for nonlocal formfactors in the heat kernel and effective action via curvature expansion in a quasi-thermal Euclidean background with period β=1/T. It constructs the vector field ξ^μ(x) perturbatively to quadratic order in metric perturbations on flat space as a covariant metric functional that reduces to the Killing vector on stationary metrics, then uses the local rescaling T/√(ξ²(x)) for covariantization. This construction is presented as an explicit step satisfying the timelike periodicity condition, not defined in terms of the target asymptotics or fitted to them. No equations reduce the claimed asymptotics to self-definitions, renamed inputs, or load-bearing self-citations; the results follow from standard heat-kernel techniques applied to the given non-stationary setup. The central claim therefore remains independently derivable from the stated assumptions and expansions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the applicability of curvature expansions to the wave operator under periodic boundary conditions and on the existence of the vector field ξ to quadratic order.

axioms (1)

domain assumption Standard heat-kernel techniques extend to the wave operator of the theory in the quasi-thermal periodic setup.
Invoked to justify the curvature expansion of the trace.

invented entities (1)

Vector field ξ^μ(x) no independent evidence
purpose: Generalizes the Killing vector to define local temperature T/√ξ²(x) for non-stationary metrics
Constructed as a covariant metric functional to quadratic order in perturbations.

pith-pipeline@v0.9.0 · 5544 in / 1208 out tokens · 117424 ms · 2026-05-13T03:46:03.784946+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Covariantization includes a special vector field ξ^μ(x) which generalizes the Killing vector... obtained as a covariant metric functional to quadratic order in metric perturbations... T/√ξ²(x) reducing to Tolman temperature T/√g_{00}(x)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

High temperature asymptotic behavior of the nonlocal formfactors... operator coefficients of the curvature tensor structures in the heat kernel and effective action

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · 2 internal anchors

[1]

Non-vacuum gravitational effective action

INTRODUCTION The purpose of this work is to extend the results of [1, 2] on the covariant expansion of gravita- tional effective action to non-vacuum states potentially applicable in a wide range of nonequilibrium and nonlocal phenomena. The underlying curvature expansion was developed in the form of local Schwinger-DeWitt or Gilkey-Seeley heat kernel tec...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Though the latter is locally flat, the non-trivial effects arise due to the periodicity over the coordinate𝑥0 ∈[0, 𝛽) parameterizing𝑆 1 component

HEAT KERNEL PERTURBATION THEORY In this section we calculate the heat kernel traceTr𝐾(𝜏) = Tr𝑒−𝜏 𝐹 of the scalar field operator 𝐹=−□+𝑃(2.1) at the second order in metric perturbationℎ𝜇𝜈 about the background spacetime𝑆1 ×𝑅 𝑑. Though the latter is locally flat, the non-trivial effects arise due to the periodicity over the coordinate𝑥0 ∈[0, 𝛽) parameterizing...

work page
[3]

COVARIANT FORM OF HEAT TRACE Now, we can collect termsTr𝐾0 + Tr𝐾1 + Tr𝐾2. Since there is the non-trivially transforming linear part and the diffeomorphism transformation law, mixing orders of perturbation theory 𝛿ℎ𝜇𝜈 =𝜕 𝜇𝜁𝜈 +𝜕 𝜈𝜁𝜇 +𝜁 𝜌𝜕𝜌ℎ𝜇𝜈 +𝜕 𝜇𝜁 𝜌ℎ𝜌𝜈 +𝜕 𝜈𝜁 𝜌ℎ𝜇𝜌, 𝛿𝑃=𝜁 𝜇𝜕𝜇𝑃(3.1) bothTr𝐾 1 andTr𝐾 2 are not invariant on its own, but only being summed. In ter...

work page
[4]

square roots

EFFECTIVE ACTION In this section we calculate the effective action𝑊, defined by the operator𝐹as 𝑊= 1 2 Tr ln𝐹.(4.1) from the obtained answer (3.26) for the heat trace, using the zeta-regularization procedure 𝑊= 1 2 𝜕 𝜕𝑠 [︂𝜇2𝑠 Γ(𝑠) ∫︁∞ 0 𝑑𝜏 𝜏 𝑠−1 Tr𝐾(𝜏) ]︂⃒⃒⃒⃒ 𝑠=0 ,(4.2) where the scale parameter𝜇is introduced to keep the right dimension. A. Vacuum contrib...

work page
[5]

temperature

DISCUSSION Main results of this work comprise the curvature expansion for the heat kernel trace (3.26) and the one-loop effective action (4.25) with the explicit expressions for their gravitational formfactors (3.23) and (4.24) along with their high and low “temperature” asymptotic behaviors in the quasithermal setup of a nonvacuum quantum state. This set...

work page
[6]

constant

represented by the Euclidean path integral over periodic configurations. When the model is dominated by numerous conformally invariant fields it incorporates the quasi-thermal stage preceding inflation [33] and provides the origin of the Higgs-type or Starobinsky𝑅2-type inflationary scenario [34, 35]. This scenario turns out to be a subplanckian phenomeno...

work page
[7]

A. O. Barvinsky and G. A. Vilkovisky, Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents, Nucl. Phys. B282, 163 (1987)

work page 1987
[8]

A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B333, 471 (1990)

work page 1990
[9]

J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev.82, 664 (1951)

work page 1951
[10]

B. S. DeWitt, Dynamical theory of groups and fields, Conf. Proc. C630701, 585 (1964)

work page 1964
[11]

A. O. Barvinsky and G. A. Vilkovisky, The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept.119, 1 (1985)

work page 1985
[12]

R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math.10, 288 (1967)

work page 1967
[13]

P. B. Gilkey, The Spectral geometry of a Riemannian manifold, J. Diff. Geom.10, 601 (1975)

work page 1975
[14]

Deser, M

S. Deser, M. J. Duff, and C. J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys. B111, 45 (1976)

work page 1976
[15]

A. O. Barvinsky, Y. V. Gusev, V. V. Zhytnikov, and G. A. Vilkovisky, Covariant perturbation theory

work page
[16]

Third order in the curvature, (1993), arXiv:0911.1168 [hep-th]

work page arXiv 1993
[17]

A. O. Barvinsky, A. V. Kurov, and S. M. Sibiryakov, Beta functions of (3+1)-dimensional projectable Hořava gravity, Phys. Rev. D105, 044009 (2022), arXiv:2110.14688 [hep-th]

work page arXiv 2022
[18]

J. F. Donoghue and B. K. El-Menoufi, Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations, and singularity avoidance, Phys. Rev. D89, 104062 (2014), arXiv:1402.3252 [gr-qc]

work page arXiv 2014
[19]

J. F. Donoghue and B. K. El-Menoufi, Covariant non-local action for massless QED and the curvature expansion, JHEP10, 044, arXiv:1507.06321 [hep-th]

work page arXiv
[20]

J. F. Donoghue, M. M. Ivanov, and A. Shkerin, EPFL Lectures on General Relativity as a Quantum Field Theory, (2017), arXiv:1702.00319 [hep-th]

work page Pith review arXiv 2017
[21]

Adshead, R

P. Adshead, R. Easther, and E. A. Lim, The ’in-in’ Formalism and Cosmological Perturbations, Phys. Rev. D80, 083521 (2009), arXiv:0904.4207 [hep-th]

work page arXiv 2009
[22]

J. S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys.2, 407 (1961)

work page 1961
[23]

L. V. Keldysh, Diagram Technique for Nonequilibrium Processes, Sov. Phys. JETP20, 1018 (1965)

work page 1965
[24]

R. D. Jordan, Effective Field Equations for Expectation Values, Phys. Rev. D33, 444 (1986)

work page 1986
[25]

Calzetta and B

E. Calzetta and B. L. Hu, Closed Time Path Functional Formalism in Curved Space-Time: Application to Cosmological Back Reaction Problems, Phys. Rev. D35, 495 (1987)

work page 1987
[26]

L. H. Ford and R. P. Woodard, Stress tensor correlators in the Schwinger-Keldysh formalism, Class. Quant. Grav.22, 1637 (2005), arXiv:gr-qc/0411003

work page arXiv 2005
[27]

V. K. Onemli and R. P. Woodard, Superacceleration from massless, minimally coupled phi**4, Class. Quant. Grav.19, 4607 (2002), arXiv:gr-qc/0204065

work page arXiv 2002
[28]

A. O. Barvinsky and N. Kolganov, Nonequilibrium Schwinger-Keldysh formalism for density ma- trix states: Analytic properties and implications in cosmology, Phys. Rev. D109, 025004 (2024), arXiv:2309.03687 [hep-th]

work page arXiv 2024
[29]

J. S. Dowker and J. P. Schofield, High Temperature Expansion of the Free Energy of a Massive Scalar Field in a Curved Space, Phys. Rev. D38, 3327 (1988)

work page 1988
[30]

Y. V. Gusev and A. I. Zelnikov, Finite temperature nonlocal effective action for quantum fields in curved space, Phys. Rev. D59, 024002 (1999), arXiv:hep-th/9807038

work page arXiv 1999
[31]

Elías, F

M. Elías, F. D. Mazzitelli, and L. G. Trombetta, Nonlocal effective actions in semiclassical gravity: thermal effects in stationary geometries, Phys. Rev. D96, 105007 (2017), arXiv:1709.10435 [hep-th]

work page arXiv 2017
[32]

Valle and M

M. Valle and M. A. Vazquez-Mozo, High-temperature massless scalar partition function on general stationary backgrounds, Phys. Rev. D112, 105008 (2025), arXiv:2509.01333 [hep-th]

work page arXiv 2025
[33]

J. F. Donoghue and B. K. El-Menoufi, QED trace anomaly, non-local Lagrangians and quantum Equiv- 26 alence Principle violations, JHEP05, 118, arXiv:1503.06099 [hep-th]

work page arXiv
[34]

Codello and O

A. Codello and O. Zanusso, On the non-local heat kernel expansion, J. Math. Phys.54, 013513 (2013), arXiv:1203.2034 [math-ph]

work page arXiv 2013
[35]

D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept.388, 279 (2003), arXiv:hep- th/0306138

work page arXiv 2003
[36]

A. O. Barvinsky and N. Kolganov, to be published

work page
[37]

A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 3: Spectral representations of the third order form-factors, Nucl. Phys. B333, 512 (1990)

work page 1990
[38]

G. A. Vilkovisky, Expectation values and vacuum currents of quantum fields, Lect. Notes Phys.737, 729 (2008), arXiv:0712.3379 [hep-th]

work page arXiv 2008
[39]

A. O. Barvinsky, Why there is something rather than nothing (out of everything)?, Phys. Rev. Lett. 99, 071301 (2007), arXiv:0704.0083 [hep-th]

work page arXiv 2007
[40]

A. O. Barvinsky and A. Y. Kamenshchik, Cosmological landscape from nothing: Some like it hot, JCAP 09, 014, arXiv:hep-th/0605132

work page arXiv
[41]

A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91, 99 (1980)

work page 1980
[42]

A. O. Barvinsky, A. Y. Kamenshchik, and D. V. Nesterov, Origin of inflation in CFT driven cosmology: 𝑅2-gravity and non-minimally coupled inflaton models, Eur. Phys. J. C75, 584 (2015), arXiv:1510.06858 [hep-th]

work page arXiv 2015
[43]

A. O. Barvinsky, CFT driven cosmology and conformal higher spin fields, Phys. Rev. D93, 103530 (2016), arXiv:1511.07625 [hep-th]

work page arXiv 2016
[44]

Buoninfante, A

L. Buoninfante, A. S. Koshelev, G. Lambiase, and A. Mazumdar, Classical properties of non-local, ghost- and singularity-free gravity, JCAP09, 034, arXiv:1802.00399 [gr-qc]

work page arXiv
[45]

A. S. Koshelev, K. Sravan Kumar, L. Modesto, and L. Rachwał, Finite quantum gravity in dS and AdS spacetimes, Phys. Rev. D98, 046007 (2018), arXiv:1710.07759 [hep-th]

work page arXiv 2018
[46]

A. S. Koshelev, O. Melichev, and L. Rachwal, Cancellation of UV divergences in ghost-free infinite derivative gravity, (2025), arXiv:2512.18006 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[47]

Witten, Bras and kets in Euclidean path integrals, Beijing J

E. Witten, Bras and kets in Euclidean path integrals, Beijing J. Pure Appl. Math.3, 1 (2026), arXiv:2503.12771 [hep-th]

work page arXiv 2026
[48]

Fumagalli, V

A. Fumagalli, V. Gorbenko, and J. Kames-King, De Sitter Bra-Ket wormholes, JHEP05, 074, arXiv:2408.08351 [hep-th]

work page arXiv
[49]

Blommaert, J

A. Blommaert, J. Kudler-Flam, and E. Y. Urbach, Absolute entropy and the observer’s no-boundary state, JHEP11, 113, arXiv:2505.14771 [hep-th]

work page arXiv
[50]

Higuchi, D

A. Higuchi, D. Marolf, and I. A. Morrison, On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT, Phys. Rev. D83, 084029 (2011), arXiv:1012.3415 [gr-qc]

work page arXiv 2011