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Cosmological horizon thermodynamics in Gauss-Bonnet quasi-dilaton Massive Gravity

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Pith's one-line read This paper claims that Gauss-Bonnet quasi-dilaton massive gravity satisfies the first and second laws of horizon thermodynamics and the holographic entropy bound, with the non-negativity of the Gauss-Bonnet coupling being the key consistenc

desk verdict First thermodynamic analysis of GBQD massive gravity, but the central GSL proof collapses on a dimensional inconsistency in the Gibbs equation. read the letter →

arxiv 2607.20112 v1 pith:FIZJWVSK submitted 2026-07-22 gr-qc

Sobhan Kazempour , Orlando Luongo , Sabahat , Sichun Sun , Chengye Yu This is my paper · ORCID
classification gr-qc MSC 83F0583C5780A10 PACS 04.70.Dy98.80.-k04.50.Kd
keywords cosmologicalapparenthorizonGauss-Bonnetquasi-dilatonmassivegravityWaldentropygeneralizedsecondlawholographicboundmodifiedFriedmannequationsnullenergyconditionthermodynamicstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that Gauss-Bonnet quasi-dilaton massive gravity — a modified gravity theory combining a massive graviton, a dilaton-like scalar, and a Gauss-Bonnet curvature term — is thermodynamically consistent at the cosmological apparent horizon. The authors rewrite the modified Friedmann equations in standard form and show that, in the equilibrium picture, the first law keeps the usual area-law entropy while the generalized second law holds under the null energy condition. In the non-equilibrium picture, the Wald entropy picks up a Gauss-Bonnet correction and the first law gains an entropy-production term; they claim the total entropy never decreases when the null energy condition, a positive Hawking temperature, and a nonnegative Gauss-Bonnet coupling hold. They also argue that the holographic entropy bound is preserved across all epochs once realistic physical fluid temperatures replace the idealized horizon-temperature assumption. A sympathetic reader would care because this tests whether the theory survives basic consistency checks expected of any candidate cosmology.

What carries the argument

The argument runs on the apparent-horizon radius r̃_A = 1/H, the Hawking temperature T_H = (2H²+Ḣ)/(4πH), the reformulation of the modified Friedmann equations as H² = ρ_T/3 and Ḣ = -N(ρ_T+P_T)/2, the Wald entropy formula for the Gauss-Bonnet sector, and the Gibbs equation for the total fluid inside the horizon. These objects convert the gravitational field equations into thermodynamic balance laws; the parameter ϵ = -Ḣ/H² organizes the sign conditions.

What would settle it

A direct dimensional check of Eq. (32) — where ρ_T and P_T have mass dimension 2 — settles whether the Gibbs equation is consistent. Alternatively, one can numerically integrate a viable background solution of the modified Friedmann equations and evaluate S_sum with physical units; if it dips below zero, the theorem is false.

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Extended reading notes

Core claim

The central claim is that the apparent-horizon thermodynamics of Gauss-Bonnet quasi-dilaton massive gravity is internally consistent. In the equilibrium description, all modifications are absorbed into effective energy density and pressure, so the first law T_H dS = -dE + W dV retains the Bekenstein-Hawking area law and the entropy grows monotonically when the null energy condition holds. In the non-equilibrium description, the Wald entropy becomes S = A/4G + 4πξ(σ)/G and the first law acquires an entropy-production term d_i S; the authors show the total entropy production is nonnegative provided the null energy condition, 2H²+Ḣ>0, and ξ(σ)≥0 hold. They further claim the holographic bound S_

Load-bearing premise

The load-bearing premise is that the rescaled cosmic fluid obeys the Gibbs equation with the total pressure and energy density as written; if that relation is not correctly normalized, the second-law and entropy-bound conclusions do not follow.

Editorial extensions

If this is right

  • If correct, the theory passes the first and second laws of horizon thermodynamics in both the equilibrium and non-equilibrium formulations.
  • The condition ξ(σ) ≥ 0 becomes a concrete physical constraint linking thermodynamic consistency, holography, and perturbative stability.
  • The apparent holographic-bound violation during the dust era is diagnosed as an artifact of the idealized local-equilibrium assumption, not as a failure of the theory.
  • The equilibrium area-law result suggests that the effective-fluid reformulation hides the gravitational modifications while preserving the standard thermodynamic structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: the local-equilibrium saturation at ϵ = 3/2 in the holographic bound could be turned into a quantitative test by computing S_inside with measured CMB or matter temperatures at several redshifts and comparing with S_horizon.
  • Editorial: the dimensional inconsistency flagged in the Gibbs equation implies the non-equilibrium entropy-production expression should be rederived with physical dimensions; this may tighten or relax the conditions for the generalized second law.
  • Editorial: the same thermodynamic consistency check could be applied to other scalar-coupled higher-curvature theories, where the positivity of the coupling function would play a similar role.
  • Editorial: the entropy-production term in the non-equilibrium picture may admit an effective bulk-viscosity interpretation whose magnitude could be constrained by CMB or large-scale-structure observations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper studies the thermodynamics of the cosmological apparent horizon in Gauss-Bonnet quasi-dilaton massive gravity. The authors rewrite the modified Friedmann equations in a standard form with total variables ρ_T and P_T, derive an equilibrium first law and entropy production rate, and claim that the generalized second law holds under the null energy condition and positive horizon temperature. They then formulate a non-equilibrium description with the Wald entropy S = A/(4G) + 4πξ(σ)/G, claim a generalized first law with an entropy-production term, and assert the GSL under additional conditions, including ξ(σ) ≥ 0. Finally, they check the holographic bound S_inside ≤ S_horizon and claim it is preserved once realistic fluid temperatures are used. The paper concludes that the model is thermodynamically and holographically consistent.

Significance. If the central claims were established, the paper would fill a gap in the horizon thermodynamics of this particular modified-gravity model and would give a thermodynamic consistency condition ξ(σ) ≥ 0 compatible with tensor-stability results from the authors' previous work. The use of both equilibrium and non-equilibrium pictures, the explicit Wald-entropy correction, and the attempt to connect the holographic bound with perturbation stability are reasonable and potentially useful directions. However, the key derivations are not sound as written: the Gibbs equation is dimensionally inconsistent, the displayed equilibrium GSL expression contradicts the stated claim, the non-equilibrium GSL proof relies on an unproved and essentially assumed positivity of an unspecified function F, and the holographic resolution is not quantitatively demonstrated. The claimed results are therefore not established.

major comments (4)
  1. [Sec. III.B, Eq. (32)] The Gibbs equation is dimensionally inconsistent as written. With ρ_T = 8πG ρ_phys and P_T = 8πG P_phys, ρ_T and P_T have mass dimension 2, while V = 4π/(3H^3) has dimension M^{-3}; hence d(ρ_T V) + P_T dV has dimension M^{-1}. Since S_T is dimensionless and T_H has dimension M, T_H dS_T has dimension M, so the two sides of Eq. (32) cannot be equal. The physical relation is T_H dS_T = M_Pl^2 [V dρ_T + (ρ_T + P_T) dV]. The missing M_Pl^2 factor propagates through Eqs. (33)-(36) into Eq. (38) and hence into Eq. (39), so the paper does not actually derive the equilibrium GSL inequality; the claimed positivity is an artifact of an incorrect normalization.
  2. [Sec. III.B, Eq. (39)] Independently of the previous point, Eq. (39) contradicts the claim it is supposed to support. For 0 < ε < 1 the bracket is ε + 16π ε(ε-1)/(2-ε), which is negative; e.g. at ε = 1/2 it is approximately 0.5 - 16.76 < 0. Thus S_sum < 0 in a parameter range the paper itself accepts (ε ≥ 0, ε < 2). The text states that the GSL is "robustly obeyed" in this regime, but the displayed formula shows the opposite. This is a load-bearing internal inconsistency, not a typographical issue.
  3. [Sec. IV.B, Eqs. (61)-(63)] The non-equilibrium GSL is not demonstrated. Eq. (61) is presented after an unexplained "systematic simplification" and no derivation of d(d_i S)/dt is given, so it cannot be independently checked. Eq. (63) separates a positive Einstein-Hilbert term from a function F that is never explicitly defined. The text then asserts that F is positive when "the additional cross-terms in F are positive along the viable cosmological branches"—this is precisely the point to be proved. Since F encodes all Gauss-Bonnet and quasi-dilaton contributions, the non-negativity of S_sum in Eq. (61) remains an assumption, not a consequence.
  4. [Sec. VI, Eqs. (78)-(84)] The holographic claim is not quantitatively established. The check yields Eq. (84), which the paper admits fails in a dust-dominated era (ε = 3/2) when H is small. The proposed resolution—using "realistic physical fluid temperatures"—is not carried out. In addition, Eq. (79) relies on T_H s = ρ_phys + P_phys, which is the Euler relation for a zero-chemical-potential fluid and is not valid for cold dust; for a non-relativistic dust component the entropy density is fixed by the phase-space distribution, not simply by ρ/T. Replacing T_H with a lower fluid temperature does not by itself suppress S_inside, since S = E/T can increase as T decreases at fixed E. Thus the conclusion that the holographic bound is "robustly preserved across all cosmological epochs" is unsupported.
minor comments (4)
  1. [Sec. IV, terminology] Calling this a non-equilibrium description is confusing because Eq. (43) shows that the dark-energy component is conserved; the only non-equilibrium ingredient is the Wald-entropy representation. Please clarify the terminology and its relation to the usual non-equilibrium treatments in f(R) and f(T) gravity.
  2. [Eqs. (24)-(25), (35)] The sign of the surface gravity is taken without discussing the absolute value; T_H > 0 also requires 2H^2 + RH > 0. This should be stated as an explicit assumption before using Eq. (35).
  3. [Sec. V, Eq. (75)] The claim that C_W > 0 for the self-accelerating attractor solutions is not shown; either provide the computation or phrase it as a conjecture. This is less central than the GSL issues but should be corrected.
  4. [Throughout] There are several presentation issues: inconsistent spacing in "FLRW", "H_th" written as "Hth", very long expressions in Eqs. (18) and (61) with no derivation, and a few unbalanced parentheses. These should be cleaned up in a revision.

Circularity Check

1 steps flagged · score 6.0 of 10

Non-equilibrium GSL proof reduces to an assumed positivity of the residual F; the rest is standard reformulation plus correctness gaps.

  1. self definitional [Section IV.B, Eq. (63) and the bullet list following it]
    "˙Ssum = 2π/GH [ ε(2ε²−3ε+2)/(2−ε) + F(H,X,ξ,ξ′) ] ... The function F collects the Gauss-Bonnet contributions ... Its positivity is guaranteed when the additional branch conditions listed below are satisfied. ... the additional cross-terms in F are positive along the viable cosmological branches. Therefore, ˙Ssum ≥ 0 holds manifestly for all physically admissible configurations, rigorously establishing that the generalized second law ... is satisfied."

    Eq. (63) defines ˙Ssum as a strictly positive Einstein-Hilbert term plus an uncomputed residual F. The subsequent proof of GSL consists of asserting F≥0 under 'viable cosmological branches'; those branches are not characterized by any independent condition, only by the required positivity of the Gauss-Bonnet cross-terms. Thus the central non-equilibrium second-law claim is not a consequence of NEC, T_H>0, and ξ≥0; it is an input assumption equivalent, up to the already-positive term, to the desired inequality ˙Ssum≥0. The conclusion is therefore true by construction for the 'admissible' configurations.

full rationale

Most of the paper is a standard reformulation: ρ_T and P_T are defined (Eqs. 17-18) so that the modified Friedmann equations take the GR form, and the equilibrium first law then follows algebraically; the paper explicitly frames this as an equilibrium redefinition, so this is not a hidden circularity. The self-citations to [48] supply the background equations and tensor-stability results; they are prior published work and do not constitute a uniqueness-import or ansatz-smuggling chain. The principal circular step is in the non-equilibrium second law: Eq. (63) splits ˙Ssum into a positive Einstein-Hilbert term plus a residual F, and the proof of GSL then reduces to asserting F≥0 on 'viable cosmological branches' that are not independently characterized. That makes the central non-equilibrium GSL claim true by construction rather than derived. Separately, there are serious non-circular correctness gaps: Eq. (32) omits the M_Pl^2 factor and is dimensionally inconsistent, invalidating Eqs. (33)-(39), and the holographic 'resolution' after Eq. (84) invokes unspecified realistic fluid temperatures without recomputing S_inside. These are correctness risks, not circularity. Overall circularity score 6.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the validity of the background equations from the authors' prior self-cited work, the standard horizon thermodynamics formalism, the null energy condition, positive horizon temperature, and an inconsistently normalized Gibbs equation. The paper introduces no new entities but imposes conditions (ξ≥0, positive cross-terms) that are effectively assumptions.

free parameters (2)
  • ξ(σ) sign = ≥0 (imposed)
    The proofs of the GSL and holographic bound require ξ(σ) ≥ 0; the paper imposes this condition rather than deriving it from data.
  • Physical fluid temperatures = unspecified
    To resolve the dust-era holographic bound failure, the paper invokes realistic fluid temperatures much lower than T_H without specifying their values or evolution.
assumptions (5)
  • domain assumption Background Friedmann equations (10)-(11) from Ref. [48]
    The paper uses the background equations from the authors' previous work without re-deriving them; if those equations are incorrect, the thermodynamics is baseless.
  • domain assumption Null energy condition ρ_phys + P_phys ≥ 0
    Used to guarantee ϵ ≥ 0 for the GSL proofs.
  • domain assumption Positive horizon temperature 2H^2 + \dot H > 0
    Ensures T_H > 0 and 0 ≤ ϵ < 2.
  • ad hoc to paper Standard Gibbs equation T_H dS_T = d(ρ_T V) + P_T dV with rescaled variables
    This equation is dimensionally inconsistent as written; the paper assumes it without the M_Pl^2 factors.
  • domain assumption Euler relation T_H s = ρ_phys + P_phys
    Used to compute the entropy density inside the horizon.

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Pith. "Pith review of Cosmological horizon thermodynamics in Gauss-Bonnet quasi-dilaton Massive Gravity." pith.science (2026). https://pith.science/paper/FIZJWVSK

@misc{pith2026260720112,
  author       = {Pith},
  title        = {Pith review of: Cosmological horizon thermodynamics in Gauss-Bonnet quasi-dilaton Massive Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIZJWVSK}},
  note         = {Machine review of arXiv:2607.20112}
}
abstract

We investigate the thermodynamic properties of the cosmological apparent horizon in Gauss-Bonnet quasi-dilaton massive gravity. We derive the modified Friedmann equations and reformulate them in standard form, thereby allowing us to study the first and second laws of thermodynamics for the apparent horizon. Both equilibrium and non-equilibrium states are considered. In the equilibrium description, the first law retains the conventional form with the Bekenstein-Hawking area law for the horizon entropy, and we show that the generalized second law is satisfied under the null energy condition. In the non-equilibrium description, the Wald entropy receives a correction from the Gauss-Bonnet coupling, and the first law acquires an additional term associated with using the Wald entropy representation of the Gauss-Bonnet sector. We demonstrate that the total entropy change is non-negative provided the null energy condition, the positive horizon temperature condition, and the Gauss-Bonnet positivity constraints $\xi(\sigma)\ge0$ are simultaneously satisfied. Furthermore, we investigate the holographic entropy bound $S_{\text{inside}} \le S_{\text{horizon}}$. We demonstrate that while the idealized local thermal equilibrium assumption leads to a formal saturation or apparent breakdown during dust-dominated eras, the bound is robustly preserved across all cosmological epochs when utilizing realistic physical fluid temperatures. The condition $\xi(\sigma)\ge0$ is shown to be compatible with the stability constraints derived from tensor perturbations in our previous work. Our results establish that Gauss-Bonnet quasi-dilaton massive gravity is a consistent modified gravity theory from the perspective of horizon thermodynamics and the holographic principle.

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Reference graph

Works this paper leans on

66 extracted references · 50 linked inside Pith

  1. [1]

    Aghanimet al.[Planck], Astron

    N. Aghanimet al.[Planck], Astron. Astrophys.641, A6 (2020) [erratum: Astron. Astrophys.652, C4 (2021)] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]]

    arXiv 2020

  2. [2]

    A. G. Riesset al.[Supernova Search Team], Astron. J. 116, 1009-1038 (1998) doi:10.1086/300499 [arXiv:astro- ph/9805201 [astro-ph]]

    arXiv 1998

  3. [3]

    Perlmutteret al.[Supernova Cosmology Project], Astrophys

    S. Perlmutteret al.[Supernova Cosmology Project], Astrophys. J.517, 565-586 (1999) doi:10.1086/307221 [arXiv:astro-ph/9812133 [astro-ph]]

    arXiv 1999

  4. [4]

    Weinberg, Rev

    S. Weinberg, Rev. Mod. Phys.61, 1-23 (1989) doi:10.1103/RevModPhys.61.1

    doi:10.1103/revmodphys.61.1 1989

  5. [5]

    Martin, Comptes Rendus Physique13, 566-665 (2012) doi:10.1016/j.crhy.2012.04.008 [arXiv:1205.3365 [astro- ph.CO]]

    J. Martin, Comptes Rendus Physique13, 566-665 (2012) doi:10.1016/j.crhy.2012.04.008 [arXiv:1205.3365 [astro- ph.CO]]

    arXiv 2012

  6. [6]

    P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559-606 (2003) doi:10.1103/RevModPhys.75.559 [arXiv:astro-ph/0207347 [astro-ph]]

    arXiv 2003

  7. [7]

    Verde, T

    L. Verde, T. Treu and A. G. Riess, Nature Astron.3, 891 (2019) doi:10.1038/s41550-019-0902-0 [arXiv:1907.10625 [astro-ph.CO]]

    arXiv 2019

  8. [8]

    Di Valentino, O

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk, Class. Quant. Grav.38, no.15, 153001 (2021) doi:10.1088/1361- 6382/ac086d [arXiv:2103.01183 [astro-ph.CO]]

    arXiv 2021

Show all 66 references
  1. [9]

    J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys.31, 161-170 (1973) doi:10.1007/BF01645742

    1973 doi

  2. [10]

    J. D. Bekenstein, Phys. Rev. D7, 2333-2346 (1973) doi:10.1103/PhysRevD.7.2333

    1973 doi

  3. [11]

    Jacobson, Phys

    T. Jacobson, Phys. Rev. Lett.75, 1260-1263 (1995) doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]]

    1995 arXiv

  4. [12]

    R. G. Cai and S. P. Kim, JHEP02, 050 (2005) doi:10.1088/1126-6708/2005/02/050 [arXiv:hep- th/0501055 [hep-th]]. 15

    2005

  5. [13]

    Akbar and R

    M. Akbar and R. G. Cai, Phys. Lett. B648, 243- 248 (2007) doi:10.1016/j.physletb.2007.03.005 [arXiv:gr- qc/0612089 [gr-qc]]

    2007

  6. [14]

    R. G. Cai, B. Hu and S. Koh, Phys. Lett. B 671, 181-186 (2009) doi:10.1016/j.physletb.2008.11.053 [arXiv:0806.2508 [hep-th]]

    2009 arXiv

  7. [15]

    R. G. Cai, L. M. Cao and N. Ohta, JHEP04, 082 (2010) doi:10.1007/JHEP04(2010)082 [arXiv:0911.4379 [hep-th]]

    2010 arXiv

  8. [16]

    Akbar and R

    M. Akbar and R. G. Cai, Phys. Rev. D75, 084003 (2007) doi:10.1103/PhysRevD.75.084003 [arXiv:hep-th/0609128 [hep-th]]

    2007 arXiv

  9. [17]

    Bamba and C

    K. Bamba and C. Q. Geng, JCAP06, 014 (2010) doi:10.1088/1475-7516/2010/06/014 [arXiv:1005.5234 [gr-qc]]

    2010 arXiv

  10. [18]

    R. M. Wald, Phys. Rev. D48, no.8, R3427-R3431 (1993) doi:10.1103/PhysRevD.48.R3427 [arXiv:gr-qc/9307038 [gr-qc]]

    1993 arXiv

  11. [19]

    Iyer and R

    V. Iyer and R. M. Wald, Phys. Rev. D50, 846-864 (1994) doi:10.1103/PhysRevD.50.846 [arXiv:gr-qc/9403028 [gr- qc]]

    1994 arXiv

  12. [20]

    Bamba and C

    K. Bamba and C. Q. Geng, JCAP11, 008 (2011) doi:10.1088/1475-7516/2011/11/008 [arXiv:1109.1694 [gr-qc]]

    2011 arXiv

  13. [21]

    Bamba, C

    K. Bamba, C. Q. Geng and S. Tsujikawa, Int. J. Mod. Phys. D20, 1363-1371 (2011) doi:10.1142/S0218271811019542 [arXiv:1101.3628 [gr-qc]]

    2011 arXiv

  14. [22]

    Fischler and L

    W. Fischler and L. Susskind, [arXiv:hep-th/9806039 [hep-th]]

    arXiv

  15. [23]

    Bak and S

    D. Bak and S. J. Rey, Class. Quant. Grav.17, L83 (2000) doi:10.1088/0264-9381/17/15/101 [arXiv:hep- th/9902173 [hep-th]]

    2000

  16. [24]

    Banihashemi and T

    B. Banihashemi and T. Jacobson, JHEP07, 042 (2022) doi:10.1007/JHEP07(2022)042 [arXiv:2204.05324 [hep- th]]

    2022 arXiv

  17. [25]

    Anninos, D

    D. Anninos, D. A. Galante and C. Maneerat, Class. Quant. Grav.41, no.16, 165009 (2024) doi:10.1088/1361- 6382/ad5824 [arXiv:2402.04305 [hep-th]]

    2024 arXiv

  18. [26]

    G. G. Luciano and J. Gin´ e, Phys. Dark Univ.41, 101256 (2023) doi:10.1016/j.dark.2023.101256 [arXiv:2210.09755 [gr-qc]]

    2023

  19. [27]

    Belfiglio, O

    A. Belfiglio, O. Luongo and S. Mancini, Phys. Rept. 1146, 1-47 (2025) doi:10.1016/j.physrep.2025.09.001 [arXiv:2506.03841 [gr-qc]]

    2025

  20. [28]

    R. B. Mann, Int. J. Mod. Phys. D34, no.09, 2542001 (2025) doi:10.1142/S0218271825420015 [arXiv:2508.01830 [gr-qc]]

    2025 arXiv

  21. [29]

    Cespedes, S

    S. Cespedes, S. de Alwis and F. Quevedo, [arXiv:2509.07077 [hep-th]]

  22. [30]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and T. Paul, Universe 10, no.9, 352 (2024) doi:10.3390/universe10090352 [arXiv:2409.01090 [gr-qc]]

    2024 arXiv

  23. [31]

    de Rham and G

    C. de Rham and G. Gabadadze, Phys. Rev. D 82, 044020 (2010) doi:10.1103/PhysRevD.82.044020 [arXiv:1007.0443 [hep-th]]

    2010 arXiv

  24. [32]

    de Rham, G

    C. de Rham, G. Gabadadze and A. J. Tol- ley, Phys. Rev. Lett.106, 231101 (2011) doi:10.1103/PhysRevLett.106.231101 [arXiv:1011.1232 [hep-th]]

    2011 arXiv

  25. [33]

    D. G. Boulware and S. Deser, Phys. Rev. D6, 3368-3382 (1972) doi:10.1103/PhysRevD.6.3368

    1972 doi

  26. [34]

    De Felice, A

    A. De Felice, A. E. Gumrukcuoglu and S. Muko- hyama, Phys. Rev. Lett.109, 171101 (2012) doi:10.1103/PhysRevLett.109.171101 [arXiv:1206.2080 [hep-th]]

    2012 arXiv

  27. [35]

    De Felice, A

    A. De Felice, A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, C. Lin and S. Mukohyama, Class. Quant. Grav.30, 184004 (2013) doi:10.1088/0264-9381/30/18/184004 [arXiv:1304.0484 [hep-th]]

    2013 arXiv

  28. [36]

    D’Amico, G

    G. D’Amico, G. Gabadadze, L. Hui and D. Pirt- skhalava, Phys. Rev. D87, 064037 (2013) doi:10.1103/PhysRevD.87.064037 [arXiv:1206.4253 [hep-th]]

    2013 arXiv

  29. [37]

    Gabadadze, R

    G. Gabadadze, R. Kimura and D. Pirt- skhalava, Phys. Rev. D90, no.2, 024029 (2014) doi:10.1103/PhysRevD.90.024029 [arXiv:1401.5403 [hep-th]]

    2014 arXiv

  30. [38]

    A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Hinterbichler, C. Lin, S. Mukohyama and M. Trodden, Phys. Rev. D88, no.2, 024023 (2013) doi:10.1103/PhysRevD.88.024023 [arXiv:1304.0449 [hep-th]]

    2013 arXiv

  31. [39]

    Zwiebach, Phys

    B. Zwiebach, Phys. Lett. B156, 315-317 (1985) doi:10.1016/0370-2693(85)91616-8

    1985 doi

  32. [40]

    D. J. Gross and J. H. Sloan, Nucl. Phys. B291, 41-89 (1987) doi:10.1016/0550-3213(87)90465-2

    1987 doi

  33. [41]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D71, 123509 (2005) doi:10.1103/PhysRevD.71.123509 [arXiv:hep-th/0504052 [hep-th]]

    2005 arXiv

  34. [42]

    Cognola, E

    G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D73, 084007 (2006) doi:10.1103/PhysRevD.73.084007 [arXiv:hep-th/0601008 [hep-th]]

    2006 arXiv

  35. [43]

    Z. K. Guo and D. J. Schwarz, Phys. Rev. D 80, 063523 (2009) doi:10.1103/PhysRevD.80.063523 [arXiv:0907.0427 [hep-th]]

    2009 arXiv

  36. [44]

    S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fronimos and E. C. Lymperiadou, Symme- try15, no.9, 1701 (2023) doi:10.3390/sym15091701 [arXiv:2307.16308 [gr-qc]]

    2023 arXiv

  37. [45]

    Dodelson, C

    M. Dodelson, C. Iossa, R. Karlsson and A. Zhiboedov, JHEP01, 036 (2024) doi:10.1007/JHEP01(2024)036 [arXiv:2304.12339 [hep-th]]

    2024 arXiv

  38. [46]

    F. L. Juli´ e, L. Pompili and A. Buonanno, Phys. Rev. D111, no.2, 024016 (2025) doi:10.1103/PhysRevD.111.024016 [arXiv:2406.13654 [gr-qc]]

    2025 arXiv

  39. [47]

    A. K. W. Chung and N. Yunes, Phys. Rev. D110, no.6, 064019 (2024) doi:10.1103/PhysRevD.110.064019 [arXiv:2406.11986 [gr-qc]]

    2024 arXiv

  40. [48]

    A. R. Akbarieh, S. Kazempour and L. Shao, Phys. Rev. D103, 123518 (2021) doi:10.1103/PhysRevD.103.123518 [arXiv:2105.03744 [gr-qc]]

    2021 arXiv

  41. [49]

    Karami and A

    K. Karami and A. Abdolmaleki, JCAP04, 007 (2012) doi:10.1088/1475-7516/2012/04/007 [arXiv:1201.2511 [gr-qc]]

    2012 arXiv

  42. [50]

    Q. J. Cao, Y. X. Chen and K. N. Shao, JCAP 05, 030 (2010) doi:10.1088/1475-7516/2010/05/030 [arXiv:1001.2597 [hep-th]]

    2010 arXiv

  43. [51]

    Y. F. Cai, F. Duplessis and E. N. Sari- dakis, Phys. Rev. D90, no.6, 064051 (2014) doi:10.1103/PhysRevD.90.064051 [arXiv:1307.7150 [hep-th]]

    2014 arXiv

  44. [52]

    Y. P. Hu and H. Zhang, Phys. Rev. D92, no.2, 024006 (2015) doi:10.1103/PhysRevD.92.024006 [arXiv:1502.00069 [hep-th]]

    2015 arXiv

  45. [53]

    Blake and D

    M. Blake and D. Tong, Phys. Rev. D88, no.10, 106004 (2013) doi:10.1103/PhysRevD.88.106004 16 [arXiv:1308.4970 [hep-th]]

    2013 arXiv

  46. [54]

    R. A. Davison, Phys. Rev. D88, 086003 (2013) doi:10.1103/PhysRevD.88.086003 [arXiv:1306.5792 [hep- th]]

    2013 arXiv

  47. [55]

    M. A. Scheel, S. L. Shapiro and S. A. Teukol- sky, Phys. Rev. D51, 4208-4235 (1995) doi:10.1103/PhysRevD.51.4208 [arXiv:gr-qc/9411025 [gr-qc]]

    1995 arXiv

  48. [56]

    Christodoulakis, N

    T. Christodoulakis, N. Dimakis and P. A. Terzis, J. Phys. A47, 095202 (2014) doi:10.1088/1751-8113/47/9/095202 [arXiv:1304.4359 [gr-qc]]

    2014 arXiv

  49. [57]

    Arkani-Hamed, H

    N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys.305, 96-118 (2003) doi:10.1016/S0003- 4916(03)00068-X [arXiv:hep-th/0210184 [hep-th]]

    2003 arXiv

  50. [58]

    Capozziello, V

    S. Capozziello, V. F. Cardone and A. Troisi, Phys. Rev. D71, 043503 (2005) doi:10.1103/PhysRevD.71.043503 [arXiv:astro-ph/0501426 [astro-ph]]

    2005 arXiv

  51. [59]

    C. W. Misner and D. H. Sharp, Phys. Rev.136, B571- B576 (1964) doi:10.1103/PhysRev.136.B571

    1964 doi

  52. [60]

    S. A. Hayward, Class. Quant. Grav.15, 3147-3162 (1998) doi:10.1088/0264-9381/15/10/017 [arXiv:gr-qc/9710089 [gr-qc]]

    1998 arXiv

  53. [61]

    Padmanabhan, Mod

    T. Padmanabhan, Mod. Phys. Lett. A17, 1147- 1158 (2002) doi:10.1142/S0217732302007260 [arXiv:hep- th/0205278 [hep-th]]

    2002

  54. [62]

    S. F. Wu, B. Wang, G. H. Yang and P. M. Zhang, Class. Quant. Grav.25, 235018 (2008) doi:10.1088/0264- 9381/25/23/235018 [arXiv:0801.2688 [hep-th]]

    2008 arXiv

  55. [63]

    S. A. Tsilioukas, N. Petropoulos and E. N. Sari- dakis, Phys. Rev. D111, no.10, 103514 (2025) doi:10.1103/PhysRevD.111.103514 [arXiv:2412.21146 [gr-qc]]

    2025 arXiv

  56. [64]

    Bousso, JHEP07, 004 (1999) doi:10.1088/1126- 6708/1999/07/004 [arXiv:hep-th/9905177 [hep-th]]

    R. Bousso, JHEP07, 004 (1999) doi:10.1088/1126- 6708/1999/07/004 [arXiv:hep-th/9905177 [hep-th]]

    1999 arXiv

  57. [65]

    Y. Gong, B. Wang and A. Wang, JCAP01, 024 (2007) doi:10.1088/1475-7516/2007/01/024 [arXiv:gr- qc/0610151 [gr-qc]]

    2007

  58. [66]

    Bamba, C

    K. Bamba, C. Q. Geng and S. Tsujikawa, Phys. Lett. B 688, 101-109 (2010) doi:10.1016/j.physletb.2010.03.070 [arXiv:0909.2159 [gr-qc]]

    2010 arXiv

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