Pith. sign in

REVIEW 4 major objections 4 minor 55 references

Thermodynamics of FLRW universe in Quadratic Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In D-dimensional FLRW quadratic gravity, the apparent horizon acquires a thermodynamic equation of state whose critical points are fixed by one quadratic-coupling parameter, sigma, altering the stability of the early universe compared…

desk verdict Central Eq. (27) is algebraically inconsistent with the paper's own criticality conditions, so the phase-transition analysis collapses; the D-dimensional framework is otherwise competent. read the letter →

arxiv 2507.12055 v1 pith:Q2QIBZO3 submitted 2025-07-16 gr-qc

Navid Safarzadeh Ilkhchi , Amin Rezaei Akbarieh , Yaghoub Heydarzade This is my paper
classification gr-qc MSC 83F0583D05 PACS 04.50.Kd98.80.-k
keywords quadraticgravityFLRWuniverseapparenthorizonMisner-SharpenergythermodynamicphasetransitioncriticaltemperatureWaldentropyhigher-dimensionalcosmology
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 tries to show that the apparent horizon of a D-dimensional FLRW universe in quadratic gravity is a genuine thermodynamic system with pressure, temperature, and entropy, and that this system can undergo phase transitions at critical points set by the quadratic couplings. It derives an equation of state for the horizon and obtains closed-form expressions for the critical radius and critical temperature, finding that all quadratic corrections enter through a single combination $\sigma$. The paper then computes Wald entropy, specific heat, enthalpy, and Gibbs free energy, arguing that the signs of these quantities depend on the quadratic parameters and the horizon radius, so the early universe can be unstable and later become stable. If correct, this would give observable thermodynamic signatures of higher-curvature corrections that general relativity does not possess.

What carries the argument

The load-bearing object is the generalized Misner-Sharp energy $E_{\mathrm{eff}}$, defined by integrating the unified first law $dE = A\Psi_a dx^a + W\,dV$ after imposing the integrability condition $\partial_r A(t,r) = \partial_t B(t,r)$. The apparent-horizon radius $R_A = 1/\sqrt{H^2 + k/a^2}$ sets the geometric scale, the Hawking temperature $T = |\kappa_g|/(2\pi)$ comes from the surface gravity, and the thermodynamic pressure is identified with the work density $W = (\rho - p)/2$. All quadratic corrections are packed into the single parameter $\sigma$, which controls the critical radius, the critical temperature, and the signs of the entropy, specific heat, enthalpy, and Gibbs free energy. Criticality is imposed through the standard conditions $(\partial P/\partial R_A)_T = 0$ and $(\partial^2 P/\partial R_A^2)_T = 0$, which produce the closed-form critical point.

What would settle it

Compute $\partial_r A(t,r) - \partial_t B(t,r)$ explicitly using the energy-momentum components in Eq. (5); if this quantity is nonzero for generic $D$, $\alpha$, $\beta$, $\gamma$, and $a(t)$, then the unified first law is not integrable and the effective Misner-Sharp energy, the thermodynamic pressure, and the critical points (26)-(27) are not defined for those backgrounds.

Watch

Extended reading notes

Core claim

The central claim is that the thermodynamics of the apparent horizon in D-dimensional FLRW quadratic gravity is fully described by the effective Misner-Sharp energy obtained from the unified first law, and that the resulting equation of state $P(R_A,T)$ has critical points at the radius $R_c^{(\pm)}$ given by Eq. (27) and critical temperature $T_c$ given by Eq. (26). The quadratic couplings $\alpha$, $\beta$, $\gamma$ appear in every thermodynamic quantity only through $\sigma = (D-4)(D-2)[(D-2)(D-1)\alpha + (D-3)D\gamma]$, which acts as a pseudo-charge. The paper shows that $\sigma$ controls whether the critical radius is real and positive: for $\sigma < 0$ the branch $R_c^{(-)}$ is physical for $D>4$, for $\sigma > 0$ the branch $R_c^{(+)}$ is physical for $D>4$, and for $\sigma = 0$ or $D = 4$ the critical radius collapses to zero, matching general relativity with phase transitions only at $R_A = 0$. On this basis the paper concludes that the quadratic terms change the stability conditions of the cosmic horizon, allowing new phases that are absent in Einstein gravity.

Load-bearing premise

The whole thermodynamic construction assumes that the differential form $A(t,r)\,dt + B(t,r)\,dr$ is integrable, so the effective Misner-Sharp energy exists; the paper states this as an assumption after Eq. (10) and never verifies $\partial_r A = \partial_t B$ for the quadratic-gravity stress tensor it derives.

Editorial extensions

If this is right

  • If $\sigma < 0$ and $D > 4$, the apparent horizon has a finite critical radius $R_c^{(-)}$, so a genuine phase transition can occur in the early universe where general relativity predicts none.
  • For $\sigma > 0$ and $D > 4$, the physical critical branch is $R_c^{(+)}$, while for $\sigma = 0$ or $D = 4$ the critical radius shrinks to zero, limiting phase transitions to the very beginning of the universe.
  • The signs of the Gibbs free energy and specific heat depend on the signs of $\alpha$ and $\gamma$, so the early universe may be thermodynamically unstable and then become stable as the horizon grows.
  • Because $\sigma$ multiplies a factor $R_A^{D-4}$ in the entropy and enthalpy, higher dimensions amplify the deviation from general relativity, making quadratic effects easier to detect in $D > 4$ cosmological models.

Reading between the lines

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

  • If the integrability condition $\partial_r A = \partial_t B$ fails for generic FLRW backgrounds, the formalism would be restricted to a special subclass of spacetimes, and the critical-point calculation would need to start from a different quasi-local energy definition.
  • Treating $\sigma$ as an independent thermodynamic variable would extend the system into an extended phase space with a $\sigma$-conjugate work term, in analogy with black-hole chemistry, potentially revealing additional phase structure beyond the critical points reported here.
  • Connecting the critical temperature $T_c$ to a specific cosmological epoch would require an explicit solution for $a(t)$; a natural next step is to see whether the predicted phase transition leaves imprints in the primordial spectrum or in the horizon-scale observables of early-universe cosmology.
Share X Bluesky LinkedIn Reddit HN

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. The paper studies the thermodynamics of the apparent horizon in a D-dimensional FLRW universe in quadratic gravity. It derives field equations from an action with R^2, R_{\mu\nu}^2, and Gauss-Bonnet-type couplings, constructs a generalized Misner-Sharp energy, and then uses the unified first law to obtain a thermodynamic pressure P(R_A,T). The central claim is that the criticality conditions \partial P/\partial R_A=0 and \partial^2 P/\partial R_A^2=0 yield a critical temperature (26) and critical radii (27), from which the paper derives Wald entropy, enthalpy, specific heat, and Gibbs free energy, and concludes that quadratic couplings change thermodynamic stability relative to general relativity. The manuscript is clearly organized and explicitly checks several GR limits, but the main critical-point calculation is internally inconsistent with the paper's own equation of state.

Significance. If the claimed critical points were correct, the paper would provide a concrete, falsifiable prediction for phase transitions in a higher-dimensional FLRW setting, and the coupling combination \sigma would be a useful thermodynamic pseudo-charge. The authors also correctly stress the dimensional reduction to GR when D=4 or \sigma=0, and they do not tune free constants to force conclusions. However, the central algebraic derivation is not sound: the critical radii Eq. (27) do not solve the manuscript's own criticality conditions when applied to Eq. (23), and the integrability condition on which the whole Misner-Sharp construction rests is assumed rather than verified. In its present form the paper does not establish the phase structure that forms its main conclusion.

major comments (4)
  1. [Sec. 4, Eqs. (23)-(27)] Solving the criticality conditions (24) and (25) with the equation of state (23) does not lead to the critical radius (27). Writing y=R_A^2 and A=D(D-2), the first condition gives Eq. (26), and substituting it into \partial^2 P/\partial R_A^2=0 yields A^2(D-3)y^2 - 12A\kappa\sigma y + 12(\kappa\sigma)^2(D-5)=0. This quadratic has roots y=0 and y=2\kappa\sigma/5 at D=5 and a double root y=\kappa\sigma/12 at D=6, whereas Eq. (27) gives y=0 or y=-14\kappa\sigma/5 at D=5 and y\approx \kappa\sigma/225 at D=6. The two sets of roots do not agree, so the claimed critical radii R_c^{(\pm)}, the bifurcation structure, and the subsequent stability analysis in Sections 4 and 5 are not supported by the manuscript's own equations.
  2. [Sec. 3, Eq. (10)] The integrability condition (10) is stated as an assumption after Eq. (9) and is never checked for the A(t,r) and B(t,r) computed from Eq. (5). The generalized Misner-Sharp energy (11), the work density, the pressure P(R_A,T), and all thermodynamic quantities derived from them depend on this integration. Since the paper itself notes that the condition can fail in modified gravity, as in f(R) gravity cited in [38], an explicit verification is required; without it the thermodynamic construction is conditional whenever the background fields are not shown to satisfy Eq. (10).
  3. [Sec. 4, Eq. (21) and Sec. 2, Eq. (5)] The constraint used to eliminate higher time derivatives is inconsistent with the condition stated earlier. Section 2 states that the higher-derivative terms in Eq. (5) do not appear when 4(D-1)\alpha + D\beta = 0, whereas Eq. (21) imposes \beta = +4(D-1)\alpha/D, which gives 4(D-1)\alpha + D\beta = 8(D-1)\alpha. The sign is opposite. Since Eqs. (22) and (23), and therefore all subsequent thermodynamic results, are derived under Eq. (21), this discrepancy must be resolved; if the intended condition is \beta = -4(D-1)\alpha/D, then the expressions for \rho, p, and P in Section 4 need to be recomputed.
  4. [Sec. 5, Eqs. (32)-(34)] The formula (34) is presented as the Wald entropy but it is obtained by integrating V'(R_A)C(R_A), and no derivation is given showing that this integral equals the Wald entropy of the quadratic-gravity action. In addition, the displayed expression contains an explicit factor 2\sigma/(D-4), which is ambiguous at D=4, the dimension on which much of the discussion focuses. If the D-dependent prefactor in \sigma is retained, the ratio has a finite limit, but as written Eq. (34) is not well defined at D=4.
minor comments (4)
  1. [Sec. 1] In the last paragraph of the introduction, the sentence 'Finally, we to show discuss how these results...' contains a typographical error and should be corrected.
  2. [Sec. 3] The heading and the opening sentence use 'Meisner-Sharp energy' instead of 'Misner-Sharp energy'; this should be corrected throughout.
  3. [Figs. 1-6] The figures are discussed qualitatively in the text but no axis labels, units, or fixed parameter values are provided in the captions; the temperature values and the chosen D should also be stated so that the plots can be checked against Eq. (23).
  4. [Sec. 2, Eq. (4) and Sec. 4, Eq. (15)] The symbol \kappa is used both for the gravitational constant 8\pi G_D and for the surface gravity \kappa_g in Eq. (15); the double use is confusing and should be noted or the surface gravity relabelled.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity: the EoS and critical-point program follow from the action through external published inputs; the only definitional element is the 'Wald entropy', constructed from the same equation of state it later interprets. The central critical-radius formula (27) fails the paper's own conditions (24)-(25), an internal consistency defect rather than circularity.

  1. self definitional [Section 5, Eqs. (31)-(34)]
    "The Wald entropy can be obtained as follows S = Z V ′(RA)C(RA)dRA, (32) where V ′(RA) represents the derivative of the volume with respect to RA, and C(RA) is obtained from the equation of state as [52] C(RA) = 1 DκR3 A (4πκσ + 2D(D − 2)πR2 A). (33) By integrating (32), we have S = 4π (D+1)/2 DΓ[(D − 1)/2] RD−4 A ( DR2 A κ + 2σ D − 4 ). (34) The above relation shows that the entropy has a direct dependence on the parameter σ, which is due to the presence of quadratic terms in the gravitational action."

    The quantity labeled 'Wald entropy' (34) is not computed from Wald's Noether-charge formula for action (1); it is defined by construction as the first-law integral (32) of the temperature coefficient C(RA) read off from the same equation of state (23) used in all later analysis. The specific heat (36), enthalpy (35), and Gibbs free energy (37) therefore add no information beyond the σ-dependence already in (23); the stability conclusions drawn from their signs restate the EoS in new variables. The 'Wald' label imports an independent status the quantity lacks. This is a definitional/labelling limitation rather than a fitted-parameter loop, and it does not feed back into Section 4's critical-point derivation, which uses only (23).

full rationale

The paper's derivation chain is essentially self-contained and, in the load-bearing sense, non-circular. The field equations (2)-(3) and the perfect-fluid form of the effective energy-momentum tensor for the FLRW metric are imported from the group's earlier published work [10,33]; these are external, statable results—the components (5) can be verified directly from (2)-(4)—so under the reviewing rules they count as independent support rather than circularity. The restriction (21) that removes higher derivatives is presented openly as a choice ('we can eliminate the contribution of the higher derivative of the scale factor by the following choice [10]'), so no ansatz is smuggled in by citation. Section 3's generalized Misner-Sharp energy (11)-(12) is obtained by integrating the unified first law under an integrability condition (10) that the paper states ('assuming that integrability condition holds') but never verifies; that is an unresolved assumption to be weighed in the correctness pass, not a circular step. Section 4's equation of state (23) follows algebraically from (5) with (21) and the identifications (14)-(20), and the critical temperature (26) does follow from the first criticality condition (24). The claimed critical radius (27), however, does not solve the paper's own equations: substituting (26) into ∂²P/∂RA²=0 built from (23) gives [D(D−2)]²(D−3)y² − 12D(D−2)κσy + 12(κσ)²(D−5) = 0, which at D=5 has roots y=0 and y=2κσ/5, and at D=6 a double root y=κσ/12, whereas (27) gives y=0 or −14κσ/5 and y≈κσ/225 respectively. This is an internal algebraic inconsistency in the paper's central prediction—a correctness risk for the referee report—not circularity: (27) is neither fitted nor defined in terms of the phase-transition claim, it is simply not entailed by (24)-(25). A further unstated sign assumption converts the absolute value in (16) into T=(1−ṘA/2)/(2πRA) when (5) becomes (22). The one genuinely definitional element is in Section 5: the 'Wald entropy' (34) is constructed by Eq. (32) from the same EoS (23) that it later interprets, so the stability language ('quadratic terms change the stability conditions') partly restates the EoS's own σ-dependence under the unsupported 'Wald' label. No data are fitted and no constants are tuned, so nothing reduces to a fit. Overall circularity is therefore minor.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The derivations rest on the unified-first-law dictionary and the asserted integrability of the Misner-Sharp energy, together with a constraint on the quadratic couplings. These are stated in the paper as assumptions rather than derived consequences.

free parameters (4)
  • alpha (Ricci-scalar-squared coupling) = not fitted; varied by hand, e.g. +1 or -1 in Figures 1-6
    A free coupling in action (1). Its sign controls the thermodynamic stability claims around Eqs. (28)-(30), but it is not determined by any data.
  • gamma (Gauss-Bonnet-type coupling) = not fitted; varied by hand, e.g. +1 or -1
    Appears in sigma via Eq. (6) and in the critical radius (27) and entropy (34).
  • D (spacetime dimension) = not fitted; chosen integer
    The whole analysis is D-dimensional; results change qualitatively for D = 4 versus D > 4.
  • Lambda (cosmological constant) = not fitted; input parameter
    Appears in the pressure offset Eq. (23) and saturated energy Eq. (13), but cancels from the critical-point conditions.
assumptions (6)
  • domain assumption For quadratic gravity the unified first law holds in the form dE = A Psi dx + W dV (Eq. 7).
    This is the standard Hayward framework imported to a higher-derivative theory without re-derivation.
  • ad hoc to paper The integrability condition (10) is satisfied by the computed A(t,r) and B(t,r).
    Stated as 'assuming that integrability condition holds' in Section 3; never demonstrated.
  • domain assumption Comparison of dE = -T dS + W dV with dU = T dS - P dV identifies U = -E and P = W (Eqs. 17-19).
    This dictionary is standard in apparent-horizon thermodynamics but is a modeling assumption for modified gravity.
  • domain assumption The apparent-horizon temperature is given by the GR surface-gravity formula (Eqs. 14-16).
    Used without modification for quadratic gravity.
  • ad hoc to paper Entropy can be obtained by integrating V'(R_A) C(R_A) (Eq. 32), and this equals Wald entropy.
    The integration defines a thermodynamic entropy from the equation of state; identification with Wald entropy is asserted, not derived.
  • ad hoc to paper The constraint beta = -4(D-1) alpha / D is imposed to eliminate higher time derivatives.
    Imported from reference [10] and restricts the quadratic-gravity parameter space; the central results depend on it.
invented entities (1)
  • sigma as a thermodynamic pseudo-charge
    purpose: Encodes the quadratic-coupling contribution to entropy, pressure, enthalpy, and stability; the paper says it can play the role of a new degree of freedom similar to electric charge in black-hole thermodynamics (Section 5).
    sigma is a combination of alpha and gamma defined in Eq. (6), so it is not an independently measurable entity; calling it a pseudo-charge is interpretative and provides no falsifiable handle outside the model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Thermodynamics of FLRW universe in Quadratic Gravity." pith.science (2026). https://pith.science/paper/Q2QIBZO3

@misc{pith2026250712055,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of FLRW universe in Quadratic Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2QIBZO3}},
  note         = {Machine review of arXiv:2507.12055}
}
abstract

In this paper, we investigate the thermodynamic aspects of quadratic gravity in a $D$-dimensional Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. First, we derive the field equations and the effective energy-momentum tensor for quadratic gravity. Then, using these equations, we obtain the generalized Misner-Sharp energy within the framework of this model. We consider the thermodynamic behavior of the apparent horizon and derive the equations of state related to the pressure, temperature, and radius of the apparent horizon. Using the thermodynamic pressure, we obtain the critical points corresponding to phase transitions. We determine the critical temperature and critical radius in terms of model parameters, including the quadratic coupling and the cosmological constant. We also examine key thermodynamic quantities, such as Wald entropy, specific heat at constant pressure, enthalpy, Gibbs free energy. By examining the behavior of these quantities, we can gain insight into the thermodynamic stability of the quadratic gravity model. In particular, we find that quadratic terms change the stability conditions and can lead to new thermodynamic behaviors compared to general relativity.

Figures

Figures reproduced from arXiv: 2507.12055 by the authors.

Figure 1
Figure 1. Pressure P as a function of RA for α = 1 and γ = 1 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Pressure P as a function of RA for α = −1 and γ = 1. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Pressure P as a function of RA for α = −1 and γ = −1 pressure P as a function of the apparent radius RA for different temperatures and model parameters. The diagrams (1), (2) and (3) show this dependence. Figures (1), (2) and (3) depict the thermodynamic pressure (P) as a function of the apparent horizon radius (RA) for varying quadratic gravity parameters α and γ in a D-dimensional FLRW universe. In Fig. (1) (α = 1… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Gibbs free energy G as a function of RA for α = 1 and γ = 1 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Gibbs free energy G as a function of RA for α = −1 and γ = 1 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Gibbs free energy G as a function of RA for α = −1 and γ = −1 6 Conclusion In this paper, we investigate the thermodynamic properties of the quadratic gravity model in the D-dimensional FLRW spacetime. First, by adding second-order curvature terms to the action, we der…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 19 canonical work pages

  1. [38]

    R. G. Cai and L. M. Cao, Phys. Rev. D 75 (2007), 064008 doi:10.1103/PhysRevD.75.064008 [arXiv:gr-qc/0611071 [gr-qc]]

    arXiv 2007

  2. [1]

    Clifton, P

    T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513 (2012), 1-189 doi:10.1016/j.physrep.2012.01.001 [arXiv:1106.2476 [astro-ph.CO]]

    arXiv 2012

  3. [2]

    Berti, E

    E. Berti, E. Barausse, V . Cardoso, L. Gualtieri, P. Pani, U. Sperhake, L. C. Stein, N. Wex, K. Yagi and T. Baker, et al. Class. Quant. Grav. 32 (2015), 243001 doi:10.1088/0264-9381/32/24/243001 [arXiv:1501.07274 [gr-qc]]

    arXiv 2015

  4. [3]

    Y . S. Myung, Phys. Rev. D 105 (2022) no.12, 124015 doi:10.1103/PhysRevD.105.124015 [arXiv:2204.06750 [gr-qc]]

    work page Pith review arXiv 2022

  5. [4]

    Kaneta, K

    K. Kaneta, K. y. Oda and M. Yoshimura, [arXiv:2503.02409 [hep-ph]]

    arXiv

  6. [5]

    T. P. Sotiriou and V . Faraoni, Rev. Mod. Phys. 82 (2010), 451-497 doi:10.1103/RevModPhys.82.451 [arXiv:0805.1726 [gr-qc]]

    arXiv 2010

  7. [6]

    De Felice and S

    A. De Felice and S. Tsujikawa, Living Rev. Rel. 13 (2010), 3 doi:10.12942/lrr- 2010-3 [arXiv:1002.4928 [gr-qc]]

    arXiv 2010

  8. [7]

    Quantum symmetry, the cosmological constant and Planck scale phenomenology

    G. Amelino-Camelia, L. Smolin and A. Starodubtsev, Class. Quant. Grav. 21 (2004), 3095-3110 doi:10.1088/0264-9381/21/13/002 [arXiv:hep-th/0306134 [hep- th]]

    work page Pith review arXiv 2004

Show all 55 references
  1. [8]

    A. K. Sanyal and B. Modak, Class. Quant. Grav. 19 (2002), 515-526 doi:10.1088/0264-9381/19/3/307 [arXiv:gr-qc/0107070 [gr-qc]]

    2002 arXiv

  2. [9]

    Salvio, Phys

    A. Salvio, Phys. Rev. D 99 (2019) no.10, 103507 doi:10.1103/PhysRevD.99.103507 [arXiv:1902.09557 [gr-qc]]

    2019 arXiv

  3. [10]

    G ¨urses, Y

    M. G ¨urses, Y . Heydarzade and C ¸ . S ¸ent¨urk, Phys. Rev. D110 (2024) no.2, 024073 doi:10.1103/PhysRevD.110.024073 [arXiv:2401.09784 [gr-qc]]

    2024 arXiv

  4. [11]

    J. F. Donoghue and G. Menezes, Nuovo Cim. C 45 (2022) no.2, 26 doi:10.1393/ncc/i2022-22026-7 [arXiv:2112.01974 [hep-th]]

    2022 arXiv

  5. [12]

    Mohaupt, [arXiv:hep-th/0512048 [hep-th]]

    T. Mohaupt, [arXiv:hep-th/0512048 [hep-th]]

    arXiv

  6. [13]

    Alonso-Serrano, L

    A. Alonso-Serrano, L. J. Garay and M. Li ˇska, Phys. Rev. D 111 (2025) no.6, 064019 doi:10.1103/PhysRevD.111.064019 [arXiv:2409.06645 [gr-qc]]

    2025

  7. [14]

    Cecotti and R

    S. Cecotti and R. Kallosh, JHEP 05 (2014), 114 doi:10.1007/JHEP05(2014)114 [arXiv:1403.2932 [hep-th]]

    2014 arXiv

  8. [15]

    Ashtekar and P

    A. Ashtekar and P. Singh, Class. Quant. Grav. 28 (2011), 213001 doi:10.1088/0264-9381/28/21/213001 [arXiv:1108.0893 [gr-qc]]

    2011 arXiv

  9. [16]

    Bojowald and A

    M. Bojowald and A. Skirzewski, eConf C0602061 (2006), 03 doi:10.1142/S0219887807001941 [arXiv:hep-th/0606232 [hep-th]]

    2006 arXiv

  10. [17]

    Date and S

    G. Date and S. Sengupta, Class. Quant. Grav. 26 (2009), 105002 doi:10.1088/0264-9381/26/10/105002 [arXiv:0811.4023 [gr-qc]]. 17

    2009 arXiv

  11. [18]

    H. Lu, A. Perkins, C. N. Pope and K. S. Stelle, Phys. Rev. Lett. 114 (2015) no.17, 171601 doi:10.1103/PhysRevLett.114.171601 [arXiv:1502.01028 [hep-th]]

    2015 arXiv

  12. [19]

    E. N. Saridakis et al. [CANTATA], Springer, 2021, ISBN 978-3-030- 83714-3, 978-3-030-83717-4, 978-3-030-83715-0 doi:10.1007/978-3-030-83715-0 [arXiv:2105.12582 [gr-qc]]

    2021 arXiv

  13. [20]

    Ayzenberg, K

    D. Ayzenberg, K. Yagi and N. Yunes, Phys. Rev. D 89 (2014) no.4, 044023 doi:10.1103/PhysRevD.89.044023 [arXiv:1310.6392 [gr-qc]]

    2014 arXiv

  14. [21]

    J. D. Barrow and S. Hervik, Phys. Rev. D 74 (2006), 124017 doi:10.1103/PhysRevD.74.124017 [arXiv:gr-qc/0610013 [gr-qc]]

    2006 arXiv

  15. [22]

    J. D. Barrow and J. D. Middleton, Phys. Rev. D 75 (2007), 123515 doi:10.1103/PhysRevD.75.123515 [arXiv:gr-qc/0702098 [gr-qc]]

    2007 arXiv

  16. [23]

    Chakraborty and D

    S. Chakraborty and D. Gregoris, Eur. Phys. J. C 81 (2021) no.10, 944 doi:10.1140/epjc/s10052-021-09697-2 [arXiv:2103.07718 [gr-qc]]

    2021 arXiv

  17. [24]

    Beltran Jimenez, L

    J. Beltran Jimenez, L. Heisenberg and T. S. Koivisto, JCAP 04 (2016), 046 doi:10.1088/1475-7516/2016/04/046 [arXiv:1602.07287 [hep-th]]

    2016 arXiv

  18. [25]

    Alvarez-Gaume, A

    L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. L ¨ust and A. Riotto, Fortsch. Phys. 64 (2016) no.2-3, 176-189 doi:10.1002/prop.201500100 [arXiv:1505.07657 [hep-th]]

    2016 arXiv

  19. [26]

    Sebastiani, Phys

    L. Sebastiani, Phys. Dark Univ. 42 (2023), 101296 doi:10.1016/j.dark.2023.101296 [arXiv:2307.04509 [gr-qc]]

    2023

  20. [27]

    Padmanabhan, Curr

    T. Padmanabhan, Curr. Sci. 109 (2015), 2236-2242 doi:10.18520/v109/i12/2236- 2242 [arXiv:1512.06546 [gr-qc]]

    2015 arXiv

  21. [28]

    Bamba, C

    K. Bamba, C. Q. Geng and S. Tsujikawa, Int. J. Mod. Phys. D 20 (2011), 1363- 1371 doi:10.1142/S0218271811019542 [arXiv:1101.3628 [gr-qc]]

    2011 arXiv

  22. [29]

    Sen, JHEP 09 (2005), 038 doi:10.1088/1126-6708/2005/09/038 [arXiv:hep- th/0506177 [hep-th]]

    A. Sen, JHEP 09 (2005), 038 doi:10.1088/1126-6708/2005/09/038 [arXiv:hep- th/0506177 [hep-th]]

    2005

  23. [30]

    Camps, JHEP 03 (2014), 070 doi:10.1007/JHEP03(2014)070 [arXiv:1310.6659 [hep-th]]

    J. Camps, JHEP 03 (2014), 070 doi:10.1007/JHEP03(2014)070 [arXiv:1310.6659 [hep-th]]

    2014 arXiv

  24. [31]

    Saavedra and F

    J. Saavedra and F. Tello-Ortiz, Eur. Phys. J. Plus 139 (2024) no.7, 657 doi:10.1140/epjp/s13360-024-05467-7 [arXiv:2310.08781 [gr-qc]]

    2024 arXiv

  25. [32]

    Bellucci, S

    S. Bellucci, S. Ferrara, A. Shcherbakov and A. Yeranyan, JHEP 10 (2009), 024 doi:10.1088/1126-6708/2009/10/024 [arXiv:0906.4910 [hep-th]]

    2009 arXiv

  26. [33]

    Gurses and Y

    M. Gurses and Y . Heydarzade, Eur. Phys. J. C 80 (2020) no.11, 1061 doi:10.1140/epjc/s10052-020-08641-0 [arXiv:2009.12825 [gr-qc]]

    2020 arXiv

  27. [34]

    Pravda, A

    V . Pravda, A. Pravdova, J. Podolsky and R. Svarc, Phys. Rev. D 95 (2017) no.8, 084025 doi:10.1103/PhysRevD.95.084025 [arXiv:1606.02646 [gr-qc]]. 18

    2017 arXiv

  28. [35]

    R. G. Cai, L. M. Cao, Y . P. Hu and N. Ohta, Phys. Rev. D 80 (2009), 104016 doi:10.1103/PhysRevD.80.104016 [arXiv:0910.2387 [hep-th]]

    2009 arXiv

  29. [36]

    A. R. Akbarieh, N. S. Ilkhchi and Y . Heydarzade, [arXiv:2506.04469 [gr-qc]]

    arXiv

  30. [37]

    S. A. Hayward, S. Mukohyama and M. C. Ashworth, Phys. Lett. A 256 (1999), 347-350 doi:10.1016/S0375-9601(99)00225-X [arXiv:gr-qc/9810006 [gr-qc]]

    1999 arXiv

  31. [39]

    I. D. Gialamas and K. Tamvakis, JCAP 03 (2023), 042 doi:10.1088/1475- 7516/2023/03/042 [arXiv:2212.09896 [gr-qc]]

    2023 arXiv

  32. [40]

    W. P. F. de Medeiros, D. M ¨uller, O. F. Piattella, M. J. Lazo and D. A. Sales, [arXiv:2502.03653 [gr-qc]]

  33. [41]

    Abdusattar, S

    H. Abdusattar, S. B. Kong, Y . Yin and Y . P. Hu, JCAP 08 (2022) no.08, 060 doi:10.1088/1475-7516/2022/08/060 [arXiv:2203.10868 [gr-qc]]

    2022 arXiv

  34. [42]

    D. W. Tian and I. Booth, Phys. Rev. D 92 (2015) no.2, 024001 doi:10.1103/PhysRevD.92.024001 [arXiv:1411.6547 [gr-qc]]

    2015 arXiv

  35. [43]

    Abdolmaleki and T

    A. Abdolmaleki and T. Najafi, Int. J. Mod. Phys. D 25 (2016) no.04, 1650040 doi:10.1142/S0218271816500401 [arXiv:1504.03544 [gr-qc]]

    2016 arXiv

  36. [44]

    Akbar and R

    M. Akbar and R. G. Cai, Phys. Rev. D 75 (2007), 084003 doi:10.1103/PhysRevD.75.084003 [arXiv:hep-th/0609128 [hep-th]]

    2007 arXiv

  37. [45]

    Cognola, R

    G. Cognola, R. Myrzakulov, L. Sebastiani and S. Zerbini, Phys. Rev. D88 (2013) no.2, 024006 doi:10.1103/PhysRevD.88.024006 [arXiv:1304.1878 [gr-qc]]

    2013 arXiv

  38. [46]

    I. Bena, M. Berkooz, J. de Boer, S. El-Showk and D. Van den Bleeken, JHEP 11 (2012), 171 doi:10.1007/JHEP11(2012)171 [arXiv:1205.5023 [hep-th]]

    2012 arXiv

  39. [47]

    Y . P. Hu, H. A. Zeng, Z. M. Jiang and H. Zhang, Phys. Rev. D 100 (2019) no.8, 084004 doi:10.1103/PhysRevD.100.084004 [arXiv:1812.09938 [gr-qc]]

    2019 arXiv

  40. [48]

    Arcadi, A

    G. Arcadi, A. Djouadi and M. Kado, Eur. Phys. J. C 81 (2021) no.7, 653 doi:10.1140/epjc/s10052-021-09411-2 [arXiv:2101.02507 [hep-ph]]

    2021 arXiv

  41. [49]

    Abdusattar, S

    H. Abdusattar, S. B. Kong, H. Zhang and Y . P. Hu, Phys. Dark Univ. 42 (2023), 101330 doi:10.1016/j.dark.2023.101330 [arXiv:2301.01938 [gr-qc]]

    2023

  42. [50]

    Hansen, D

    D. Hansen, D. Kubiznak and R. Mann, Phys. Lett. B 771 (2017), 277-280 doi:10.1016/j.physletb.2017.04.076 [arXiv:1610.03079 [gr-qc]]

    2017 arXiv

  43. [51]

    R. J. Yang, Entropy 16 (2014), 4483-4488 doi:10.3390/e16084483 [arXiv:1405.6909 [gr-qc]]

    2014 arXiv

  44. [52]

    Zheng and R

    Y . Zheng and R. J. Yang, Universe 6 (2020) no.3, 47 doi:10.3390/universe6030047 [arXiv:1907.05868 [gr-qc]]. 19

    2020 arXiv

  45. [53]

    Rudra and B

    P. Rudra and B. Pourhassan, Phys. Dark Univ. 33 (2021), 100849 doi:10.1016/j.dark.2021.100849 [arXiv:2008.11034 [gr-qc]]

    2021

  46. [54]

    Cs ´aki, M

    C. Cs ´aki, M. Geller, Z. Heller-Algazi and A. Ismail, JHEP 06 (2023), 202 doi:10.1007/JHEP06(2023)202 [arXiv:2301.10247 [hep-ph]]

    2023 arXiv

  47. [55]

    Mahdavian Yekta and H

    D. Mahdavian Yekta and H. Golchin, Eur. Phys. J. C 80 (2020) no.5, 473 doi:10.1140/epjc/s10052-020-8045-0 [arXiv:2004.00866 [hep-th]]. 20

    2020 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.