Scalarization of Charged Black Hole in Gauss-Bonnet Extended Starobinsky-Maxwell Gravity

Hai-Shan Liu; Rui-Xi Zhu

REVIEW 4 major objections 4 minor 31 references

Scalarization of Charged Black Hole in Gauss-Bonnet Extended Starobinsky-Maxwell Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Hairy black holes in Gauss-Bonnet extended Starobinsky gravity form folded, multi-branched solution families that cross the Schwarzschild curve and connect at a common minimal horizon radius.

desk verdict Plausible new branch structure and charged extension, but the numerical foundation for the headline topology is unverified and one perturbation equation is wrong as printed. read the letter →

arxiv 2607.14620 v1 pith:IQVNKRB5 submitted 2026-07-16 gr-qc

Rui-Xi Zhu , Hai-Shan Liu This is my paper

classification gr-qc MSC 83C5783C2283D05 PACS 04.70.-s04.50.Kd04.40.Nr

keywords blackholescalarizationGauss-BonnetextendedStarobinskygravityspontaneoushairyholeschargedfirstlawofthermodynamicsmulti-branchsolutionshigher-curvature

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 establish that black hole scalarization in pure higher-curvature gravity — where the scalar degree of freedom comes from the Starobinsky R² term coupled to the Gauss-Bonnet invariant — is geometrically richer than previously known. The authors claim that hairy solutions do not merely bifurcate off the Schwarzschild branch and sit on one side; they cross the Schwarzschild curve, flip the sign of the scalar hair, and split into two smooth branches that meet at a common minimal horizon radius. With a Maxwell field added, they map the full existence domain in the (M, Q) plane, find that an extra disconnected branch appears for certain charges, and show that the single-to-disconnected transition is governed by whether constant-mass or constant-charge contours hit a new curve of interior critical endpoints. A sympathetic reader would care because multi-valued hairy black hole families mean several distinct black holes with the same mass and charge, a potentially observable signature for strong-field gravity, and because, on the thermodynamic side, the paper verifies the standard first law — dM = T dS + Φ dQ — for all charged hairy solutions.

What carries the argument

The load-bearing object is the action S = (1/16π) ∫√−g [ R + φ R − ½ μ² φ² + ½ β φ² E_GB ], whose algebraic scalar equation ties φ to the curvature invariants, effectively reducing the theory to a pure higher-curvature gravity that still carries a dynamical scalar mode. Numerically, the paper integrates the field equations inward from asymptotic Yukawa data and identifies a horizon when both metric functions fall below 10⁻⁶ — this criterion locates each solution point and defines the branch structure. The new geometric objects are the 'existence line' (the bifurcation curve off Schwarzschild/Reissner–Nordström), the inner and outer boundaries of the existence domain, and the 'purple dashed l

What would settle it

Recompute the (M, r₊) curves and the (Q, M) existence domain with a tighter horizon threshold (say 10⁻¹⁰) and higher radial resolution; if the crossing of the Schwarzschild/Reissner–Nordström curve, the common minimal horizon, or the purple dashed endpoints shift beyond the plotted curve widths, the multi-branch claim collapses. Independently, verify that the printed perturbation equation (21) — which contains a term 4 R^μν_(0) ∇_μ∇_ν U(δφ) — is genuinely of order (δφ)² under the stated linearization; if it is not, the charged bifurcation (existence) line in Fig. 5 may be mislocated.

Watch

Extended reading notes

Core claim

The central claim is that in Gauss-Bonnet extended Starobinsky gravity, specifically with the quadratic scalar–Gauss-Bonnet coupling U(φ) = ½ β φ², static, spherically symmetric hairy black holes organize into a folded, multivalued curve in the mass–horizon-radius plane. The hairy branch emerges from the Schwarzschild solution at a critical mass, crosses the Schwarzschild curve with the scalar hair passing through zero, reaches a minimum horizon radius, and then bends back, so the two arms connect smoothly into a single hooked curve. With electric charge, the same folding persists and, for intermediate charges, a genuinely disconnected second branch appears whose endpoints lie inside the exi

Load-bearing premise

The entire branch topology — the crossings, the common minimal horizon radius, and the interior critical endpoints — rests on the untested numerical proxy that a horizon exists where the two metric functions happen to fall below 10⁻⁶, with no convergence or resolution analysis, so the multi-branch structure could in principle be an artifact of the integration scheme.

Editorial extensions

If this is right

A fixed black-hole mass and charge can admit several distinct hairy solutions, since the scalar hair is a multi-valued function of (M, Q).
The multi-branch structure persists from the neutral to the charged case, and adding charge produces an additional disconnected branch for certain charge ranges.
The scalar charge is secondary hair: it does not enter the first law, and hairy entropy nearly equals Reissner–Nordström entropy at the same (M, Q).
The transition from one smooth branch to two disconnected branches is controlled by whether constant-mass or constant-charge curves intersect the interior critical-endpoint line (the purple dashed curve).
The existence domain is bounded not only by extremality and bifurcation points, but also by a new kind of boundary made of interior critical points.

Reading between the lines

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

If the folded branch structure proves generic, then scalarization theories with algebraic scalar equations — where φ is a function of curvature invariants — may all exhibit multi-valued hairy black hole families; this could be tested by applying the same inward-integration analysis to standard Einstein-scalar-Gauss-Bonnet and Einstein-Maxwell-scalar models.
The purple dashed line of critical endpoints resembles the projection of a fold catastrophe (like a swallowtail caustic); the single-to-disconnected transition may be a cusp in the solution manifold, which a local perturbation analysis around the critical points could confirm.
The near-equality of hairy and hairless entropy at fixed (M, Q) suggests the scalar hair is thermodynamically 'silent,' carrying no independent conserved charge; showing generally that the conjugate variables of scalar charges vanish on regular solutions would extend this to other secondary-hair theories.
The paper's horizon proxy (both metric functions below 10⁻⁶) is untested; a high-resolution convergence study with a tighter threshold would settle whether the branch crossings and critical endpoints are genuine or artifacts of the integration scheme.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new content is the negative-scalar-hair neutral branch that folds the (M, r+) curve, and the charged extension with disconnected branch pairs ending on an interior critical curve. Both are new relative to the authors' earlier work [26], and the physical reasoning around them (the phi -> -phi symmetry at the bifurcation point, the near-coincidence of hairy and bald entropies, the structure of the first law) is sensible. The paper does a broad numerical scan and maps the existence domain carefully. Credit where it's due: this is a concrete advance in the scalarization subfield, if the results hold. Now the soft spots, in proportion. First, equation (21), used to compute the charged existence line, includes a term 4 R^mu nu_(0) grad_mu grad_nu U(delta phi). With U = 1/2 beta phi^2, that term is O(delta phi^2) and does not belong in the stated linearization. As printed, the existence line in Fig. 5 lacks a valid derivation. It may be a typo, but it has to be fixed or removed. Second, the central branch-topology claim rests entirely on the numerical pipeline: inward integration from truncated asymptotic data, with a horizon identified when h and f both fall below 10^-6. There is no convergence test, no resolution study, and no code or data released. Near the branch crossings and the minimal-radius connection, the horizon slope is small, so that threshold can shift r+ by an amount comparable to the separation between branches. Conversely, a solution with f_min around 5e-6 would be misclassified as a black hole. The paper itself concedes 'due to finite numerical precision, the functions cannot be made to reach exactly zero.' The stress test is not off the mark: this is a real gap, not a formality. Third, the abstract says the first law is obeyed 'exactly,' but the evidence is two Maxwell-relation checks at fixed Q=1.0 and fixed S=325.5, using fitted M(S) and M(Q). That is evidence, but not proof for all charged hairy solutions. Restate the claim with residuals and a broader sample. Fourth, the manuscript is contaminated by duplicated fragments from [26] (repeated figure captions and text blocks) and several bibliography entries are never cited. That is not a scientific flaw, but it signals carelessness. Who this is for: people working on black hole scalarization, especially in higher-curvature pure gravity. The new branch structure, if real, is a meaningful advance. I would send this to a serious referee rather than desk reject, but I would ask for a major revision: fix Eq. (21), provide convergence studies (at minimum vary the 10^-6 threshold and grid spacing and show the branches persist), release code or data, and clean up the text. The physics is plausible; the evidence as printed is not yet at the claimed standard.

Referee Report

4 major / 4 minor

Summary. The paper studies static, spherically symmetric scalarized black holes in Gauss-Bonnet extended Starobinsky gravity with quadratic scalar coupling U=1/2 β φ², both in the neutral theory and after adding a Maxwell field. For the neutral case the authors claim that the hairy solutions form two smooth branches joined at a common minimal horizon radius and that the branch crosses the Schwarzschild curve, giving multiple hairy solutions for a fixed mass. In the charged case they map an existence domain in the (Q,M) plane, identify a green bifurcation line from Reissner-Nordström, outer/inner boundaries, and an interior 'purple dashed line' of critical endpoints, and describe how constant-mass and constant-charge slices exhibit one branch, two disconnected branches, or a merging process. They also verify the first law dM = T dS + Φ dQ by checking two Maxwell relations numerically. The central new claim is the multi-branch topology, previously unreported for static spherically symmetric scalarization.

Significance. If the numerical results are robust, the claimed multi-branch structure would be a genuinely new feature in scalarized black hole solutions of pure higher-curvature gravity, and the charged extension with its rich existence domain would be of considerable interest to the strong-field gravity community. The paper builds on the authors' earlier framework [26] and extends it in a natural direction, with explicit asymptotics, horizon expansions, Wald entropy, and a thermodynamic consistency check. However, the central claims rest entirely on an undocumented numerical pipeline, and one of the printed perturbation equations contains a term that is not linear in the perturbative field. These issues currently prevent the result from being accepted as stated.

major comments (4)

[Sec. 5, Eq. (21)] Equation (21) is presented as the linear perturbation equation on a Reissner-Nordström background. For the model under study, U(δφ)=1/2 β (δφ)², so ∇_μ∇_ν U(δφ) = β[(∇_μ δφ)(∇_ν δφ)+δφ ∇_μ∇_ν δφ], which is O(δφ²). The term 4 R^μν_(0) ∇_μ∇_ν U(δφ) therefore does not belong in a linearized equation and should be dropped if the stated O(δφ²) truncation is used. Since this equation is the basis for locating the charged bifurcation curve (the green 'existence line' in Fig. 5), the printed derivation of that line is not valid as it stands. Please correct the equation, recompute the zero mode, and confirm whether the existence line in Fig. 5 changes; if the term was in fact omitted in the numerical code, state that explicitly.
[Sec. 3, Figs. 2, 5-9] The central multi-branch and disconnected-branch claims rely on identifying a horizon as the point where both h(r) and f(r) drop below 10^{-6}. The paper itself concedes that 'due to finite numerical precision, the functions cannot be made to reach exactly zero.' No convergence study, threshold-variation test, or grid-resolution analysis is provided. Near the branch crossings, the minimal-radius connection, and the purported critical endpoints, the separation between branches can be comparable to the uncertainty introduced by this threshold, and a solution whose minimum of f is slightly above 10^{-6} would be misclassified as horizonless. A resolution study varying the threshold (e.g. 10^{-6}, 10^{-8}, 10^{-10}) and the scanning grid in (M,φ0) is essential to establish that the branch topology is not a numerical artifact. Without it, the paper's headline claim of a previously unreported
[Sec. 5, Fig. 5 and Figs. 6-9] The 'purple dashed line' of interior critical endpoints is a central structural element: branches terminate on it and constant-mass/charge contours that intersect it split into disconnected families. Yet the paper gives no equation, no physical characterization, and no independent numerical test of these endpoints. They could be genuine critical points of the solution family, but they could also be places where the inward integration from the truncated asymptotic data stops finding a horizon due to the 10^{-6} criterion or to the grid spacing. Please define the purple dashed line in terms of the solution (e.g. as a turning point of r_+ or φ0 along a branch) and show that its location converges as resolution and threshold are improved.
[Sec. 5, Eq. (22)-(23) and Fig. 10] The abstract states that 'all charged hairy solutions obey the first law exactly,' but the verification consists of two numerical fits: M(S) at fixed Q=1.0 and M(Q) at fixed S=325.5. The agreement shown is a useful check, but it is neither exact nor a proof for all solutions. In addition, the exclusion of the scalar charge from the first law relies on the vanishing-conjugate argument taken from the authors' previous work [26], and this argument is not re-derived for the charged case. Please either soften the claim to 'numerically verified for representative solutions' or provide a derivation of the first-law form that covers the full charged solution space.

minor comments (4)

[General text] The manuscript contains several garbled passages and repeated blocks, e.g. the paragraph beginning 'We can form dimensionless parameters...' appears twice with 'uni03BC' artifacts, and a portion of text from Phys. Rev. D 103, 084043 (2021) appears verbatim. The text needs thorough cleaning.
[Fig. 2] The figure caption says 'the left panel' and 'right panel' but the panels are not labeled clearly in the body text; the reader has to infer which curve corresponds to the newly found negative-φ branch. Please add explicit panel labels and a consistent description.
[Sec. 2, Eq. (3)-(9)] The transition from the general action (3) to the specific model (9) is made through an algebraic equation for φ, but the relation φ=2αR/(1−2αβ E_GB) is stated without showing the intermediate steps. Since this relation is needed to justify the action (9), a brief derivation or a reference to the corresponding section of [26] would help.
[Sec. 4, text near Fig. 3] The text says 'the existence of multiple distinct scalar parameter values corresponding to a single mass indicates the multibranched structure.' This is true, but the multivaluedness is also visible in Fig. 3 only for a limited mass range; a quantitative statement of the mass interval in which three branches coexist would strengthen the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multi-branch and charged branch claims are new numerical constructions, not reductions to fitted inputs or self-citation chains.

full rationale

I walked the claimed derivation chain. The theory action (3)/(9) is introduced as a review of the authors' prior work [26], but the central new claims—the negative-hair branch crossing the Schwarzschild curve, the minimal-radius connection of two branches, and the charged disconnected branches—are not imported from [26]; they are obtained in this paper by numerical integration of the field equations using the asymptotic data (12)/(20) and the horizon criterion h,f < 10^-6. That numerical pipeline is a construction, not a fit of the claimed output. The bifurcation analysis in Eq. (16)/(21) is a standard linear zero-mode computation; the existence line in Fig. 5 is then used as a background for locating hairy branches, and the branch topology itself comes from the numerical solution families, so there is no definitional equivalence between input and result. The paper's first-law 'verification' is a self-consistency check: the Wald entropy (15), temperature, and potential are computed from the numerical solutions, and the Maxwell relations (23) are compared with derivatives of fitted M(S) and M(Q). This is not a prediction forced by construction; it is a post-hoc numerical consistency test and could fail if the solutions or thermodynamic quantities were inaccurate. The admitted limitation 'due to finite numerical precision, the functions cannot be made to reach exactly zero' is a numerical robustness concern and a correctness risk, not circularity; the absence of convergence tests does not make the branch structure equal to its inputs. The apparent O(delta phi^2) term in Eq. (21) is an internal consistency issue in the perturbation equation, not a circular step. Self-citation to [26] is present and the model choice follows that work, but [26] is a published, externally checkable numerical study and is not being used to assert the new branch structure; it supplies the framework and the previously known positive branch, while the new branches are constructed here. No load-bearing step reduces to its own input, so the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the theory framework and entropy formalism taken from the authors' prior work [26], on the standard scalarization assumption that linear zero modes mark bifurcations, on discarding the growing Yukawa mode, on asserting that the scalar charge's conjugate vanishes, and on a single benchmark point (mu, beta) = (0.01, 50). No new entities are invented. The only numeric fits are the M(S) and M(Q) functions used in the first-law check, whose forms are unreported.

free parameters (3)

M(S) fit used in the first Maxwell-relation check = unspecified function; Q = 1.0, S in [250, 375]
The claimed verification (dM/dS)_Q = T differentiates a curve fitted to the same numerical solutions whose temperatures are already computed; fit form, order, and residuals are not reported, so the agreement is not quantified.
M(Q) fit used in the second Maxwell-relation check = unspecified function; S = 325.5, Q in [0, 2.5]
Same issue as above: the reported 'excellent match' (Fig. 10) has no error budget or fit specification.
horizon detection thresholds = h < 10^-6 and f < 10^-6 simultaneously
Ad hoc numerical definition of the event horizon (Sec. 3); no sensitivity analysis shows the claimed branch topology is invariant under tightening or loosening the threshold.

assumptions (5)

domain assumption The zero mode of the linearized scalar equation around Schwarzschild/RN marks the bifurcation point where hairy branches emerge
Secs. 4-5; the existence line in Fig. 5 is obtained this way. Standard scalarization criterion, but an assumption about the global solution structure.
domain assumption The growing Yukawa mode e^{+mu r / sqrt(3)} is unphysical and discarded; only the decaying mode (11) is imposed at infinity
Sec. 3; standard for massive fields, but rules out boundary conditions that could host other branches.
domain assumption The scalar charge phi_0 has a vanishing thermodynamic conjugate and is excluded from the first law
Sec. 4: 'the massive scalar charge phi_0 cannot enter the first law... its thermodynamic conjugate variable corresponds to a divergent solution component and must consequently be constrained to vanish.' Assumed rather than derived at that point; no citation given in text.
standard math Wald entropy formula gives S = pi r_+^2 (1 + phi_+) + 2 pi beta phi_+^2
Sec. 3, Eq. (15), referencing Wald [29]; standard formalism.
domain assumption Results at (mu, beta) = (0.01, 50) are representative of the theory's behavior elsewhere in parameter space
All figures use these couplings; the abstract's general statements about multi-branch and disconnected structures presuppose this generality, which is asserted but never demonstrated.

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Cite this review

Pith. "Pith review of Scalarization of Charged Black Hole in Gauss-Bonnet Extended Starobinsky-Maxwell Gravity." pith.science (2026). https://pith.science/paper/IQVNKRB5

@misc{pith2026260714620,
  author       = {Pith},
  title        = {Pith review of: Scalarization of Charged Black Hole in Gauss-Bonnet Extended Starobinsky-Maxwell Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IQVNKRB5}},
  note         = {Machine review of arXiv:2607.14620}
}

read the original abstract

We re-examine spontaneous scalarization of black holes within Starobinsky gravity supplemented by the Gauss-Bonnet invariant. A novel feature is uncovered: scalarized solutions split into two smooth branches that connect at a common minimal horizon radius, a multi-branch structure previously unreported for static, spherically symmetric scalarization. Extending the theory with a Maxwell field, we find that this multi-branch structure persists, while a new additional disconnected branch emerges within certain parameter ranges. We thoroughly analyse the transition from a single smooth branch to two disconnected branches. Lastly, we verify, employing the standard Maxwell thermodynamic relations, that all charged hairy solutions obey the first law exactly.

Figures

Figures reproduced from arXiv: 2607.14620 by the authors.

**Figure 1.** Metric functions h(r) and f(r) for a scalarized hairy black hole solutions with asymptotic parameters (M, ϕ0) = (5.0, 0.3967) and theory couplings (µ, β) = (0.01, 50). The event horizon at r+ = 9.487 is identified by the simultaneous vanishing of h(r) and f(r). An obvious distinction from the Schwarzschild black hole, where h(r) = f(r), is that here h(r) ̸= f(r) for the hairy case. 7 [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 3.** FIG. 3. The left panel shows the range of the allowed mass G. 3. dependence FIG. 3. Fig. 2: The mass as a function of horizon radius [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** FIG. 4. The first two graphs show the (dimensionless) entropy/mass relation for the new hairy black holes, indistinguishable from that shows the numerical deviation of the new black hole entropy from the Schwarzschild where ΔS=S ≡ ðS SÞ=S shows the numerical deviation of the new black hole entropy from the Schwarzsch [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (8 more)

**Figure 3.** Figure 3: The figure illustrates that the scalar quantities ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png]

**Figure 4.** Figure 4: The left panel compares the entropy/mass relation for the hairy black hole with ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: This figure shows the existence domain in the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: This figure investigates the charge dependence of hairy black hole solutions evaluated at four [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: This figure presents hairy black hole solutions for three distinct fixed charge values and illustrates [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Summary of hairy black hole solutions in the mass-horizon radius plane for five selected charge [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: This figure depicts the coalescence process of two disconnected hairy black hole solution branches. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: The left panel verifies the first Maxwell relation. The green dashed line represents ( [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

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Works this paper leans on

31 extracted references · 25 linked inside Pith

[26]

H. S. Liu, H. Lu, Z. Y. Tang and B. Wang, Phys. Rev. D103, no.8, 084043 (2021) doi:10.1103/PhysRevD.103.084043 [arXiv:2004.14395 [gr-qc]]

arXiv 2021
[1]

Antoniou, A

G. Antoniou, A. Bakopoulos and P. Kanti, Phys. Rev. Lett.120, no.13, 131102 (2018) doi:10.1103/PhysRevLett.120.131102 [arXiv:1711.03390 [hep-th]]

arXiv 2018
[2]

H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Phys. Rev. Lett.120, no.13, 131104 (2018) doi:10.1103/PhysRevLett.120.131104 [arXiv:1711.02080 [gr-qc]]

arXiv 2018
[3]

D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett.120, no.13, 131103 (2018) doi:10.1103/PhysRevLett.120.131103 [arXiv:1711.01187 [gr-qc]]

arXiv 2018
[4]

C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, Phys. Rev. Lett.121, no.10, 101102 (2018) doi:10.1103/PhysRevLett.121.101102 [arXiv:1806.05190 [gr-qc]]

arXiv 2018
[5]

P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, Class. Quant. Grav.36, no.13, 134002 (2019) [erratum: Class. Quant. Grav.37, no.4, 049501 (2020)] doi:10.1088/1361-6382/ab23a1 [arXiv:1902.05079 [gr-qc]]

arXiv 2019
[6]

H. S. Liu and L. Zhang, JHEP10, 067 (2024) doi:10.1007/JHEP10(2024)067 [arXiv:2407.08208 [gr-qc]]

arXiv 2024
[7]

Li and H

Z. Li and H. S. Liu, Phys. Rev. D112, no.6, 064001 (2025) doi:10.1103/d2gl-2y6q [arXiv:2505.02340 [gr-qc]]

arXiv 2025

Show all 31 references

[8]

Li and H

Z. Li and H. S. Liu, Phys. Lett. B879, 140661 (2026) doi:10.1016/j.physletb.2026.140661 [arXiv:2508.01790 [gr-qc]]. 20

2026
[9]

Z. Li, H. S. Liu and H. Lu, Phys. Rev. D113, no.4, 044032 (2026) doi:10.1103/9h7h-686p [arXiv:2509.20755 [gr-qc]]

2026
[10]

Zhang and H

L. Zhang and H. S. Liu, [arXiv:2603.18562 [gr-qc]]
[11]

K. S. Stelle, Phys. Rev. D16, 953-969 (1977) doi:10.1103/PhysRevD.16.953

1977 doi
[12]

W. Li, W. Song and A. Strominger, JHEP04, 082 (2008) doi:10.1088/1126- 6708/2008/04/082 [arXiv:0801.4566 [hep-th]]

2008 arXiv
[13]

E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett.102, 201301 (2009) doi:10.1103/PhysRevLett.102.201301 [arXiv:0901.1766 [hep-th]]

2009 arXiv
[14]

Liu and Y

Y. Liu and Y. w. Sun, JHEP04, 106 (2009) doi:10.1088/1126-6708/2009/04/106 [arXiv:0903.0536 [hep-th]]

2009 arXiv
[15]

Deser, H

S. Deser, H. Liu, H. Lu, C. N. Pope, T. C. Sisman and B. Tekin, Phys. Rev. D83, 061502 (2011) doi:10.1103/PhysRevD.83.061502 [arXiv:1101.4009 [hep-th]]

2011 arXiv
[16]

Lu and C

H. Lu and C. N. Pope, Phys. Rev. Lett.106, 181302 (2011) doi:10.1103/PhysRevLett.106.181302 [arXiv:1101.1971 [hep-th]]

2011 arXiv
[17]

E. A. Bergshoeff, O. Hohm, J. Rosseel and P. K. Townsend, Phys. Rev. D83, 104038 (2011) doi:10.1103/PhysRevD.83.104038 [arXiv:1102.4091 [hep-th]]

2011 arXiv
[18]

Porrati and M

M. Porrati and M. M. Roberts, Phys. Rev. D84, 024013 (2011) doi:10.1103/PhysRevD.84.024013 [arXiv:1104.0674 [hep-th]]

2011 arXiv
[19]

R. C. Myers and B. Robinson, JHEP08, 067 (2010) doi:10.1007/JHEP08(2010)067 [arXiv:1003.5357 [gr-qc]]

2010 arXiv
[20]

Oliva and S

J. Oliva and S. Ray, Class. Quant. Grav.27, 225002 (2010) doi:10.1088/0264- 9381/27/22/225002 [arXiv:1003.4773 [gr-qc]]

2010 arXiv
[21]

Bueno and P

P. Bueno and P. A. Cano, Phys. Rev. D94, no.10, 104005 (2016) doi:10.1103/PhysRevD.94.104005 [arXiv:1607.06463 [hep-th]]

2016 arXiv
[22]

Y. Z. Li, H. S. Liu and H. Lu, JHEP02, 166 (2018) doi:10.1007/JHEP02(2018)166 [arXiv:1708.07198 [hep-th]]

2018 arXiv
[23]

A. A. Starobinsky, Phys. Lett. B91, 99-102 (1980) doi:10.1016/0370-2693(80)90670-X

1980 doi
[24]

Nelson, Phys

W. Nelson, Phys. Rev. D82, 104026 (2010) doi:10.1103/PhysRevD.82.104026 [arXiv:1010.3986 [gr-qc]]. 21

2010 arXiv
[25]

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

2015 arXiv
[27]

Z. H. Yang, G. Fu, X. M. Kuang and J. P. Wu, Eur. Phys. J. C82, no.10, 868 (2022) doi:10.1140/epjc/s10052-022-10834-8 [arXiv:2112.15052 [gr-qc]]

2022 arXiv
[28]

Z. Liu, Z. Li, L. Liang, S. Li and H. Yu, Phys. Rev. D110, no.12, 124052 (2024) doi:10.1103/PhysRevD.110.124052 [arXiv:2410.14108 [gr-qc]]

2024 arXiv
[29]

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
[30]

G. Y. Lu, M. N. Yang and H. Lu, JHEP11, 066 (2025) doi:10.1007/JHEP11(2025)066 [arXiv:2508.14158 [hep-th]]

2025
[31]

Astefanesei, C

D. Astefanesei, C. Herdeiro, A. Pombo and E. Radu, JHEP10, 078 (2019) doi:10.1007/JHEP10(2019)078 [arXiv:1905.08304 [hep-th]]. 22

2019 arXiv

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Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [26]

H. S. Liu, H. Lu, Z. Y. Tang and B. Wang, Phys. Rev. D103, no.8, 084043 (2021) doi:10.1103/PhysRevD.103.084043 [arXiv:2004.14395 [gr-qc]]

arXiv 2021

[2] [1]

Antoniou, A

G. Antoniou, A. Bakopoulos and P. Kanti, Phys. Rev. Lett.120, no.13, 131102 (2018) doi:10.1103/PhysRevLett.120.131102 [arXiv:1711.03390 [hep-th]]

arXiv 2018

[3] [2]

H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Phys. Rev. Lett.120, no.13, 131104 (2018) doi:10.1103/PhysRevLett.120.131104 [arXiv:1711.02080 [gr-qc]]

arXiv 2018

[4] [3]

D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett.120, no.13, 131103 (2018) doi:10.1103/PhysRevLett.120.131103 [arXiv:1711.01187 [gr-qc]]

arXiv 2018

[5] [4]

C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, Phys. Rev. Lett.121, no.10, 101102 (2018) doi:10.1103/PhysRevLett.121.101102 [arXiv:1806.05190 [gr-qc]]

arXiv 2018

[6] [5]

P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, Class. Quant. Grav.36, no.13, 134002 (2019) [erratum: Class. Quant. Grav.37, no.4, 049501 (2020)] doi:10.1088/1361-6382/ab23a1 [arXiv:1902.05079 [gr-qc]]

arXiv 2019

[7] [6]

H. S. Liu and L. Zhang, JHEP10, 067 (2024) doi:10.1007/JHEP10(2024)067 [arXiv:2407.08208 [gr-qc]]

arXiv 2024

[8] [7]

Li and H

Z. Li and H. S. Liu, Phys. Rev. D112, no.6, 064001 (2025) doi:10.1103/d2gl-2y6q [arXiv:2505.02340 [gr-qc]]

arXiv 2025