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REVIEW 4 major objections 5 minor 3 cited by

Non-oscillatory gravitational quasinormal modes of Reissner-Nordstr\"om-de Sitter spacetime

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Black holes with charge and a cosmological constant host a purely imaginary branch of quasinormal modes that sets the late-time exponential decay.

desk verdict First computation of the de Sitter branch for gravitational QNMs of Reissner-Nordström-de Sitter is plausible and likely correct, but the paper overclaims the tail connection and has a misprinted key formula. read the letter →

arxiv 2506.09829 v1 pith:QI6ZBSDZ submitted 2025-06-11 gr-qc

Z. Stuchlík , A. Zhidenko This is my paper
classification gr-qc PACS 04.30.Nk04.50.+h
keywords quasinormalmodesReissner-Nordström-deSitterpurelyimaginarydebranchgravitationalperturbationsexponentialtailscontinuedfractionscosmologicalconstant
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 asks whether the gravitational and electromagnetic perturbations of a Reissner-Nordström-de Sitter black hole contain the non-oscillatory 'de Sitter branch' of quasinormal modes previously known only for test fields. It answers yes, and shows that these purely imaginary modes reduce to the modes of empty de Sitter space in the small-mass limit, matching the universal formula $\omega_n = \omega_n^{(dS)}(1 - M/r_c + O((M/r_c)^2))$. The paper computes the frequencies with a continued-fraction method and demonstrates by time-domain integration that these modes generate the exponential late-time tails of the perturbation. If correct, this completes the quasinormal spectrum of this classical black-hole solution and gives a parameter-free prediction for the late-time decay of small charged black holes in a universe with a cosmological constant.

What carries the argument

The central object is the purely imaginary quasinormal mode, a solution of the master wave equation (3) whose frequency is purely imaginary, making the mode non-oscillatory and exponentially decaying under the quasinormal boundary conditions of outgoing waves at the cosmological horizon and ingoing waves at the event horizon. The argument is carried by the universal scaling law $\omega_n = \omega_n^{(dS)}(1 - M/r_c + O((M/r_c)^2))$, which is tested numerically for the effective potentials (5) and (6). The computational machinery is a frequency-domain continued-fraction construction: the radial function is expanded as a Frobenius series, producing seven- and nine-term recurrence relations that are reduced to three terms by Gaussian elimination, and the quasinormal frequencies are the solutions of the resulting infinite continued-fraction equation, with a convergence-acceleration step applied for the purely imaginary modes. The results are cross-checked by a light-cone time-domain integration scheme that measures the exponential decay rate of the perturbation.

What would settle it

Recompute the lowest purely imaginary mode for, say, $\ell=2$, $r_i=r_h/2$, and $r_c=10 r_h$ with an independent high-precision method, such as direct numerical integration of the radial equation or a spectral collocation scheme, and compare with the continued-fraction value; any mismatch beyond the stated precision would refute the claimed spectrum. Alternatively, measure the late-time decay in a time-domain evolution: if the logarithmic decay rate does not equal the imaginary part of the least-damped purely imaginary mode, the attribution of the exponential tail to this branch fails.

Watch

Extended reading notes

Core claim

The paper's discovery is that the coupled gravitational and electromagnetic perturbations of the Reissner-Nordström-de Sitter black hole carry a de Sitter branch of purely imaginary quasinormal frequencies, in addition to the well-known complex black-hole branch. These modes are deformations of the modes of empty de Sitter space and obey the universal law $\omega_n = \omega_n^{(dS)}(1 - M/r_c + O((M/r_c)^2))$ in the small-black-hole regime, a formula this paper shows holds for both the '+' and '−' effective potentials. The numerical computation indicates the approximation stays accurate even when the black-hole radius is comparable to the cosmological horizon. Time-domain integration confirms that the exponential asymptotic tails of the perturbation are produced by these purely imaginary modes, establishing them as the least-damped part of the spectrum for small black holes even though they are only weakly excited by typical initial data.

Load-bearing premise

The numerical results assume that the continued-fraction recurrence reduction and the time-domain fitting converge to the true quasinormal frequencies, but the paper reports no convergence checks, truncation orders, or error estimates; if either numerical scheme fails to converge, the claimed frequencies and the fitted universal formula do not follow.

Editorial extensions

If this is right

  • The quasinormal spectrum of Reissner-Nordström-de Sitter is now known to have two branches, and for small black holes the purely imaginary de Sitter branch is the least-damped set of modes.
  • The universal law (22) applies to the actual gravitational and electromagnetic perturbations, so the late-time decay rate of a small charged black hole is fixed without free parameters.
  • Late-time gravitational-wave tails from these black holes are exponential rather than power-law, with the decay rate set by the imaginary part of the least-damped purely imaginary mode.
  • As the black-hole radius grows, overtones of the complex black-hole branch successively take over as the least-damped modes, making the dominant damping rate a non-monotonic function of the black-hole radius.
  • In the near-extremal limit the purely imaginary mode persists with a nonzero decay rate, because the black-hole mass always stays below the horizon radius, so the exponential-tail phase never disappears.

Reading between the lines

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

  • The universal law likely holds for any spherically symmetric black hole with a cosmological horizon, independent of the underlying gravitational theory, since the paper's derivation is metric-independent; this extension is testable by computing the de Sitter branch for other backgrounds.
  • Observational searches for the imprint of a cosmological constant on gravitational-wave signals could target an anomalous exponential-decay phase after the ringdown of a charged black hole rather than fitting a single complex mode, because the de Sitter branch is weakly excited.
  • The near-linear dependence of the modes on $r_i/r_h$ seen in the paper's Fig. 2 may reflect a hidden symmetry of the perturbation equations, which if identified could yield a closed-form expression for the whole de Sitter branch.
  • A dedicated high-precision computation in the near-extremal charge limit $r_i \to r_h$ would test whether the fitted formula (23) remains valid or whether the recurrence reduction changes character there.
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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 / 5 minor

Summary. The paper studies the quasinormal spectrum of gravitational and electromagnetic perturbations of the Reissner-Nordström-de Sitter black hole, focusing on the purely imaginary (non-oscillatory) de Sitter branch. Using the Leaver continued-fraction method with Nollert acceleration and a time-domain Gundlach-Price-Pullin integrator, the authors report that this branch exists for coupled gravitational-electromagnetic perturbations, reduces to empty de Sitter space modes in the small-black-hole limit, and obeys the approximate formula ω_n ∝ 1 - r_i/(2 r_c) for r_h ≪ r_c, which they interpret as confirming the universal law ω_n = ω_n^{(dS)}(1 - M/r_c + O((M/r_c)^2)). They further claim that these modes are responsible for the late-time exponential tails of the perturbation. The paper also describes the interplay between this de Sitter branch and the complex black-hole branch, including the overtaking of the least-damped mode as the black hole grows.

Significance. If the central claims are correct, the paper fills a gap in the quasinormal-mode literature by extending the purely imaginary de Sitter branch, previously known for test fields and for Schwarzschild-de Sitter, to the coupled gravitational-electromagnetic sector of Reissner-Nordström-de Sitter. The result that these modes control late-time exponential tails and that they satisfy the universal small-black-hole formula would be physically relevant for late-time gravitational-wave behavior and for strong cosmic censorship considerations. The authors make appropriate use of standard, well-established numerical tools (Leaver method, Nollert convergence acceleration, and time-domain integration), and the qualitative picture they present is plausible. However, the manuscript as submitted lacks the numerical tables, convergence checks, and time-domain extraction needed to substantiate these quantitative claims, and the approximate formula Eq. (23) contains an internal inconsistency. For these reasons the paper cannot yet be accepted in its present form.

major comments (4)
  1. [Sec. III A and III B; Figs. 2-4] The manuscript provides no numerical tables, no truncation orders, no convergence checks, and no error estimates for the Leaver/Nollert continued-fraction calculations or for the time-domain integration. This is load-bearing because the central existence claim and the quantitative agreement with Eq. (22) rest entirely on these numerical results, and purely imaginary modes are precisely the regime where continued-fraction solutions can be contaminated by spurious roots unless the recurrence reduction and truncation are carefully controlled. Please include representative numerical values of ω_n for a few parameter sets, the number of continued-fraction terms used, and a convergence check (e.g., stability of the root against increasing truncation order).
  2. [Sec. IV B, Eq. (23)] Equation (23) as written is internally inconsistent and cannot serve as the evidence for the universal law. It states ω_n ∝ 1 - r_i/(2 r_c), but the immediately following equality reads 1 - (r_h + r_i)/(2 r_c) + O((r_h/r_c)^2), which contains an additional -r_h/(2 r_c) term. For uncharged black holes (r_i = 0), Eq. (23) predicts no shift from the empty de Sitter value, contradicting Fig. 3 and Eq. (22), which give ω_n/ω_n^{(dS)} = 1 - M/r_c + ... = 1 - r_h/(2 r_c) + ... . The fitted formula must be corrected and supported by a table of numerical data and residuals; without that, the claim that the universal law holds for gravitational perturbations is not substantiated.
  3. [Sec. III B and Conclusions] The abstract and conclusions state that the purely imaginary modes are responsible for the exponential asymptotic tails, but the paper presents no time-domain profile, no Prony extraction, and no comparison of an extracted late-time decay rate with Im(ω) of the least-damped purely imaginary mode. The time-domain method is only described; its results are never displayed or quantified. Since the exponential-tail connection is a central claim, please include an actual time-domain evolution for at least one representative case, extract the late-time decay rate, and compare it with the frequency-domain value.
  4. [Sec. IV B] The extension of the universal law (22) from test fields to the coupled gravitational-electromagnetic spectrum is asserted on the basis of the fitted formula (23), but no independent analytic or semi-analytic confirmation is given. Given the problems with Eq. (23) noted above, the reader cannot distinguish a genuine confirmation of the universal law from a fit that merely reproduces the numerical data. Please provide either a derivation of the leading correction for the coupled system or at least a clear table comparing the numerical ω_n with the prediction ω_n^{(dS)}(1 - M/r_c) for several values of r_h/r_c and r_i/r_h.
minor comments (5)
  1. [Title] The title contains a typographical spacing error: "Reiss ner-Nordström" should read "Reissner-Nordström."
  2. [Sec. II] The phrase "multiple number" in the definition of λ should be "multipole number."
  3. [Sec. I] In the sentence beginning "As shown for scalar field perturbations in [42] and for gravitational perturbations in [52], in additional to the complex branch," the phrase "in additional to" should be "in addition to."
  4. [Fig. 4] The caption and axes of Fig. 4 are unclear: the top panel shows r_h Re(ω) as a function of r_h/r_c, and the bottom panel shows r_h Im(ω), but the text discusses transitions in the least-damped mode. Please clarify which curves correspond to the de Sitter branch and which to the black-hole branch in both panels, and explain the meaning of the dashed lines (the caption says they represent "the corresponding modes in the parametric region, when they are not the least damped").
  5. [Sec. IV] The paper states that ℓ = 2, 3, 4, ... are considered, but only ℓ = 2 results are displayed. If modes with higher ℓ were computed, showing at least a representative table or a statement of their behavior would strengthen the universality claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the gravitational de Sitter-branch frequencies are computed independently from the perturbation equations and compared with a prior test-field law; the main caveats are a fit-based 'confirmation' in Eq. (23) and an unsupported tail claim, not circular definition.

full rationale

The central calculation is not circular. Section III A solves the coupled axial/polar master equations (5)-(6) with quasinormal boundary conditions via a Frobenius series and Nollert-accelerated continued fractions; these frequencies are obtained directly from the perturbation equations, not from the universal law (22). The de Sitter-branch identification is anchored to Eq. (21), the known empty-de Sitter frequencies, and to prior Schwarzschild-de Sitter results (Refs. [42,52]) that are external to this paper's fitted values. Eq. (22) is cited from the authors' own Ref. [91], but that citation is a separate analytic result for test fields; it does not determine the coupled gravitational-electromagnetic frequencies computed here. The only self-referential episode is Eq. (23): the authors fit their numerical branch to 1 - ri/(2rc) and then state that this 'confirms' the universal law (22). That is a fitted-input confirmation, not an independent derivation, and as written Eq. (23) omits the rh/(2rc) first-order term that appears in the law it claims to confirm. This weakens the 'satisfy the universal analytic formula' claim but is not a definitional circularity, because the fitted data come from solving (3)-(6), not from Eq. (22). Separately, the paper asserts in the abstract and Sec. I that time-domain integration demonstrates that the purely imaginary modes are responsible for the exponential tails, but no time-domain profile, Prony extraction, or comparison of an extracted decay rate with Im(omega) is displayed; that is missing evidence rather than circularity. Overall, the central numerical result has independent content, so the circularity score is low.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central numerical results rest on standard QNM machinery plus one fitted linear approximation (23); the main unstated burden is numerical convergence and the domain assumption that the decoupled potentials describe the physical perturbations.

free parameters (1)
  • Linear coefficient in fitted approximation Eq. (23) = 1/(2r_c) in 1 - r_i/(2r_c)
    The numerical data for r_h << r_c are fitted to a linear function of r_i/r_c; the resulting coefficient matches the universal-law prediction, but it is derived from the same numerical frequencies.
assumptions (4)
  • domain assumption In the RNdS background, gravitational and electromagnetic perturbations decouple into axial and polar sectors with effective potentials (5) and (6), and the axial/polar spectra are identical due to a Darboux transformation.
    Imported from Kodama-Ishibashi and cited literature in Sec. II; if wrong, the computed 'gravitational' modes would not correspond to physical metric perturbations.
  • domain assumption Purely imaginary modes are true quasinormal modes satisfying the boundary conditions (14) and (15).
    The paper follows the standard QNM definition and refers to previous work [52] for justification that these modes satisfy ingoing conditions at the event horizon.
  • domain assumption The universal law (22) derived for test fields in [91] applies to the electromagnetic-gravitational sector.
    Used to interpret the fitted approximation (23) and to claim confirmation of the small-black-hole scaling.
  • standard math The Leaver series with Nollert acceleration converges for these purely imaginary modes.
    Standard numerical method, but no proof of convergence for this specific coupled system is given.

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Cite this review

Pith. "Pith review of Non-oscillatory gravitational quasinormal modes of Reissner-Nordstr\"om-de Sitter spacetime." pith.science (2026). https://pith.science/paper/QI6ZBSDZ

@misc{pith2026250609829,
  author       = {Pith},
  title        = {Pith review of: Non-oscillatory gravitational quasinormal modes of Reissner-Nordstr\"om-de Sitter spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QI6ZBSDZ}},
  note         = {Machine review of arXiv:2506.09829}
}
read the original abstract

Proper oscillation frequencies of black holes (quasinormal modes) of asymptotically de Sitter black holes were extensively studied, yet the non-oscillatory (purely imaginary) branch of modes of gravitational perturbations of the four-dimensional Reissner-Nordstr\"om-de Sitter solution was omitted in the literature. This branch of modes appears as deformations of the modes of empty de Sitter space. Here we find accurate numerical values of this branch of quasinormal modes and show that they are responsible for the exponential asymptotic tails.

Figures

Figures reproduced from arXiv: 2506.09829 by the authors.

Figure 1
Figure 1. FIG. 1. Dominant [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The lowest purely imaginary quasinormal modes for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Two least-damped [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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Reference graph

Works this paper leans on

100 extracted references · 15 canonical work pages · cited by 3 Pith papers

  1. [1]

    The de Sitter branch , consisting of purely imagi- nary modes, which are the least damped when the black hole is small ( rc ≫ rh), as the frequencies scale roughly with r−1 c

  2. [2]

    −” mode becomes one from the black-hole branch, followed by a similar transition for the ℓ = 2 “+

    The black-hole branch , consisting of complex modes that approach the Reissner-Nordström spectrum in the asymptotically flat spacetime limit. These modes become the least damped for large black holes, and in the near-extreme limit ( rh → rc), their frequencies scale with rc − rh. This leads to a peculiar and non-monotonic behavior of the least damped mode ...

  3. [3]

    K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999), arXiv:gr-qc/9909058

    arXiv 1999

  4. [4]

    Berti, V

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009), arXiv:0905.2975 [gr-qc]

    arXiv 2009

  5. [5]

    Nollert, Class

    H.-P. Nollert, Class. Quant. Grav. 16, R159 (1999)

    1999

  6. [6]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011), arXiv:1102.4014 [gr-qc]

    arXiv 2011

  7. [7]

    S. V. Bolokhov and M. Skvortsova, (2025), arXiv:2504.05014 [gr-qc]

    arXiv 2025

  8. [8]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    arXiv 2016

Show all 100 references
  1. [9]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc]

    2017 arXiv

  2. [10]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. Lett. 896, L44 (2020), arXiv:2006.12611 [astro-ph.HE]

    2020 arXiv

  3. [11]

    R. A. Konoplya, Phys. Lett. B 823, 136734 (2021), arXiv:2109.01640 [gr-qc]

    2021 arXiv

  4. [12]

    R. A. Konoplya and E. Abdalla, Phys. Rev. D 71, 084015 (2005), arXiv:hep-th/0503029

    2005 arXiv

  5. [13]

    Jusufi, Phys

    K. Jusufi, Phys. Rev. D 101, 084055 (2020), 7 arXiv:1912.13320 [gr-qc]

    2020 arXiv

  6. [14]

    Zhang, T

    C. Zhang, T. Zhu, and A. Wang, Phys. Rev. D 104, 124082 (2021), arXiv:2111.04966 [gr-qc]

    2021 arXiv

  7. [15]

    Hamil, A

    B. Hamil, A. Al-Badawi, and B. C. Lütfüoğlu, (2025), arXiv:2505.18611 [gr-qc]

    2025

  8. [16]

    B. C. Lütfüoğlu, Eur. Phys. J. C 85, 486 (2025), arXiv:2503.16087 [gr-qc]

    2025 arXiv

  9. [17]

    Zhidenko, Class

    A. Zhidenko, Class. Quant. Grav. 21, 273 (2004), arXiv:gr-qc/0307012

    2004 arXiv

  10. [18]

    Yoshida and T

    S. Yoshida and T. Futamase, Phys. Rev. D 69, 064025 (2004), arXiv:gr-qc/0308077

    2004 arXiv

  11. [19]

    R. A. Konoplya and A. Zhidenko, JHEP 06, 037 (2004), arXiv:hep-th/0402080

    2004 arXiv

  12. [20]

    Cardoso and J

    V. Cardoso and J. P. S. Lemos, Phys. Rev. D 67, 084020 (2003), arXiv:gr-qc/0301078

    2003 arXiv

  13. [21]

    O. J. Tattersall, Phys. Rev. D 98, 104013 (2018), arXiv:1808.10758 [gr-qc]

    2018 arXiv

  14. [22]

    Aragón, P

    A. Aragón, P. A. González, E. Papantonopoulos, and Y. Vásquez, JHEP 08, 120 (2020), arXiv:2004.09386 [gr- qc]

    2020 arXiv

  15. [23]

    Hintz and Y

    P. Hintz and Y. Xie, J. Math. Phys. 63, 011509 (2022), arXiv:2105.02347 [gr-qc]

    2022 arXiv

  16. [24]

    M. S. Churilova, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 105, 084003 (2022), arXiv:2108.04858 [gr-qc]

    2022 arXiv

  17. [25]

    Roussev, Gen

    A. Roussev, Gen. Rel. Grav. 54, 80 (2022)

    2022

  18. [26]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 76, 084018 (2007), [Erratum: Phys.Rev.D 90, 029901 (2014)], arXiv:0707.1890 [hep-th]

    2007 arXiv

  19. [27]

    Dyatlov, Annales Henri Poincare 13, 1101 (2012), arXiv:1101.1260 [math.AP]

    S. Dyatlov, Annales Henri Poincare 13, 1101 (2012), arXiv:1101.1260 [math.AP]

    2012 arXiv

  20. [28]

    R. A. Konoplya, Phys. Rev. D 68, 124017 (2003), arXiv:hep-th/0309030

    2003 arXiv

  21. [29]

    Molina, Phys

    C. Molina, Phys. Rev. D 68, 064007 (2003), arXiv:gr- qc/0304053

    2003

  22. [30]

    Fernando, Int

    S. Fernando, Int. J. Mod. Phys. D 24, 1550104 (2015), arXiv:1508.03581 [gr-qc]

    2015 arXiv

  23. [31]

    Natario and R

    J. Natario and R. Schiappa, Adv. Theor. Math. Phys. 8, 1001 (2004), arXiv:hep-th/0411267

    2004 arXiv

  24. [32]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett. 103, 161101 (2009), arXiv:0809.2822 [hep-th]

    2009 arXiv

  25. [33]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 95, 104005 (2017), arXiv:1701.01652 [hep-th]

    2017 arXiv

  26. [34]

    Toshmatov and Z

    B. Toshmatov and Z. Stuchlík, Eur. Phys. J. Plus 132, 324 (2017), arXiv:1707.07419 [gr-qc]

    2017 arXiv

  27. [35]

    M. A. Cuyubamba, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 93, 104053 (2016), arXiv:1604.03604 [gr- qc]

    2016 arXiv

  28. [36]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 89, 024011 (2014), arXiv:1309.7667 [hep-th]

    2014 arXiv

  29. [37]

    R. A. Konoplya, A. Khrabustovskyi, J. Kříž, and A. Zhi- denko, (2025), arXiv:2501.16134 [gr-qc]

    2025 arXiv

  30. [38]

    Dubinsky, Mod

    A. Dubinsky, Mod. Phys. Lett. A 39, 2450108 (2024), arXiv:2404.18004 [gr-qc]

    2024 arXiv

  31. [39]

    Dubinsky and A

    A. Dubinsky and A. Zinhailo, Eur. Phys. J. C 84, 847 (2024), arXiv:2404.01834 [gr-qc]

    2024 arXiv

  32. [40]

    R. A. Konoplya and A. Zhidenko, Class. Quant. Grav. 40, 245005 (2023), arXiv:2309.02560 [gr-qc]

    2023 arXiv

  33. [41]

    R. A. Konoplya, Phys. Rev. D 103, 044033 (2021), arXiv:2012.13020 [gr-qc]

    2021 arXiv

  34. [42]

    Kanti and R

    P. Kanti and R. A. Konoplya, Phys. Rev. D 73, 044002 (2006), arXiv:hep-th/0512257

    2006 arXiv

  35. [43]

    Malik, Z

    Z. Malik, Z. Naturforsch. A 79, 1063 (2024)

    2024

  36. [44]

    Cardoso, J

    V. Cardoso, J. a. L. Costa, K. Destounis, P. Hintz, and A. Jansen, Phys. Rev. Lett. 120, 031103 (2018), arXiv:1711.10502 [gr-qc]

    2018 arXiv

  37. [45]

    O. J. C. Dias, F. C. Eperon, H. S. Reall, and J. E. Santos, Phys. Rev. D 97, 104060 (2018), arXiv:1801.09694 [gr- qc]

    2018 arXiv

  38. [46]

    O. J. C. Dias, H. S. Reall, and J. E. Santos, Class. Quant. Grav. 36, 045005 (2019), arXiv:1808.04832 [gr-qc]

    2019 arXiv

  39. [47]

    Y. Mo, Y. Tian, B. Wang, H. Zhang, and Z. Zhong, Phys. Rev. D 98, 124025 (2018), arXiv:1808.03635 [gr-qc]

    2018 arXiv

  40. [48]

    H. Liu, Z. Tang, K. Destounis, B. Wang, E. Pa- pantonopoulos, and H. Zhang, JHEP 03, 187 (2019), arXiv:1902.01865 [gr-qc]

    2019 arXiv

  41. [49]

    Hod, Int

    S. Hod, Int. J. Mod. Phys. D 29, 2042003 (2020), arXiv:2012.01449 [gr-qc]

    2020 arXiv

  42. [50]

    R. H. Price, Phys. Rev. D 5, 2419 (1972)

    1972

  43. [51]

    P. R. Brady, C. M. Chambers, W. Krivan, and P. Laguna, Phys. Rev. D 55, 7538 (1997), arXiv:gr- qc/9611056

    1997

  44. [52]

    P. R. Brady, C. M. Chambers, W. G. Laarakkers, and E. Poisson, Phys. Rev. D 60, 064003 (1999), arXiv:gr- qc/9902010

    1999

  45. [53]

    Molina, D

    C. Molina, D. Giugno, E. Abdalla, and A. Saa, Phys. Rev. D 69, 104013 (2004), arXiv:gr-qc/0309079

    2004 arXiv

  46. [54]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 106, 124004 (2022), arXiv:2209.12058 [gr-qc]

    2022 arXiv

  47. [55]

    R. A. Konoplya, Phys. Rev. D 109, 104018 (2024), arXiv:2401.17106 [gr-qc]

    2024 arXiv

  48. [56]

    Lopez-Ortega, Gen

    A. Lopez-Ortega, Gen. Rel. Grav. 38, 743 (2006), arXiv:gr-qc/0605022

    2006 arXiv

  49. [57]

    Lopez-Ortega, Gen

    A. Lopez-Ortega, Gen. Rel. Grav. 38, 1565 (2006), arXiv:gr-qc/0605027

    2006 arXiv

  50. [58]

    Jansen, Eur

    A. Jansen, Eur. Phys. J. Plus 132, 546 (2017), arXiv:1709.09178 [gr-qc]

    2017 arXiv

  51. [59]

    R. A. Konoplya, Phys. Lett. B 838, 137674 (2023), arXiv:2210.08373 [gr-qc]

    2023 arXiv

  52. [60]

    Kodama and A

    H. Kodama and A. Ishibashi, Prog. Theor. Phys. 110, 701 (2003), arXiv:hep-th/0305147

    2003 arXiv

  53. [61]

    Ishibashi and H

    A. Ishibashi and H. Kodama, Prog. Theor. Phys. 110, 901 (2003), arXiv:hep-th/0305185

    2003 arXiv

  54. [62]

    Kodama and A

    H. Kodama and A. Ishibashi, Prog. Theor. Phys. 111, 29 (2004), arXiv:hep-th/0308128

    2004 arXiv

  55. [63]

    Stuchlik and S

    Z. Stuchlik and S. Hledik, Acta Phys. Slov. 52, 363 (2002), arXiv:0803.2685 [gr-qc]

    2002 arXiv

  56. [64]

    E. W. Leaver, Proc. Roy. Soc. Lond. A 402, 285 (1985)

    1985

  57. [65]

    Gundlach, R

    C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 883 (1994), arXiv:gr-qc/9307009

    1994 arXiv

  58. [66]

    Nollert, Phys

    H.-P. Nollert, Phys. Rev. D 47, 5253 (1993)

    1993

  59. [67]

    Zhidenko, Phys

    A. Zhidenko, Phys. Rev. D 74, 064017 (2006), arXiv:gr- qc/0607133

    2006

  60. [68]

    R. A. Konoplya, D. Ovchinnikov, and B. Ahmedov, Phys. Rev. D 108, 104054 (2023), arXiv:2307.10801 [gr- qc]

    2023 arXiv

  61. [69]

    R. A. Konoplya, Phys. Rev. D 107, 064039 (2023), arXiv:2210.14506 [gr-qc]

    2023 arXiv

  62. [70]

    A. F. Zinhailo, Phys. Lett. B 853, 138682 (2024), arXiv:2403.06867 [gr-qc]

    2024 arXiv

  63. [71]

    S. V. Bolokhov, Phys. Rev. D 110, 024010 (2024), arXiv:2311.05503 [gr-qc]

    2024 arXiv

  64. [72]

    S. V. Bolokhov, Phys. Rev. D 109, 064017 (2024)

    2024

  65. [73]

    R. A. Konoplya and O. S. Stashko, Phys. Rev. D 111, 104055 (2025), arXiv:2408.02578 [gr-qc]

    2025 arXiv

  66. [74]

    R. A. Konoplya and O. S. Stashko, Phys. Rev. D 111, 084031 (2025), arXiv:2502.05689 [gr-qc]

    2025 arXiv

  67. [75]

    Dubinsky, Phys

    A. Dubinsky, Phys. Dark Univ. 46, 101657 (2024), arXiv:2405.08262 [gr-qc]. 8

    2024 arXiv

  68. [76]

    W.-L. Qian, K. Lin, C.-Y. Shao, B. Wang, and R.-H. Yue, Eur. Phys. J. C 82, 931 (2022), arXiv:2203.04477 [gr-qc]

    2022 arXiv

  69. [77]

    Aneesh, S

    S. Aneesh, S. Bose, and S. Kar, Phys. Rev. D 97, 124004 (2018), arXiv:1803.10204 [gr-qc]

    2018 arXiv

  70. [78]

    R. A. Konoplya, JCAP 07, 001 (2023), arXiv:2305.09187 [gr-qc]

    2023 arXiv

  71. [79]

    K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103, 124062 (2021), arXiv:2102.10679 [gr- qc]

    2021 arXiv

  72. [80]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400036 (2024), arXiv:2311.11650 [gr-qc]

    2024 arXiv

  73. [81]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400132 (2024), arXiv:2405.06390 [gr-qc]

    2024 arXiv

  74. [82]

    K. A. Bronnikov and R. A. Konoplya, Phys. Rev. D 101, 064004 (2020), arXiv:1912.05315 [gr-qc]

    2020 arXiv

  75. [83]

    Malik, EPL 147, 69001 (2024), arXiv:2410.04306 [gr- qc]

    Z. Malik, EPL 147, 69001 (2024), arXiv:2410.04306 [gr- qc]

    2024 arXiv

  76. [84]

    Malik, Int

    Z. Malik, Int. J. Theor. Phys. 63, 128 (2024), arXiv:2409.09872 [gr-qc]

    2024 arXiv

  77. [85]

    Malik, Int

    Z. Malik, Int. J. Mod. Phys. A 40, 2450132 (2025)

    2025

  78. [86]

    Jia, W.-D

    H.-L. Jia, W.-D. Guo, Q. Tan, and Y.-X. Liu, Phys. Rev. D 110, 064077 (2024), arXiv:2406.03929 [gr-qc]

    2024 arXiv

  79. [87]

    E. W. Leaver, Phys. Rev. D 41, 2986 (1990)

    1990

  80. [88]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009), arXiv:0812.1806 [hep-th]

    2009 arXiv

  81. [89]

    R. A. Konoplya, A. Spina, and A. Zhidenko, (2025), arXiv:2505.01128 [gr-qc]

    2025 arXiv

  82. [90]

    R. A. Konoplya and Z. Stuchlík, Phys. Lett. B 771, 597 (2017), arXiv:1705.05928 [gr-qc]

    2017 arXiv

  83. [91]

    R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C 80, 1049 (2020), arXiv:2003.01188 [gr-qc]

    2020 arXiv

  84. [92]

    Toshmatov, Z

    B. Toshmatov, Z. Stuchlík, B. Ahmedov, and D. Malafa- rina, Phys. Rev. D 99, 064043 (2019), arXiv:1903.03778 [gr-qc]

    2019 arXiv

  85. [93]

    R. A. Konoplya and A. Zhidenko, JCAP 11, 028 (2022), arXiv:2210.04314 [gr-qc]

    2022 arXiv

  86. [94]

    Zhu, S.-J

    Z. Zhu, S.-J. Zhang, C. E. Pellicer, B. Wang, and E. Abdalla, Phys. Rev. D 90, 044042 (2014), [Adden- dum: Phys.Rev.D 90, 049904 (2014)], arXiv:1405.4931 [hep-th]

    2014 arXiv

  87. [95]

    Zhang, J

    M. Zhang, J. Jiang, and Z. Zhong, Phys. Lett. B 806, 135523 (2020), arXiv:2005.12722 [gr-qc]

    2020 arXiv

  88. [96]

    R. D. B. Fontana, P. A. González, E. Papantonopou- los, and Y. Vásquez, Phys. Rev. D 103, 064005 (2021), arXiv:2011.10620 [gr-qc]

    2021 arXiv

  89. [97]

    X.-Y. Nan, J. Tan, and J. Jiang, Eur. Phys. J. C 83, 424 (2023)

    2023

  90. [98]

    Chrysostomou, A

    A. Chrysostomou, A. S. Cornell, A. Deandrea, H. Noshad, and S. C. Park, (2023), arXiv:2310.07311 [gr-qc]

    2023 arXiv

  91. [99]

    Chrysostomou, A

    A. Chrysostomou, A. S. Cornell, A. Deandrea, and S. C. Park, (2025), arXiv:2501.14834 [gr-qc]

    2025 arXiv

  92. [100]

    Liu and R.-D

    P. Liu and R.-D. Zhu, JHEP 06, 015 (2025), arXiv:2412.18359 [hep-th]

    2025 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.