Pith. sign in

REVIEW 2 major objections 5 minor 6 cited by

Gravitational quasinormal modes of the Hayward spacetime

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Increasing the quantum parameter gamma shifts the axial gravitational quasinormal spectrum of the Hayward spacetime to higher frequencies and weaker damping, with the first overtone affected most.

desk verdict First gravitational QNM spectrum for Hayward with a nice analytic expansion; numbers are cross-checked and plausible, but the perturbation equation rests on an unproven decoupling of anisotropic-fluid axial modes. read the letter →

arxiv 2508.19989 v1 pith:XX3ABDDG submitted 2025-08-27 gr-qc

S. V. Bolokhov , Milena Skvortsova This is my paper · ORCID
classification gr-qc MSC 83C5783C3581Q20 PACS 04.30.-w04.50.Kd
keywords quasinormalmodesHaywardspacetimeregularblackholesaxialgravitationalperturbationsasymptoticallysafegravityWKBmethodPadéapproximantsringdownovertones
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

The paper sets out to determine the gravitational ringdown spectrum of the Hayward regular black hole, a geometry that also serves as an effective quantum-corrected solution in asymptotically safe gravity. Using sixth- and eighth-order WKB formulas with Padé resummation, cross-checked by time-domain integration with Prony extraction, it computes axial gravitational quasinormal modes for multipoles $\ell=2,3$ and overtones $n=0,1$. The central finding is that raising the quantum parameter $\gamma$ increases the oscillation frequency and lowers the damping rate at every mode studied, so the ringdown becomes both faster and longer-lived; the first overtone responds more strongly than the fundamental mode. The paper also derives an analytic expansion of the frequencies beyond the eikonal limit. If the calculation is right, it provides the first systematic gravitational-perturbation spectrum for this geometry and singles out subdominant overtones as the most sensitive probe of near-horizon quantum corrections.

What carries the argument

The load-bearing object is the effective axial potential of Eq. (5), a single-barrier Regge-Wheeler-type potential built from the Hayward metric function $f(r)=1-\frac{2Mr^2}{r^3+2Ml^2}$, written in asymptotically safe variables as $f(r)=1-\frac{2r^2/M^2}{r^3/M^3+\gamma}$. As $\gamma$ grows, the barrier becomes higher and its maximum moves closer to the horizon in the radial coordinate, which suppresses transmission through the barrier and thereby raises oscillation frequencies while lowering damping. The quantitative machinery consists of the sixth-order WKB condition with Padé resummation, the double-null time-domain integration scheme, and the inverse-multipole expansion in $\kappa=\ell+1/2$ that yields the beyond-eikonal formula (17).

What would settle it

A full linearized calculation retaining all anisotropic-fluid perturbations coupled to the axial system: if the resulting effective potential differs from Eq. (5) at order $\gamma$, the reported spectrum is not the complete gravitational ringdown. On the numerical side, a Leaver-type continued-fraction computation for $\gamma=1.18$, $\ell=2$, $n=1$ that disagrees with $0.396187-0.074213i$ by more than the paper's quoted sub-percent agreement would undercut the accuracy claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a computed gravitational ringdown spectrum for the Hayward spacetime: axial gravitational perturbations obey the Regge-Wheeler-like master equation $\frac{d^2\Psi}{dr_*^2}+(\omega^2-V(r))\Psi=0$ with effective potential $V(r)=f(r)\left(\frac{2f(r)}{r^2}-\frac{f'(r)}{r}+\frac{(\ell+2)(\ell-1)}{r^2}\right)$, where $f(r)=1-\frac{2Mr^2}{r^3+2Ml^2}$. Across $\ell=2,3$ and $n=0,1$, raising $\gamma$ from $0$ toward the extremal value $1.18$ increases $\mathrm{Re}\,\omega$ by roughly six to seven percent and reduces $|\mathrm{Im}\,\omega|$ by roughly fifteen to seventeen percent, so the ringdown oscillates faster and damps more slowly. The effect is systematically larger for the first overtone than for the fundamental mode, matching the outburst of overtones reported earlier for test fields. Two independent methods, higher-order WKB with Padé approximants and time-domain integration with Prony extraction, agree to fractions of a percent for the fundamental mode and to about a percent in the imaginary part for the first overtone, and an analytic expansion beyond the eikonal limit reproduces the numerical spectrum within about one percent for $\ell=3$ except near extremality.

Load-bearing premise

The spectrum follows from the assumption that anisotropic-fluid fluctuations along the anisotropy direction decouple from the axial gravitational sector; if those fluctuations couple to the perturbation functions, the master equation, the effective potential, and all derived frequencies change.

Editorial extensions

If this is right

  • Larger $\gamma$ means the Hayward black hole rings at higher frequency and with less damping; between $\gamma=0$ and $\gamma=1.18$ the real part rises by roughly six percent and the damping falls by fifteen to seventeen percent for the modes tabulated.
  • Subdominant overtones amplify the quantum effect: the first overtone responds to $\gamma$ more strongly than the fundamental, so overtone measurements are the natural observational target for near-horizon quantum structure.
  • The eikonal photon-sphere correspondence holds for $\ell\ge2$ at small $\gamma$, and the analytic expansion (17) is accurate to about one percent for $\ell=3$, so the ringdown can be understood geometrically in that regime.
  • The two independent numerical routes agree closely for the modes studied, so the reported spectrum is stable across methods; higher overtones would require continued-fraction techniques rather than WKB.

Reading between the lines

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

  • Extension: the monotonic $\gamma$ trend for $\ell=2,3$ and $n=0,1$ suggests testing higher overtones and multipoles with a method that does not degrade there, such as continued fractions, to see whether the overtone outburst grows, saturates, or reverses near extremality.
  • Extension: because the potential barrier narrows as $\gamma$ grows, the same background may show near-extremal features such as modified late-time tails or echoes that the present waveform analysis does not target; a dedicated search in the time-domain signal could reveal them.
  • Extension: the decoupling of anisotropy-direction fluctuations from the axial sector is the step to test directly, since comparing the full linearized dynamics with Eq. (5) would settle which frequencies are the true gravitational ringdown.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper computes axial gravitational quasinormal modes of the Hayward spacetime, a regular black-hole metric that also arises as an effective solution in asymptotically safe gravity. Using a Regge-Wheeler-type master equation with potential (5), the authors obtain quasinormal frequencies for ℓ=2,3 and n=0,1 via sixth- and eighth-order WKB with Padé approximants, and they cross-check the fundamental ℓ=2 mode against time-domain integration with Prony extraction. They also derive an analytic beyond-eikonal expansion (17) and report that increasing the quantum parameter γ raises the real part of the frequency and lowers the damping rate, with a stronger effect on the first overtone, which they interpret as an 'outburst of overtones.' The γ=0 limit reproduces the known Schwarzschild frequencies, and the WKB-6/WKB-8 agreement is at the sub-percent level throughout.

Significance. If the perturbation setup is valid, this is the first detailed computation of the gravitational ringdown for the Hayward geometry in the asymptotic-safety interpretation, and the monotonic γ-dependence of the spectrum is a concrete, falsifiable prediction. The paper's numerical discipline is a clear strength: the γ=0 check, the small WKB-6/WKB-8 differences, the time-domain agreement for the fundamental mode, and the explicit analytic formula (17) give the reader several independent anchors for the results. The overtone-sensitivity claim is physically interesting and consistent with earlier test-field studies. However, the significance is conditional on the central perturbation-theory step, which is asserted but not derived.

major comments (2)
  1. [Sec. III, Eqs. (4)–(5)] The reduction to the single Regge–Wheeler-type master equation is not derived. The Hayward spacetime in the asymptotic-safety interpretation is supported by an anisotropic fluid, and the linearized Einstein equations for odd-parity perturbations will in general couple the metric functions h0 and h1 to perturbations of the fluid four-velocity and anisotropic stress. The text states that 'fluctuations along the anisotropy direction do not contribute in the axial sector' and that one 'impos[es] the condition that anisotropy perturbations vanish,' but no equations are given to demonstrate that this truncation is consistent. Since the entire spectrum, including the γ-monotonicity claim, follows from the potential (5), this assumption is load-bearing. The authors themselves acknowledge the simplification in Sec. III, yet the abstract and conclusions are worded unconditionally. The manuscript should either provide the explicit perturbation equations and the decoupling argument, or it should qualify every central claim as conditional on this truncation.
  2. [Sec. V, Table V] The independent time-domain check is limited to the fundamental mode (n=0, ℓ=2). The overtone results, which are central to the 'outburst of overtones' claim, are confirmed only by the agreement between WKB-6 and WKB-8. Since both are WKB approximations and the paper itself notes that higher overtones are less accurately captured by WKB, the lack of an independent verification for n=1 weakens the quantitative overtone claim. A time-domain extraction that includes the first overtone, or a Leaver continued-fraction computation, would substantially strengthen this part of the analysis.
minor comments (5)
  1. [Sec. IV, WKB method paragraph] The sentence beginning 'which produces usually most accurate results in most cases, though the lower order formula when applied to ℓ > s perturbations...' is duplicated and garbled; it should be rewritten for clarity.
  2. [Table VI caption] The word 'approximanst' should be 'approximants'.
  3. [Figs. 1 and 2 captions] The color legends are incomplete or confusing, e.g. 'γ = 0 (black) (green) γ = 0.9 (orange)'; the mapping between line style/color and γ values should be stated unambiguously.
  4. [Abstract] The phrase 'the analytic approximation for quasinormal modes are obtained' has a subject-verb agreement error; it should be 'is obtained' or 'approximations are obtained.'
  5. [Sec. V, Eqs. (16)–(17)] The derivations of the position expansion (16) and the analytic frequency formula (17) are not shown; a brief outline of the expansion procedure or a precise pointer to the scheme in Ref. [35] would aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: gamma is an input metric parameter and all QNM results follow from solving the stated master equation; only acknowledged physical assumptions, not circular reductions.

full rationale

The derivation chain is self-contained. The background metric f(r) in Eqs. (2)-(3) fixes the effective potential (5) through the standard Regge-Wheeler form; gamma is an input parameter of the metric, and no quasinormal frequency or damping rate is fitted to produce the reported spectrum. The master equation (4) and the boundary conditions (6)-(7) define the spectral problem, and both the WKB-Pade results and the time-domain Prony extraction are numerical or semi-analytical solutions of that same equation, so their agreement is a consistency check rather than a circular prediction. The analytic formula (17) is explicitly an asymptotic expansion beyond the eikonal limit of the same WKB approximation, not an independent input, and it is presented as an approximation rather than as a first-principles prediction. The acknowledged simplifications in Sec. III - 'Under the simplifying assumption that fluctuations along the anisotropy direction do not contribute in the axial sector' and 'imposing the condition that anisotropy perturbations vanish' - are physical modeling assumptions that could change the effective potential if invalid, but they do not make the subsequent computation circular: the spectrum is not encoded in the assumption, and the paper itself flags that such neglect 'may omit possible corrections to the complete gravitational spectrum.' The self-citations [4] and [13] are contextual (a review and a prior related geometry) and are not used as the evidence that the Hayward axial potential or the reported gamma-monotonicity is correct. No fitted-input-called-prediction, no author-imported uniqueness theorem, and no renaming of a known result was found. The only caveat is a correctness or validity risk concerning the decoupling of the anisotropic-fluid perturbations, which belongs outside circularity scoring.

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

No free parameters are fitted in this paper: gamma is an external model parameter scanned over its allowed range, and the mass M sets the scale. The main assumptions are the Hayward/asymptotic-safety interpretation taken from ref. [6] and the axial-sector decoupling approximation in Sec. III. No new entities are introduced; the anisotropic-fluid construction is taken from prior work [16,17].

assumptions (5)
  • domain assumption The Hayward metric (2)-(3) represents an effective quantum-corrected black hole solution in asymptotically safe gravity, with gamma bounded by 32/27.
    Adopted from ref. [6]; grounds the interpretation of gamma as the strength of quantum corrections and fixes the allowed parameter range (Sec. II).
  • ad hoc to paper Axial gravitational perturbations obey the Regge-Wheeler-type potential (5), with vanishing anisotropic-fluid perturbations in the axial sector.
    Introduced in Sec. III; the text states it as a simplifying assumption, and no explicit derivation of the decoupling is provided.
  • standard math The sixth-order WKB condition with Pade approximants (m=n=3) yields accurate low-lying quasinormal frequencies for this potential.
    Standard semi-analytic method (refs. [26-28,36,38]); accuracy is checked a posteriori through WKB-6/WKB-8 agreement and time-domain comparison.
  • standard math Quasinormal modes are selected by purely ingoing boundary conditions at the horizon and purely outgoing ones at infinity, Eqs. (6)-(7).
    Standard definition for asymptotically flat black-hole potentials (Sec. IV).
  • domain assumption The 1/kappa eikonal expansion is applicable to the Hayward potential for l >= 2 and small gamma.
    Required for the analytic formula (17); the paper verifies it against WKB and finds deviations only near extremal gamma (Table VI).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gravitational quasinormal modes of the Hayward spacetime." pith.science (2026). https://pith.science/paper/XX3ABDDG

@misc{pith2026250819989,
  author       = {Pith},
  title        = {Pith review of: Gravitational quasinormal modes of the Hayward spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XX3ABDDG}},
  note         = {Machine review of arXiv:2508.19989}
}
abstract

We study gravitational quasinormal modes of the Hayward spacetime, a regular black-hole geometry that also admits an interpretation as an effective quantum-corrected solution within asymptotically safe gravity. Using both the higher-order WKB method supplemented with Pade approximants and time-domain integration with Prony analysis, we obtain accurate spectra for axial perturbations and explore the impact of the quantum parameter $\gamma$. We find that increasing $\gamma$ systematically raises the oscillation frequencies while reducing the damping rates, making the ringdown longer lived. For the first overtone, the effect of $\gamma$ is noticeably stronger than for the fundamental mode, providing an indication of the so-called outburst of overtones previously observed for test fields, and pointing to the particular sensitivity of subdominant modes to near-horizon quantum corrections. In addition, the analytic approximation for quasinormal modes are obtained in the form of expansion beyond the eikonal limit.

Figures

Figures reproduced from arXiv: 2508.19989 by the authors.

Figure 1
Figure 1. FIG. 1: Potential as a function of the tortoise coordinate fo [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Semi-lograrithmic time-domain profile for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hawking Emission from Black Holes Evaporating toward Wormholes and the Accuracy of the WKB Approximation

    gr-qc 2026-06 unverdicted novelty 5.5 of 10

    Numerical greybody factors for photons and massless Dirac fields in Simpson-Visser and Casadio-Fabbri-Mazzacurati geometries reveal that WKB overestimates luminosities by orders of magnitude near wormhole endpoints, i...

  2. Quasinormal spectra of higher dimensional regular black holes in theories with infinite curvature corrections

    gr-qc 2025-09 conditional novelty 5.0 of 10

    For six families of higher-dimensional regular black holes, positive higher-curvature coupling reduces quasinormal-mode oscillation frequencies and damping rates, a trend verified at low and eikonal multipoles.

  3. Wave and particle probes of a regular T-duality-inspired black hole with gravitational self-energy

    gr-qc 2026-07 conditional novelty 4.0 of 10

    Increasing zero-point length makes photon and ISCO orbits more compact in ADM units while heavier scalars raise oscillation frequency and suppress damping on this regular self-energy black hole.

  4. Gravitational perturbations of a regular T-duality inspired black hole: Quasinormal modes, excitation factors, and time-domain evolution

    gr-qc 2026-07 conditional novelty 4.0 of 10

    Gravitational quasinormal-mode frequencies of a T-duality-inspired regular black hole are computed, showing the zero-point length makes the ringdown oscillate faster and alters damping non-monotonically.

  5. Long-lived quasinormal modes and grey-body factors of supermassive black holes with a dark matter halo

    gr-qc 2025-11 conditional novelty 4.0 of 10

    Massive scalar fields around a Schwarzschild black hole with a dark matter halo ring longer as field mass grows, while realistic halo parameters leave quasinormal frequencies and grey-body factors almost unchanged.

  6. Bonanno-Reuter regular black hole: quasi-resonances, grey-body factors and absorption cross-sections of a massive scalar field

    gr-qc 2025-10 conditional novelty 4.0 of 10

    Damping of massive scalar quasinormal modes decreases with field mass in the Bonanno-Reuter black hole, and grey-body factors shift to higher frequencies.

Reference graph

Works this paper leans on

81 extracted references · 3 canonical work pages · cited by 6 Pith papers

  1. [1]

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

    arXiv 1999

  2. [2]

    Berti, V

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009), 0905.2975

    arXiv 2009

  3. [3]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011), 1102.4014

    arXiv 2011

  4. [4]

    S. V. Bolokhov and M. Skvortsova (2025), 2504.05014

    arXiv 2025

  5. [5]

    S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006), gr- qc/0506126

    arXiv 2006

  6. [6]

    A. Held, R. Gold, and A. Eichhorn, JCAP 06, 029 (2019), 1904.07133

    arXiv 2019

  7. [7]

    R. A. Konoplya, A. F. Zinhailo, J. Kunz, Z. Stuchlik, and A. Zhidenko, JCAP 10, 091 (2022), 2206.14714

    arXiv 2022

  8. [8]

    Malik, EPL 147, 69001 (2024), 2410.04306

    Z. Malik, EPL 147, 69001 (2024), 2410.04306

    arXiv 2024

Show all 81 references
  1. [9]

    R. A. Konoplya, JCAP 07, 001 (2023), 2305.09187

    2023 arXiv

  2. [10]

    Takahashi, H

    K. Takahashi, H. Motohashi, and M. Minamitsuji, Phys. Rev. D 100, 024041 (2019), 1904.03554

    2019 arXiv

  3. [11]

    Bonanno and M

    A. Bonanno and M. Reuter, Phys. Rev. D 62, 043008 (2000), hep-th/0002196

    2000 arXiv

  4. [12]

    Platania, Eur

    A. Platania, Eur. Phys. J. C 79, 470 (2019), 1903.10411

    2019 arXiv

  5. [13]

    S. V. Bolokhov and M. Skvortsova, International Jour- nal of Gravitation and Theoretical Physics 1, 3 (2025), 2507.07196

    2025 arXiv

  6. [14]

    Bonanno, D

    A. Bonanno, D. Malafarina, and A. Panassiti, Phys. Rev. Lett. 132, 031401 (2024), 2308.10890

    2024 arXiv

  7. [15]

    Q.-L. Shi, R. Wang, W. Xiong, and P.-C. Li (2025), 2506.16217

    2025 arXiv

  8. [16]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. Lett. 121, 241301 (2018), 1806.00648

    2018 arXiv

  9. [17]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. D 98, 126003 (2018), 1806.02406

    2018 arXiv

  10. [18]

    Bouhmadi-López, S

    M. Bouhmadi-López, S. Brahma, C.-Y. Chen, P. Chen, and D.-h. Yeom, JCAP 07, 066 (2020), 2004.13061

    2020 arXiv

  11. [19]

    R. A. Konoplya and O. S. Stashko, Phys. Rev. D 111, 104055 (2025), 2408.02578

    2025 arXiv

  12. [20]

    K. A. Bronnikov, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 86, 024028 (2012), 1205.2224

    2012 arXiv

  13. [21]

    Chen and P

    C.-Y. Chen and P. Chen, Phys. Rev. D 99, 104003 (2019), 1902.01678

    2019 arXiv

  14. [22]

    Regge and J

    T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957)

    1957

  15. [23]

    Berti, K

    E. Berti, K. D. Kokkotas, and E. Papantonopoulos, Phys. Rev. D 68, 064020 (2003), gr-qc/0306106

    2003 arXiv

  16. [24]

    K. D. Kokkotas, Nuovo Cim. B 108, 991 (1993)

    1993

  17. [25]

    R. A. Konoplya and A. Zhidenko, Phys. Lett. B 648, 236 (2007), hep-th/0611226

    2007 arXiv

  18. [26]

    B. F. Schutz and C. M. Will, Astrophys. J. Lett. 291, L33 (1985)

    1985

  19. [27]

    Iyer and C

    S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987)

    1987

  20. [28]

    R. A. Konoplya, Phys. Rev. D 68, 024018 (2003), gr- qc/0303052

    2003

  21. [29]

    Antonelli, A

    T. Antonelli, A. Giusti, R. Casadio, and L. Heisenberg (2025), 2505.08415

    2025 arXiv

  22. [30]

    Pedrotti and M

    D. Pedrotti and M. Calzà, Phys. Rev. D 111, 124056 (2025), 2504.01909

    2025 arXiv

  23. [31]

    R. A. Konoplya and A. F. Zinhailo, Phys. Lett. B 810, 135793 (2020), 2004.02248

    2020 arXiv

  24. [32]

    Mahdavian Yekta, M

    D. Mahdavian Yekta, M. Karimabadi, and S. A. Alavi, Annals Phys. 434, 168603 (2021), 1912.12017

    2021 arXiv

  25. [33]

    Matyjasek, Phys

    J. Matyjasek, Phys. Rev. D 104, 084066 (2021), 2107.04815

    2021 arXiv

  26. [34]

    B. C. Lütfüoğlu, Eur. Phys. J. C 85, 630 (2025), 2504.18482

    2025 arXiv

  27. [35]

    R. A. Konoplya and A. Zhidenko, Class. Quant. Grav. 40, 245005 (2023), 2309.02560

    2023 arXiv

  28. [36]

    Matyjasek and M

    J. Matyjasek and M. Opala, Phys. Rev. D 96, 024011 (2017), 1704.00361

    2017 arXiv

  29. [37]

    Hamil and B

    B. Hamil and B. C. Lütfüoğlu (2025), 2503.17474

    2025

  30. [38]

    R. A. Konoplya, A. Zhidenko, and A. F. Zinhailo, Class. Quant. Grav. 36, 155002 (2019), 1904.10333

    2019 arXiv

  31. [39]

    R. A. Konoplya, Phys. Lett. B 823, 136734 (2021), 2109.01640

    2021 arXiv

  32. [40]

    Paul, Eur

    P. Paul, Eur. Phys. J. C 84, 218 (2024), 2312.16479

    2024 arXiv

  33. [41]

    Zhang, D.-C

    X. Zhang, D.-C. Zou, C.-M. Zhang, M. Zhang, and R.-H. Yue (2025), 2508.17736

    2025

  34. [42]

    Miyachi, R

    T. Miyachi, R. Namba, H. Omiya, and N. Oshita, Phys. Rev. D 111, 124045 (2025), 2503.17245

    2025 arXiv

  35. [43]

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

    2005 arXiv

  36. [44]

    Xiong, P

    W. Xiong, P. Liu, C.-Y. Zhang, and C. Niu, Phys. Rev. D 106, 064057 (2022), 2112.12523

    2022 arXiv

  37. [45]

    J. S. Santos, M. B. Cruz, and F. A. Brito, Eur. Phys. J. C 81, 1082 (2021), 2103.11212

    2021 arXiv

  38. [46]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400132 (2024), 2405.06390

    2024 arXiv

  39. [47]

    Malik, Int

    Z. Malik, Int. J. Theor. Phys. 63, 199 (2024), 2308.10412

    2024 arXiv

  40. [48]

    R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlik, Phys. Rev. D 102, 044023 (2020), 2006.10462. 8

    2020 arXiv

  41. [49]

    Varghese and V

    N. Varghese and V. C. Kuriakose, Gen. Rel. Grav. 43, 2757 (2011), 1011.6608

    2011 arXiv

  42. [50]

    Skvortsova, Fortsch

    M. Skvortsova, Fortsch. Phys. 72, 2400036 (2024), 2311.11650

    2024 arXiv

  43. [51]

    Dubinsky, International Journal of Gravitation and Theoretical Physics 1, 2 (2025), 2507.00256

    A. Dubinsky, International Journal of Gravitation and Theoretical Physics 1, 2 (2025), 2507.00256

    2025

  44. [52]

    S. V. Bolokhov, Eur. Phys. J. C 84, 634 (2024), 2404.09364

    2024 arXiv

  45. [53]

    K. A. Bronnikov and R. A. Konoplya, Phys. Rev. D 101, 064004 (2020), 1912.05315

    2020 arXiv

  46. [54]

    Malik, Annals Phys

    Z. Malik, Annals Phys. 479, 170046 (2025), 2409.01561

    2025 arXiv

  47. [55]

    S. V. Bolokhov, Phys. Rev. D 110, 024010 (2024), 2311.05503

    2024 arXiv

  48. [56]

    Skvortsova, Grav

    M. Skvortsova, Grav. Cosmol. 30, 279 (2024), 2405.15807

    2024 arXiv

  49. [57]

    R. A. Konoplya, Phys. Lett. B 804, 135363 (2020), 1912.10582

    2020 arXiv

  50. [58]

    B. C. Lütfüoğlu, Eur. Phys. J. C 85, 486 (2025), 2503.16087

    2025 arXiv

  51. [59]

    M. S. Churilova, R. A. Konoplya, Z. Stuchlik, and A. Zhi- denko, JCAP 10, 010 (2021), 2107.05977

    2021 arXiv

  52. [60]

    Dubinsky and A

    A. Dubinsky and A. F. Zinhailo, EPL 149, 69004 (2025), 2410.15232

    2025 arXiv

  53. [61]

    Abdalla, O

    E. Abdalla, O. P. F. Piedra, F. S. Nuñez, and J. de Oliveira, Phys. Rev. D 88, 064035 (2013), 1211.3390

    2013 arXiv

  54. [62]

    Gundlach, R

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

    1994 arXiv

  55. [63]

    de Prony, Journal de l’École Polytechnique 1, 24 (1795)

    G. de Prony, Journal de l’École Polytechnique 1, 24 (1795)

  56. [64]

    R. A. Konoplya, A. Zhidenko, and C. Molina, Phys. Rev. D 75, 084004 (2007), gr-qc/0602047

    2007 arXiv

  57. [65]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009), 0812.1806

    2009 arXiv

  58. [66]

    R. A. Konoplya and Z. Stuchlík, Phys. Lett. B 771, 597 (2017), 1705.05928

    2017 arXiv

  59. [67]

    Khanna and R

    G. Khanna and R. H. Price, Phys. Rev. D 95, 081501 (2017), 1609.00083

    2017 arXiv

  60. [68]

    R. A. Konoplya and A. F. Zinhailo, Eur. Phys. J. C 80, 1049 (2020), 2003.01188

    2020 arXiv

  61. [69]

    R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlík, Phys. Rev. D 99, 124042 (2019), 1903.03483

    2019 arXiv

  62. [70]

    R. A. Konoplya, A. Spina, and A. Zhidenko, Phys. Rev. D 112, 024060 (2025), 2505.01128

    2025 arXiv

  63. [71]

    S. V. Bolokhov, Phys. Lett. B 856, 138879 (2024), 2310.12326

    2024 arXiv

  64. [72]

    R. A. Konoplya, Phys. Lett. B 838, 137674 (2023), 2210.08373

    2023 arXiv

  65. [73]

    R. A. Konoplya, A. Khrabustovskyi, J. Kříž, and A. Zhi- denko, JCAP 04, 062 (2025), 2501.16134

    2025 arXiv

  66. [74]

    R. A. Konoplya and A. Zhidenko, JHEAp 44, 419 (2024), 2209.00679

    2024 arXiv

  67. [75]

    Toshmatov, Z

    B. Toshmatov, Z. Stuchlík, B. Ahmedov, and D. Malafa- rina, Phys. Rev. D 99, 064043 (2019), 1903.03778

    2019 arXiv

  68. [76]

    K. Lin, J. Li, and S. Yang, Int. J. Theor. Phys. 52, 3771 (2013)

    2013

  69. [77]

    Dutta Roy and S

    P. Dutta Roy and S. Kar, Phys. Rev. D 106, 044028 (2022), 2206.04505

    2022 arXiv

  70. [78]

    Malik (2024), 2412.13385

    Z. Malik (2024), 2412.13385

    2024 arXiv

  71. [79]

    Malik, Z

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

    2024

  72. [80]

    Malik, Int

    Z. Malik, Int. J. Theor. Phys. 63, 128 (2024), 2409.09872

    2024 arXiv

  73. [81]

    Malik, Int

    Z. Malik, Int. J. Mod. Phys. A 40, 2450132 (2025). 9

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.