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Discrete symmetries of modified Teukolsky equations

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Pith's one-line read Complex deformations of the Teukolsky potential break the m=0 degeneracy of black-hole quasinormal modes.

desk verdict Clean, usable note: complex δV splits m=0 modes and multipole-dependent frequency-domain potentials can inject non-physical branches in time domain. read the letter →

arxiv 2607.27053 v1 pith:N45P4IDI submitted 2026-07-29 gr-qc

Ciro De Simone This is my paper · ORCID
classification gr-qc
keywords Teukolskyequationquasinormalmodesdiscretesymmetriesmodifiedgravityblack-holeperturbationstime-domainevolutionshigher-derivative
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 Teukolsky equation that governs black-hole ringdown has discrete symmetries that force the m=0 quasinormal modes to come in degenerate pairs. This paper shows that a broad class of complex deformations of the Teukolsky potential changes that symmetry map: the equation for a deformation coefficient α is identified with the equation for its complex conjugate α*, so the two m=0 frequencies split. The result is proved both from the master frequency-domain equation and from (2+1)-dimensional time evolutions of Gaussian wave packets. The same discrete map halves the number of simulations needed to extract the m=0 fundamental modes, and it warns that promoting multipole-dependent frequency-domain potentials (as in higher-derivative gravity) into time-domain codes can inject non-physical branches of modes into the waveforms.

What carries the argument

The discrete symmetry transformation (complex conjugation + ϕ→−ϕ + a→−a, supplemented by α→α* when the deformation is complex). It leaves the modified Teukolsky operator invariant and therefore dictates exactly which pair of quasinormal modes is excited in any given time-domain run.

What would settle it

Evolve a modified Teukolsky equation with a purely imaginary α at m=0 and extract both frequencies with the Prony method; if they remain degenerate (or fail to match the predicted pair ω(α) and ω(α*)), the claimed symmetry breaking is false.

Watch

Extended reading notes

Core claim

A complex deformation δV of the Teukolsky potential replaces the ordinary discrete symmetry that pairs (ℓ,m,α) with (ℓ,−m,α) by a map that pairs (ℓ,m,α) with (ℓ,−m,α*). Consequently the m=0 modes, which are degenerate in general relativity, split into two distinct frequencies whose real parts flip sign under α→α*. The claim is established analytically from the master equation and confirmed by Prony extraction from (2+1)-dimensional numerical evolutions across spins, multipoles and black-hole spins.

Load-bearing premise

The whole deformation is assumed to sit only in a multiplicative radial potential term that does not re-mix angles with radii or couple different multipoles—an assumption that holds only inside the small-coupling, separable approximation.

Editorial extensions

If this is right

  • m=0 time-domain runs with complex α automatically supply both members of the split pair, cutting the simulation count in half.
  • When several α^(k) are nonzero, the same map collapses entire quadrants of parameter space, reducing the cost of higher-order coefficient fits.
  • Frequency-domain potentials that depend on (n,ℓ,m) cannot be dropped unchanged into a time-domain code without checking that the extra excited branches are physical.
  • In higher-derivative gravity the first-order coefficients generally violate the conjugacy condition, so the extra modes seen in simulations are non-physical contaminants.

Reading between the lines

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

  • Any future ringdown pipeline that fits complex beyond-GR parameters at m=0 will need an explicit branch-selection rule, otherwise it will mix physical and conjugate modes.
  • The same discrete map should apply, with only minor changes, to other separable master equations (Regge–Wheeler–Zerilli, Dirac) once a complex potential deformation is introduced.
  • If a concrete theory produces non-separable or frequency-dependent corrections, the clean α↔α* pairing will break and new numerical diagnostics will be required.
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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

2 major / 5 minor

Summary. The paper studies how a class of multiplicative deformations δV of the Teukolsky potential modifies the residual discrete symmetries of the Teukolsky equation. In frequency domain it shows that complex α^(k) map (ℓ,m,α) to (ℓ,−m,α*), so the m=0 modes split into the distinct pair ω(α) and ω(α*) (Eqs. 20–21). The same map is recovered in the time-domain operator under conjugation + ϕ→−ϕ + a→−a (+ α→α*). (2+1)D evolutions and Prony extractions corroborate the split and the GR limit. The symmetries are then used to halve the number of runs needed for m=0 coefficient fits, and are applied to first-order higher-derivative gravity (HDG) potentials, where multipole-dependent α^(k)_nℓm generally violate the consistency condition (26) and can inject non-physical branches into time-domain waveforms.

Significance. Within the standard parametrized/small-coupling Teukolsky framework the result is clean, elementary, and practically useful: it clarifies when m=0 degeneracy survives, gives a concrete recipe for cheaper time-domain spectroscopy of complex deformations, and flags a genuine pitfall when frequency-domain HDG potentials are promoted to time-independent PDE coefficients. The combination of analytic symmetry maps with (2+1)D simulations and public coefficient repositories is a strength. The work is incremental rather than foundational, but it is the right kind of incremental result for the beyond-GR ringdown literature.

major comments (2)
  1. [Sec. IV.B] Sec. IV.B and Eqs. (24)–(26): the central HDG claim—that condition (26) fails and therefore time-domain runs excite non-physical branches—is asserted after consulting the public coefficient tables, but the manuscript never shows an explicit numerical example (waveform, Prony spectrum, or growth rate) in which a spurious conjugate mode appears or becomes unstable. Without at least one concrete contaminated run, the load-bearing warning remains qualitative.
  2. [Sec. IV.A] Sec. IV.A: the efficiency procedure (imaginary α only, quadrant reduction in Fig. 2, identification of ±Re ω with α and α*) is described carefully, yet the paper does not actually carry it out—no fitted d_ω^(k) values, no comparison to the linear coefficients of Ref. [21], and no count of runs saved. As written, the section is a proposal rather than a demonstrated application of the symmetry.
minor comments (5)
  1. Throughout (abstract, Sec. I, III.B, etc.) the text contains broken quotation marks and missing spaces around m=0 (e.g. “them“0degeneracy”, “m“0”). These are systematic typesetting artifacts and should be cleaned.
  2. [Fig. 1] Fig. 1 caption and body: units are stated as M=0.5 while the rest of the paper works in M=1 geometric units; a single consistent convention would avoid confusion when comparing Prony numbers to Eq. (7).
  3. [Fig. 2] Fig. 2 is purely schematic. A short caption sentence stating that it is illustrative (not a measured coefficient plane) would help.
  4. [Sec. II.B] The numerical implementation is deferred almost entirely to the companion arXiv:2607.25311 [48]. A brief self-contained paragraph (grid, boundary conditions, convergence order already claimed) would make the present manuscript readable on its own.
  5. [References] Refs. [7–9] cite 2025/2026 LIGO papers with very recent arXiv numbers; verify final bibliographic data at proof stage.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: symmetry breaking follows algebraically from the modified operator; self-citations supply methods only.

  1. self citation load bearing [Sec. II.B (after Eq. 12); Sec. IV.B; Refs. [41], [48]]
    "The details on the numerical implementation of the modified Teukolsky equation can be found in Ref. [48]. ... This test has been realized by implementing the first order HDG modified Teukolsky equation in time domain using the formalism introduced in [48]"

    The numerical evolutions that corroborate the symmetry map are implemented via the author's own companion paper [48] (and related [41]). This is ordinary methods self-citation, not load-bearing for the algebraic symmetry result, which stands independently from Eqs. 11–23. Flagged only as minor scaffolding dependence; does not force the central claim.

full rationale

The load-bearing claim—that a complex multiplicative deformation δV breaks the residual discrete map so m=0 modes split into the distinct pair ω(α) and ω(α*)—is read off directly from the modified time/frequency-domain operators (Eqs. 11–12, 20–23). The residual symmetry (complex conjugation + ϕ→−ϕ + a→−a + α→α*) is an elementary invariance check, not a fit and not imported as a uniqueness theorem. Frequency-domain benchmarks (Leaver / linear d_ω coefficients) come from the external parametrized-QNM framework [21] and its public repo; GR Teukolsky limits are standard. Self-citations ([41], [48]) and the Cano–Franchini–Völkel line supply the (2+1)D integrator, Prony extraction, and HDG coefficient tables—implementation scaffolding, not the symmetry derivation itself. No step reduces a claimed prediction to its own fitted input or to an unverified self-cited uniqueness result. Score 1 only for the minor, non-load-bearing self-citation of the companion time-domain code paper.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The load-bearing content rests on the classical Teukolsky operator and its known discrete symmetries, on the parametrized δV deformation of Cano et al. (small-coupling, separable, multipole-decoupled), and on the premise that a frequency-domain potential with frozen (n,ℓ,m)-dependent coefficients may be inserted into a time-domain PDE. No new physical entities are postulated. The α^(k) are theory inputs, not fits performed in this work; the only auxiliary choices are numerical (Gaussian initial data, Prony window, sample α values).

free parameters (2)
  • α^(k) deformation coefficients (and their imaginary parts) = illustrative O(0.1) samples; linear response assumed
    Complex couplings that define δV; scanned by hand over small values (e.g. 0.1±i0.1, real/imaginary grids in [-0.1,0.1]) to illustrate splitting and to fit d_ω coefficients. Not observationally fitted here, but they are free inputs the spectra depend on.
  • truncation integer K and which k channels are active
    Chooses the radial shape of δV; HDG application activates k∈{-2,0,1,2}. Fixed by the parametrized framework / target theory, not derived in this paper.
assumptions (4)
  • domain assumption Vacuum Teukolsky equation on Kerr is separable and invariant under complex conjugation combined with ϕ→−ϕ and a→−a (or the frequency-domain map conjugation + ω→−ω* + θ→π−θ).
    Standard GR result used as the undeformed baseline throughout Sec. III.A; cited to Teukolsky and Krivan et al.
  • domain assumption Beyond-GR corrections are captured by a small, separable, multiplicative potential deformation δV=Δ^{-1} Σ_k α^{(k)}(r/r_+)^k with no induced ℓ–m mixing.
    Defines the whole modified framework (Eqs. 1, 6, 11–12); taken from the parametrized Teukolsky program and EFT references [21,30].
  • domain assumption First-order-in-coupling QNM shifts and HDG coefficient formulas remain valid benchmarks for the simulated waveforms at the quoted α amplitudes.
    Used to claim ≤1% agreement and to interpret HDG runs (Sec. IV); higher-order backreaction on the potential is neglected.
  • ad hoc to paper A frequency-domain potential whose coefficients were computed at a fixed Kerr QNM can be promoted to a time-independent coefficient function in the time-domain PDE.
    Implicit in the HDG time-domain implementation (Sec. IV.B) and in the non-physical-branch discussion; this is exactly the step the paper warns can create spurious modes, yet it is still the setup used to test the symmetries.

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Cite this review

Pith. "Pith review of Discrete symmetries of modified Teukolsky equations." pith.science (2026). https://pith.science/paper/N45P4IDI

@misc{pith2026260727053,
  author       = {Pith},
  title        = {Pith review of: Discrete symmetries of modified Teukolsky equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N45P4IDI}},
  note         = {Machine review of arXiv:2607.27053}
}
read the original abstract

The Teukolsky equation possesses discrete symmetries that constrain the properties of black hole perturbations and their quasinormal mode spectrum. In this study, we explore how a class of modifications of the Teukolsky potential can alter the symmetry structure of the equation and break the m = 0 degeneracy of quasinormal modes. We prove this result in frequency domain using the master Teukolsky equation and in time domain via (2+1)-dimensional simulations. We also show that the discrete symmetries can be leveraged for a more efficient characterization of the m = 0 quasinormal modes from time-domain evolutions. As a theory-specific application, we consider the case of higher-derivative theories of gravity, highlighting that time-domain implementations of frequency-domain potentials can give rise to additional non-physical branches of modes.

Figures

Figures reproduced from arXiv: 2607.27053 by the authors.

Figure 1
Figure 1. displays two waveforms corresponding to the multipole ℓ “ 2, m “ 0 and α p´4q “ 0.1 ˘ i 0.1. As ex￾pected, the two profiles coincide since the same pair of modes is excited in each simulation, with similar ampli￾tudes and phases. The frequencies extracted using the Prony method [37], which consists of fitting the numerical waveform with a sum of damped sinusoids, agree to less than 0.05%. Moreover, the Prony estimat… view at source ↗
Figure 2
Figure 2. FIG. 2. Quadrants of the same color are mapped by the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Works this paper leans on

58 extracted references · 44 linked inside Pith

  1. [21]

    P. A. Cano, L. Capuano, N. Franchini, S. Maenaut, and S. H. Völkel, Parametrized quasinormal mode framework for modified Teukolsky equations, Phys. Rev. D110, 104007 (2024), arXiv:2407.15947 [gr-qc]

    arXiv 2024

  2. [1]

    R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963)

    1963

  3. [2]

    Maggiore,Gravitational Waves

    M. Maggiore,Gravitational Waves. Vol. 1: Theory and Experiments(Oxford University Press, 2007)

    2007

  4. [3]

    Maggiore,Gravitational Waves

    M. Maggiore,Gravitational Waves. Vol. 2: Astrophysics and Cosmology(Oxford University Press, 2018)

    2018

  5. [4]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    arXiv 2016

  6. [5]

    Abbottet al.(LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys

    R. Abbottet al.(LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X11, 021053 (2021), arXiv:2010.14527 [gr- qc]

    arXiv 2021

  7. [6]

    Abbottet al.(KAGRA, VIRGO, LIGO Scien- tific), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run, Phys

    R. Abbottet al.(KAGRA, VIRGO, LIGO Scien- tific), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run, Phys. Rev. X13, 041039 (2023), arXiv:2111.03606 [gr-qc]

    arXiv 2023

  8. [7]

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), GWTC-4.0: Updating the Gravitational-Wave Transient Catalog with Observations from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run, (2025), arXiv:2508.18082 [gr-qc]

    arXiv 2025

Show all 58 references
  1. [8]

    A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), Black Hole Spectroscopy and Tests of General Relativity with GW250114, Phys. Rev. Lett.136, 041403 (2026), arXiv:2509.08099 [gr-qc]

    2026 arXiv

  2. [9]

    A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), GW250114: Testing Hawking’s Area Law and the Kerr Nature of Black Holes, Phys. Rev. Lett.135, 111403 (2025), arXiv:2509.08054 [gr-qc]

    2025 arXiv

  3. [10]

    Chandrasekhar,The mathematical theory of black holes(1985)

    S. Chandrasekhar,The mathematical theory of black holes(1985)

    1985

  4. [11]

    K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel.2, 2 (1999), arXiv:gr-qc/9909058

    1999 arXiv

  5. [12]

    Nollert, TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class

    H.-P. Nollert, TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav.16, R159 (1999)

    1999

  6. [13]

    R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys.83, 793 (2011), arXiv:1102.4014 [gr-qc]

    2011 arXiv

  7. [14]

    Franchini and S

    N. Franchini and S. H. Völkel, Testing General Rel- ativity with Black Hole Quasi-normal Modes (2024) arXiv:2305.01696 [gr-qc]

    2024 arXiv

  8. [15]

    S. A. Teukolsky, Rotating black holes - separable wave equations for gravitational and electromagnetic pertur- bations, Phys. Rev. Lett.29, 1114 (1972)

    1972

  9. [16]

    Cardoso and L

    V. Cardoso and L. Gualtieri, Testing the black hole ‘no- hair’ hypothesis, Class. Quant. Grav.33, 174001 (2016), arXiv:1607.03133 [gr-qc]

    2016 arXiv

  10. [17]

    M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Testing the no-hair theorem with GW150914, Phys. Rev. Lett.123, 111102 (2019), arXiv:1905.00869 [gr-qc]

    2019 arXiv

  11. [18]

    The Resonant Frequencies of Rotating Holes, Astrophys

    S.L.Detweiler,BlackHolesandGravitationalWaves.III. The Resonant Frequencies of Rotating Holes, Astrophys. J.239, 292 (1980)

    1980

  12. [19]

    Berti, V

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

    2009 arXiv

  13. [20]

    Bertiet al., Black hole spectroscopy: from theory to experiment, (2025), arXiv:2505.23895 [gr-qc]

    E. Bertiet al., Black hole spectroscopy: from theory to experiment, (2025), arXiv:2505.23895 [gr-qc]

    2025 arXiv

  14. [22]

    R. Tang, N. Franchini, S. H. Völkel, and E. Berti, Quasinormal modes of rotating black holes beyond gen- eral relativity in the WKB approximation, (2025), arXiv:2512.17786 [gr-qc]

    2025 arXiv

  15. [23]

    Yu and L.-B

    Z. Yu and L.-B. Wu, The parameterized quasinor- mal modes for modified Teukolsky equations, (2025), arXiv:2511.02274 [gr-qc]

    2025

  16. [24]

    Cardoso, M

    V. Cardoso, M. Kimura, A. Maselli, E. Berti, C. F. B. Macedo, and R. McManus, Parametrized black hole quasinormal ringdown: Decoupled equations for non- rotating black holes, Phys. Rev. D99, 104077 (2019), arXiv:1901.01265 [gr-qc]

    2019 arXiv

  17. [25]

    McManus, E

    R. McManus, E. Berti, C. F. B. Macedo, M. Kimura, A. Maselli, and V. Cardoso, Parametrized black hole quasinormal ringdown. II. Coupled equations and quadratic corrections for nonrotating black holes, Phys. Rev. D100, 044061 (2019), arXiv:1906.05155 [gr-qc]

    2019 arXiv

  18. [26]

    S. H. Völkel, N. Franchini, and E. Barausse, Theory- agnostic reconstruction of potential and couplings from quasinormal modes, Phys. Rev. D105, 084046 (2022), arXiv:2202.08655 [gr-qc]

    2022 arXiv

  19. [27]

    Franchini and S

    N. Franchini and S. H. Völkel, Parametrized quasinormal mode framework for non-Schwarzschild metrics, Phys. Rev. D107, 124063 (2023), arXiv:2210.14020 [gr-qc]

    2023 arXiv

  20. [28]

    D. Li, P. Wagle, Y. Chen, and N. Yunes, Perturbations of Spinning Black Holes beyond General Relativity: Modi- fiedTeukolskyEquation,Phys.Rev.X13,021029(2023), 8 arXiv:2206.10652 [gr-qc]

    2023 arXiv

  21. [29]

    Hussain and A

    A. Hussain and A. Zimmerman, Approach to computing spectral shifts for black holes beyond Kerr, Phys. Rev. D 106, 104018 (2022), arXiv:2206.10653 [gr-qc]

    2022 arXiv

  22. [30]

    P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Uni- versal Teukolsky equations and black hole perturbations in higher-derivative gravity, Phys. Rev. D108, 024040 (2023), arXiv:2304.02663 [gr-qc]

    2023 arXiv

  23. [31]

    G. Z. Tóth, Noether currents for the Teukolsky mas- ter equation, Class. Quant. Grav.35, 185009 (2018), arXiv:1801.04710 [gr-qc]

    2018 arXiv

  24. [32]

    Krivan, P

    W. Krivan, P. Laguna, P. Papadopoulos, and N. Ander- sson, Dynamics of perturbations of rotating black holes, Phys. Rev. D56, 3395 (1997), arXiv:gr-qc/9702048

    1997 arXiv

  25. [33]

    C. V. Vishveshwara, Scattering of Gravitational Radi- ation by a Schwarzschild Black-hole, Nature227, 936 (1970)

    1970

  26. [34]

    Pazos-Avalos and C

    E. Pazos-Avalos and C. O. Lousto, Numerical integration of the Teukolsky equation in the time domain, Phys. Rev. D72, 084022 (2005), arXiv:gr-qc/0409065

    2005 arXiv

  27. [35]

    Harms, S

    E. Harms, S. Bernuzzi, and B. Brügmann, Numerical so- lution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and applica- tion to late-time decays, Class. Quant. Grav.30, 115013 (2013), arXiv:1301.1591 [gr-qc]

    2013 arXiv

  28. [36]

    P. A. Cano, L. Capuano, N. Franchini, S. Maenaut, and S. H. Völkel, Higher-derivative corrections to the Kerr quasinormal mode spectrum, Phys. Rev. D110, 124057 (2024), arXiv:2409.04517 [gr-qc]

    2024 arXiv

  29. [37]

    Berti, V

    E. Berti, V. Cardoso, J. A. Gonzalez, and U. Sperhake, Mining information from binary black hole mergers: A Comparison of estimation methods for complex exponen- tials in noise, Phys. Rev. D75, 124017 (2007), arXiv:gr- qc/0701086

    2007

  30. [38]

    Kimura, Note on the parametrized black hole quasi- normal ringdown formalism, Phys

    M. Kimura, Note on the parametrized black hole quasi- normal ringdown formalism, Phys. Rev. D101, 064031 (2020), arXiv:2001.09613 [gr-qc]

    2020 arXiv

  31. [39]

    E. W. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A402, 285 (1985)

    1985

  32. [40]

    github, parametrized_qnm_framework (2024)

    2024

  33. [41]

    De Simone, S

    C. De Simone, S. H. Völkel, K. D. Kokkotas, V. De Falco, and S. Capozziello, Confronting eikonal and post-Kerr methods with numerical evolution of scalar field pertur- bations in spacetimes beyond Kerr, Phys. Rev. D113, 104004 (2026), arXiv:2601.09607 [gr-qc]

    2026 arXiv

  34. [42]

    Thomopoulos, S

    S. Thomopoulos, S. H. Völkel, and H. P. Pfeiffer, Ringdown spectroscopy of phenomenologically modi- fied black holes, Phys. Rev. D112, 064054 (2025), arXiv:2504.17848 [gr-qc]

    2025

  35. [43]

    R. A. Konoplya, A. Spina, and A. Zhidenko, Time evo- lution of black hole perturbations in quadratic gravity, Phys. Rev. D112, 024060 (2025), arXiv:2505.01128 [gr- qc]

    2025 arXiv

  36. [44]

    R. H. Price and L. M. Burko, Late time tails from momentarily stationary, compact initial data in Schwarzschild spacetimes, Phys. Rev. D70, 084039 (2004), arXiv:gr-qc/0408077

    2004 arXiv

  37. [45]

    R. F. Rosato and P. Pani, Universality of late-time ringdown tails, Phys. Rev. D112, 024080 (2025), arXiv:2505.08877 [gr-qc]

    2025 arXiv

  38. [46]

    D. D. Doneva, L. G. Collodel, C. J. Krüger, and S. S. Yazadjiev, Black hole scalarization induced by the spin: 2+1 time evolution, Phys. Rev. D102, 104027 (2020), arXiv:2008.07391 [gr-qc]

    2020 arXiv

  39. [47]

    Pedrotti and S

    D. Pedrotti and S. Vagnozzi, Quasinormal modes-shadow correspondence for rotating regular black holes, Phys. Rev. D110, 084075 (2024), arXiv:2404.07589 [gr-qc]

    2024 arXiv

  40. [48]

    De Simone, S

    C. De Simone, S. H. Völkel, K. D. Kokkotas, and S. Capozziello, Parametrized beyond-Teukolsky frame- work in the time domain, (2026), arXiv:2607.25311 [gr- qc]

    2026 arXiv

  41. [49]

    Berti, V

    E. Berti, V. Cardoso, and C. M. Will, On gravitational- wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D73, 064030 (2006), arXiv:gr-qc/0512160

    2006 arXiv

  42. [50]

    E. N. Dorband, E. Berti, P. Diener, E. Schnetter, and M. Tiglio, A Numerical study of the quasinormal mode excitation of Kerr black holes, Phys. Rev. D74, 084028 (2006), arXiv:gr-qc/0608091

    2006 arXiv

  43. [51]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Geodesic stability, Lyapunov exponents and quasinormal modes, Phys. Rev. D79, 064016 (2009), arXiv:0812.1806 [hep-th]

    2009 arXiv

  44. [52]

    H. Yang, D. A. Nichols, F. Zhang, A. Zimmerman, Z. Zhang, and Y. Chen, Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation, Phys. Rev. D86, 104006 (2012), arXiv:1207.4253 [gr-qc]

    2012 arXiv

  45. [53]

    Maenaut, G

    S. Maenaut, G. Carullo, P. A. Cano, A. Liu, V. Car- doso, T. Hertog, and T. G. F. Li, Ringdown analysis of rotating black holes in effective field theory extensions of general relativity, Phys. Rev. D113, 044039 (2026), arXiv:2411.17893 [gr-qc]

    2026 arXiv

  46. [54]

    github, BeyondKerrQNM (2024)

    2024

  47. [55]

    Capozziello and M

    S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept.509, 167 (2011), arXiv:1108.6266 [gr-qc]

    2011 arXiv

  48. [56]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Modi- fied Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys. Rept.692, 1 (2017), arXiv:1705.11098 [gr-qc]

    2017 arXiv

  49. [57]

    P. A. Cano and M. David, Teukolsky equation for near- extremal black holes beyond general relativity: Near- horizon analysis, Phys. Rev. D110, 064067 (2024), arXiv:2407.02017 [gr-qc]

    2024 arXiv

  50. [58]

    Ghosh, N

    R. Ghosh, N. Franchini, S. H. Völkel, and E. Barausse, Quasinormal modes of nonseparable perturbation equa- tions: The scalar non-Kerr case, Phys. Rev. D108, 024038 (2023), arXiv:2303.00088 [gr-qc]

    2023 arXiv

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Reviewed July 30, 2026 · model on record in the stance chip above.