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This paper presents the first time-domain implementation of the parametrized modified Teukolsky equation, reproducing frequency-domain quasinormal-mode coefficients to about 2% for low azimuthal numbers while exposing amplitude, phase, and

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2026-08-01 02:50 UTC pith:7ILOKJZ4

load-bearing objection First working time-domain solver for the parametrized modified Teukolsky equation, honestly validated against frequency-domain benchmarks; the boundary-reflection worry is overstated because the domain is causally large for the reported times, but the code-release link needs attention. the 2 major comments →

arxiv 2607.25311 v1 pith:7ILOKJZ4 submitted 2026-07-28 gr-qc astro-ph.HEhep-th

Parametrized beyond-Teukolsky framework in the time domain

Ciro De Simone , Sebastian H. V\"olkel , Kostas D. Kokkotas , Salvatore Capozziello This is my paper
classification gr-qc astro-ph.HEhep-th
keywords black hole perturbation theoryquasinormal modesmodified Teukolsky equationtime-domain evolutionringdown amplitudes and phaseslate-time tailsbeyond-GR black hole testsKerr spacetime
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish that the parametrized beyond-Teukolsky framework—a theory-agnostic way of encoding small deviations from Kerr black-hole dynamics as a power-series deformation of the Teukolsky potential—can be evolved directly in the time domain, not just solved as an eigenvalue problem. It claims that the new time-domain code reproduces the frequency-domain quasinormal-mode coefficients to about 2% for low azimuthal numbers, with higher azimuthal numbers limited by spherical–spheroidal mode mixing, and that it delivers quantities an eigenvalue calculation cannot: the prompt response, ringdown amplitudes and phases, quadratic and mixed coefficient diagnostics, and the onset of the late-time power-law tail. If correct, this gives black-hole spectroscopy a validated, flexible tool for translating parametrized deviations from general relativity into full linear waveforms and for diagnosing when the linear approximation itself breaks down.

Core claim

The central claim is that the parametrized modified Teukolsky equation, with the radial potential augmented by δV(r) = Δ^{-1} Σ_{k=-K}^{4} α^{(k)} (r/r_+)^{k}, can be integrated as a (2+1)-dimensional scattering problem and that the resulting waveforms reproduce the established frequency-domain quasinormal-mode results: for ℓ=2, m∈{-1,0,1} at a=0.25, the extracted linear coefficients d^{(k)}_{ω,nℓm} agree with frequency-domain benchmarks to about 2%, while for m=±2 mode mixing degrades the real part to 5–6%. The same evolutions yield quadratic and mixed coefficients d^{(k1,k2)}_{ω,nℓm} within the linearized model, which the paper proposes as a diagnostic for the validity range of the linear

What carries the argument

The load-bearing object is the modified Teukolsky potential δV(r) = Δ^{-1} Σ_{k=-K}^{4} α^{(k)} (r/r_+)^{k}, a power-series deformation of the standard Kerr potential that is linear in the parameters α^{(k)}; the framework's key identity is the expansion of the quasinormal frequency as ω ≈ ω_0 + Σ_k d^{(k)}_{ω,nℓm} α^{(k)} + ½ Σ_{k1,k2} d^{(k1,k2)}_{ω,nℓm} α^{(k1)}α^{(k2)}, whose coefficients are the objects compared between frequency and time domains. In the time domain, the deformation is inserted into the Teukolsky equation as a radial modification of the angular potential term, and the equation is advanced with a second-order finite-difference scheme on a (r*, θ) grid using ingoing Gauss

Load-bearing premise

The results rest on placing the numerical boundaries at r*=-200M and +400M with ingoing/outgoing advection conditions and trusting that they do not contaminate the computed modes, amplitudes, phases, or tail onsets, even though no domain-size convergence study is reported for the modified potentials and the k=4 deformation is unstable.

What would settle it

Re-run the same Gaussian wave-packet evolutions for representative deformations (e.g., k=-4, α=0.1 and k=1, α=-3) with the inner boundary moved to r*=-400M and the outer to r*=800M, or with hyperboloidal compactified coordinates, and check whether the extracted d^{(k)}_{ω,nℓm}, ringdown amplitudes and phases, and tail-onset times shift by more than the quoted ~2% (or ~10% for tails); if they do, the validation is compromised. A complementary check is to compute k=3 and k=4 coefficients with a frequency-domain code that includes the 1/r and constant large-distance potential terms.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, the time-domain coefficients independently validate the frequency-domain eigenvalue results for low azimuthal numbers and mark the multipole range (roughly |m|≤1 for ℓ=2 at moderate spin) where direct QNM extraction from these simulations is reliable.
  • The quadratic and mixed coefficients give a practical bound on the deformation parameters—the linear approximation remains trustworthy only while |d^{(k)}| exceeds |d^{(k,k)} α^{(k)}|—which can be used to restrict viable beyond-GR parameters before theory-specific modeling.
  • Ringdown amplitudes and phases are predicted to be strongly affected by the deformation, with a sign pattern explainable by the near-horizon plateau: negative α suppresses transmission and boosts the reflected amplitude, positive α does the opposite; this adds observables beyond mode frequencies.
  • For k<3 deformations, the late-time tail keeps the Kerr power-law exponent (≈ t^{-(2ℓ+3)} within ~10% extraction uncertainty), but the onset of the tail can shift significantly because the deformation changes the damping time of the preceding ringdown.
  • The same time-domain tool can be turned to theory-specific modified-gravity models (higher-derivative gravity, dynamical Chern-Simons), non-separable or frequency-dependent deformations, overtones, superradiance, and instabilities, since only the potential terms in the evolution system need changing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The most consequential unstated implication is that this turns the parametrized framework into a waveform generator: one could in principle fit observed ringdowns directly for amplitudes, phases, and tail onsets, not just frequencies, which may break degeneracies in parameter estimation that frequency-only analyses suffer.
  • The boundary-truncation caveat suggests a specific test that the paper leaves open: implementing hyperboloidal compactified coordinates (which the authors mention as future work) would settle whether the quoted amplitude, phase, and tail-onset shifts—especially the late-time results the paper itself calls 'not very accurate'—survive with boundaries pushed to null infinity.
  • An interesting extension would be to compare the quadratic coefficients computed inside the linear model with the full second-order modified Teukolsky equation of a concrete theory; this would show whether the linear-model validity criterion overestimates or underestimates the true regime of validity, since the paper notes the true second-order terms can be comparable or larger.
  • Because the mode-mixing degradation for m=±2 is attributed to using spin-weighted spherical harmonics rather than spin-weighted spheroidal harmonics as initial data, a testable improvement would be to iterate the spheroidal basis using the deformed QNM frequency; if the 5–6% errors drop toward the ~2% level, the physical-mode-mixing explanation is confirmed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper presents the first time-domain implementation of the parametrized modified Teukolsky equation introduced in Ref. [39]. The authors evolve Gaussian wave packets on a (2+1)-dimensional grid, extract quasinormal-mode frequencies with the Prony method, and compare the linear deformation coefficients d^{(k)}_{\omega,n\ell m} with frequency-domain Leaver benchmarks. They report ≈2% agreement for low azimuthal numbers, with larger (5–6%) errors for m=±2 that they attribute to physical mode mixing. They also compute quadratic and mixed coefficients within the linearized modified-potential model, analyze ringdown amplitudes and phases, and study the late-time tail exponent and its onset. The central claim is that this time-domain framework provides a validated, theory-agnostic tool that goes beyond eigenvalue calculations, giving access to the full linear evolution and to new diagnostics for black-hole spectroscopy.

Significance. If the validation and new results are reliable, this is a useful complement to the eigenvalue-only framework of Ref. [39]. The authors are careful to quantify uncertainties: the m=0 Kerr baseline matches to ≈0.3%, the TD–FD comparison uses 68% HDI error bars, and the degradation for m=±2 is disclosed and supported by a resolution study. The quadratic-coefficient discussion is appropriately hedged: the authors explicitly note that these are coefficients of the linear modified equation, not complete second-order corrections of a generic theory. However, the paper's most novel claims — amplitude/phase trends and tail-onset shifts — depend on the finite-domain numerical treatment, whose convergence is not demonstrated. The validation against frequency-domain QNM frequencies does not cover these new results, because amplitude, phase, and tail onset are sensitive to the entire propagation, including the outer boundary region.

major comments (2)
  1. [Sec. V and Appendix A] The finite-domain claim is load-bearing for the amplitude/phase and tail-onset results. The domain is r*∈[-200M,400M] with advection boundary conditions (Sec. IIB), and Appendix A asserts without a convergence study that the domain is large enough. For k=3 the modified potential decays as 1/r at large distance, so at r*=400M it is still ~1/400 M^{-1}; for k=4 it tends to a constant. An outgoing advection boundary condition is therefore not asymptotically exact for these modified potentials, and partial reflections can contaminate the very quantities Sec. V uses to draw physical conclusions. The QNM-frequency validation in Sec. III does not test this, since ringing is localized near the potential peak. I request either a domain-size convergence study for the k=3 (and, if possible, k=4) modified potentials, or a clear restriction of the claims to the range where boundary effects are demons
  2. [Sec. VB] The tail analysis is explicitly qualified by the statement 'the code is not very accurate at late times'. The claim that the tail exponent is unchanged but its onset is shifted is based on waveforms such as Fig. 9, whose late-time segments are admitted to be noisy. The onset shifts shown for α^(1)=±3 are outside the linear regime and are not accompanied by a quantified uncertainty. Without an estimate of how numerical noise and boundary reflections affect the extracted onset time, the quantitative statement 'can substantially shift the onset' is not yet supported. A convergence study in domain size and resolution, with a defined criterion for the onset time, would make this result robust.
minor comments (3)
  1. [General] There are several typographical and formatting issues, e.g., 'Wignerd-functionsd psq ℓm' in Eq. (19) and inconsistent use of r* vs r˚. Please also define all symbols in Eq. (5) at first use and clarify the units in the discussion of d^{(k)}_{\omega,n\ell m}.
  2. [Sec. III, Figs. 4–5] The labels in the figures and captions could be clearer about the specific values of n, ℓ, and a used. In particular, the hierarchy among m values in Fig. 4 is mentioned in the text but not visible from the figure alone; consider adding markers or a legend that encodes the m values consistently.
  3. [Appendix A] The code availability statement refers to Ref. [77], which appears to be the frequency-domain framework. Please state explicitly whether the time-domain code used for this paper is also publicly available, and if so, where.

Circularity Check

0 steps flagged

No significant circularity: the time-domain solver is cross-checked against an independent frequency-domain computation of the same modified operator.

full rationale

The paper's central derivation is a numerical time-domain integration of the modified Teukolsky equation (11) with the potential deformation δV of Eq. (5). The deformation parameters α^(k) are inputs, not outputs, and the time-domain extraction of QNM frequencies is compared with frequency-domain Leaver-method coefficients from Ref. [39] (Sec. III, Figs. 3–5). This is a cross-check of two independent numerical evaluations of the same linear operator; the reported 2% (m=0,±1) and 5–6% (m=±2) disagreements show the comparison is empirical rather than constructional. The quadratic and mixed coefficients (Sec. IV) are explicitly labeled as diagnostics of the linear model: 'the quadratic coefficients are computed in the linear modified Teukolsky equation. The correct quadratic coefficients will, in general, contain additional contributions associated with the second-order modified Teukolsky equation.' The amplitude/phase and tail-onset results (Sec. V) are direct extractions from the simulated waveforms; no parameter is fitted to those quantities. The only author-overlapping reference is Ref. [39], but it is a published independent computation using Leaver's method with a stated code repository, so it is external evidence rather than a load-bearing self-citation. Appendix A's finite-domain boundary discussion is a numerical accuracy concern, not an identity-by-construction issue. No step in the derivation chain reduces to its own input.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The paper's free inputs are the deformation parameters α^(k) and the hand-chosen numerical/initial-data settings. The domain assumptions are the physical mapping of the δV potential family to beyond-GR physics, the small-coupling structure of the modified equation, the correctness of the shared-author frequency-domain benchmark, and the adequacy of standard boundary/initial-data treatments for the modified operator. No new physical entities are introduced: δV is a phenomenological modeling term from Ref. [39], and the quadratic/mixed coefficients are fit outputs of the linear model, explicitly labeled as such.

free parameters (4)
  • α^(k) deformation parameters = scanned by hand: α ∈ [−0.1, 0.1] for linear fits; α = −3 for the tail illustration
    The independent variables of the model (Eq. 5); all results are response functions of them. Ranges are chosen by hand.
  • Gaussian initial data (width, center, extraction point) = σ = 1M, r*_0 = 12M, extraction at r* = 30M, θ = π/2
    The amplitude and phase results (Sec. VA) depend on this hand-chosen wave packet; the paper acknowledges these quantities are initial-data-dependent.
  • Numerical grid and domain = Nr = 8000, Nθ = 32; r* ∈ [−200M, 400M]
    Chosen by hand; a resolution study supports them, but no domain-size convergence test is reported for the modified potentials, on which the tail-onset claims rest.
  • Prony fit windows and α-grids = hundreds of starting times; 5 α values per fit; quadratic α-grid spacing not stated
    Extraction systematics (Sec. III) depend on these choices; the quadratic-fit grid spacing is not fully specified, limiting exact replication.
axioms (6)
  • domain assumption The modified radial Teukolsky equation (Eq. 1) with δV = (1/Δ) Σ_k α^(k)(r/r+)^k is the correct leading linearized perturbation operator for a broad class of beyond-GR theories.
    Taken from Ref. [39] (overlapping authorship); the interpretation of all results as beyond-GR signatures rests on it. Sec. IIA, Eq. (5).
  • domain assumption Small-coupling approximation: the deformation enters only the potential term, leaving the derivative structure of the equation unchanged.
    Sec. IIB: 'the deformation in Eq. (1) only affects the last term in Eq. (11)'; Sec. IV concedes full second-order theory dynamics would add terms of comparable size.
  • standard math Leaver's continued-fraction method gives correct frequency-domain eigenvalues and d^(k) coefficients for the modified radial equation.
    Used as the validation benchmark (Sec. III, Refs. [39,76]); independent of the time-domain code but computed by overlapping authors.
  • standard math The modified Lax-Wendroff scheme remains second-order convergent and stable when δV is added to the potential.
    Appendix A: hyperbolicity is preserved because matrix M is unchanged, but no stability proof or convergence study for the modified equation is given beyond one resolution check.
  • domain assumption The Price-law tail ~ t^−(2ℓ+3) governs late-time decay, and k<3 deformations do not change the exponent.
    Sec. VB; compared with ~10% uncertainty and consistent with Ref. [58]. The tail-onset shift claim depends on identifying the power-law segment in noise-limited data.
  • domain assumption The near-horizon modified field behaves as Δ^−s(r−r+)^{−iσ} with σ in Eq. (A21), so standard QNM-type advection boundary conditions at finite r* remain adequate.
    Appendix A; no domain-size convergence demonstration is provided for modified potentials.

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read the original abstract

Modifications to General Relativity can significantly alter the perturbative response of black holes, leaving imprints on quasinormal-mode spectra, waveform amplitudes and phases, and late-time tails. The parametrized beyond-Teukolsky framework was introduced to capture possible deviations from Kerr dynamics, but the ringdown has so far only been explored as an eigenvalue problem. We present the first time-domain implementation of this framework and perform (2+1)-dimensional scattering experiments with Gaussian wave packets. This approach provides the full linear evolution of the perturbation, from the initial prompt response through the ringdown and into the late-time regime. Using frequency-domain eigenvalue results as benchmarks, we find excellent agreement for low azimuthal numbers, while higher azimuthal numbers are affected by mode mixing, which limits the precision of the extracted modes. We further use the time-domain waveforms to estimate quadratic and mixed coefficients within the linearized modified-potential model, providing a diagnostic for the regime of validity of the linear approximation. Beyond mode frequencies, we show that the deformation parameters can strongly affect the ringdown amplitude and phase, with trends that can be understood from the near-horizon structure of the modified potential. We also analyze the late- time behavior, finding that near-horizon deformations leave the tail exponent unchanged but can substantially shift the onset of the power-law decay. These results demonstrate that time-domain evolutions provide a complementary and flexible framework for testing parametrized deviations from General Relativity in black-hole perturbation theory.

Figures

Figures reproduced from arXiv: 2607.25311 by Ciro De Simone, Kostas D. Kokkotas, Salvatore Capozziello, Sebastian H. V\"olkel.

Figure 1
Figure 1. Figure 1: FIG. 1. Real and imaginary parts of the modified Teukolsky potential for angular momentum [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Waveforms obtained for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relative error [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the linear coefficients [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Real and imaginary part of the quadratic coefficients [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Real and imaginary part of the mixed coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Fractional deviation of the amplitude [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Waveforms for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of the linear coefficients from frequency and time domain at [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Color map of the relative error ( [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

  1. Discrete symmetries of modified Teukolsky equations

    gr-qc 2026-07 conditional novelty 5.0

    Complex modifications of the Teukolsky potential break m=0 QNM degeneracy and can inject non-physical mode branches when frequency-domain multipole-dependent potentials are evolved in time.

  2. Constraining deviations from the Teukolsky equation with GW250114

    gr-qc 2026-07 conditional novelty 5.0

    GW250114's fundamental ringdown mode bounds theory-agnostic deviations from the Teukolsky equation to be consistent with zero at characteristic scales of 60-100 km.

Reference graph

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