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arxiv: 2601.13411 · v2 · submitted 2026-01-19 · 🌀 gr-qc · hep-th

Quasinormal modes and their excitation beyond general relativity. II: isospectrality loss in gravitational waveforms

Hector O. Silva , Giovanni Tambalo , Kostas Glampedakis , Kent Yagi This is my paper

Pith reviewed 2026-05-16 12:41 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modesblack hole ringdownbeyond general relativitygravitational wavesisospectralityeffective field theorymetric perturbations
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0 comments X

The pith

In cubic-curvature extensions of general relativity, black hole ringdown signals make the two fundamental quasinormal modes hard to identify in the time series.

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

The paper continues a study of quasinormal modes for a nonrotating black hole in the simplest effective-field-theory extension of general relativity that includes cubic-in-curvature terms. In general relativity the spectra for polar and axial perturbations are identical, but this isospectrality is lost in the extended theory. The authors perform time-domain simulations to determine how the loss appears in the gravitational-wave ringdown and whether the two fundamental modes can still be read off from the waveform. They conclude that, under their mapping from master functions to wave polarizations, identifying either mode is generally difficult, although a non-general-relativistic signature can sometimes be detected.

Core claim

The equivalence between the quasinormal-mode spectra of polar and axial metric perturbations no longer holds. In the time-domain ringdown obtained from numerical simulations, this loss of isospectrality means that neither of the two fundamental modes is readily identifiable from the waveform, although evidence for a non-general-relativistic mode can appear in some cases.

What carries the argument

The prescription that maps the gauge-invariant master functions for polar and axial metric perturbations onto the two gravitational-wave polarizations.

If this is right

  • Ringdown waveforms in this theory do not display clear, separable contributions from the two parities.
  • Mode identification in beyond-general-relativity ringdowns requires more than simple frequency matching.
  • The initial-data choice affects which modes are excited and therefore their visibility in the signal.
  • Deviations from general relativity remain detectable in principle but only under favorable conditions.

Where Pith is reading between the lines

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

  • Similar difficulties may appear in any modified-gravity model that breaks isospectrality, requiring new analysis methods for future detectors.
  • Relaxing the cubic truncation or changing the initial-data prescription could alter the degree of mode mixing observed.
  • The result suggests that ringdown tests of gravity will need to incorporate the full time-domain structure rather than isolated frequencies.

Load-bearing premise

The specific way chosen to relate the master functions of each parity to the wave polarizations, together with the cubic truncation of the effective field theory and the choice of initial data.

What would settle it

A high-signal-to-noise ringdown observation that either cleanly separates two distinct fundamental frequencies with different damping times matching the predicted polar and axial spectra, or exactly reproduces the single isospectral frequency of general relativity.

Figures

Figures reproduced from arXiv: 2601.13411 by Giovanni Tambalo, Hector O. Silva, Kent Yagi, Kostas Glampedakis.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of isospectrality breaking. Two degenerate [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Deviation from unity of the propagation speed squared [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The effective potentials [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Quadrupolar axial- and polar-parity waveforms in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Snapshots of the incident perturbation (solid line and [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between quadrupolar axial waveforms [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison between waveforms in general relativity [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison between waveforms in general relativity and in the effective field theory for various values of [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Illustration of the variable time window in which we [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Mismatch between polar (top panel) and axial (bot [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Gravitational waveforms. From left to right we show the comparable-mixing, polar-dominated, and axial-dominated [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Trajectories in the complex plane for the best-fit [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Dependence of the mismatch and the best-fit param [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Dependence of the mismatch and the best-fit pa [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The two real branches of the Lambert [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Local convergence analysis of the quadrupolar CPM [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The Schwarzian derivative and its comparison to the [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Snapshots of the incident perturbation (solid line [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Trajectories in the complex plane for the best [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Dependence of the mismatch and the best-fit param [PITH_FULL_IMAGE:figures/full_fig_p021_25.png] view at source ↗
read the original abstract

We continue our series of papers where we study the quasinormal modes, and their excitation, of black holes in the simplest beyond general relativity model in which first-principle calculations are tractable: a nonrotating black hole in an effective-field-theory extension of general relativity with cubic-in-curvature terms. In this theory, the equivalence between the quasinormal mode spectra associated with metric perturbations of polar and axial parities ("isospectrality") of the Schwarzschild black hole in general relativity no longer holds. How does this loss of isospectrality translate into the time-domain ringdown of gravitational waves? Given such a ringdown, can we identify the two "fundamental quasinormal modes" associated to the two metric-perturbation parities? We study these questions through a large suite of time-domain numerical simulations, by a prescription on how to relate the gauge-invariant master functions that describe metric perturbations of each parity with the gravitational polarizations. Under the assumptions made in our calculations, we find that it is in general difficult to identify either of the two fundamental modes from the time series, although finding evidence for a non-general-relativistic mode is possible sometimes. We discuss our results in light of our assumptions and speculate about what may occur when they are relaxed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper continues a series on quasinormal modes of nonrotating black holes in a cubic-in-curvature EFT extension of GR. It shows that isospectrality between axial and polar perturbations is lost, then uses a large suite of time-domain simulations to study how this appears in gravitational waveforms. The central claim is that, under the paper's assumptions on the mapping from gauge-invariant master functions to strain polarizations, it is generally difficult to identify either fundamental mode from the time series, although evidence for a non-GR mode can sometimes be extracted.

Significance. If the numerical results are robust, the work supplies concrete, simulation-based evidence on the practical observability of isospectrality breaking in ringdown signals. This is directly relevant to GW data analysis in modified gravity and highlights a potential obstacle for mode identification that is not present in GR. The explicit conditioning of the conclusion on the chosen mapping and initial-data family is a strength, as is the discussion of how relaxing assumptions might change the outcome.

major comments (2)
  1. [waveform extraction / methods] The prescription that maps the axial and polar master functions to the gravitational polarizations h+ and h× (described in the section on waveform extraction and used for all reported time series) is load-bearing for the headline result. Because the cubic EFT modifies the linearized equations, the standard GR relation between master functions and metric perturbations receives corrections at the same order; the paper adopts one particular truncation of this relation. A sensitivity test to plausible alternative truncations (or an explicit derivation showing the corrections are negligible) is needed to confirm that the reported difficulty of mode identification is not an artifact of this choice.
  2. [results / time-domain analysis] The claim that identification is 'in general difficult' rests on the specific initial-data family and the cubic truncation. The abstract and results sections report only qualitative statements from the simulations; quantitative fits, error budgets, and a systematic scan over initial-data parameters would be required to make the 'generally difficult' statement robust rather than dependent on the chosen setup.
minor comments (2)
  1. [abstract] The abstract states the conclusions are conditional on 'the assumptions made in our calculations' but does not list the key assumptions (mapping prescription, initial-data family, EFT truncation) explicitly; adding a short enumerated list would improve clarity.
  2. [figures] Figure captions and axis labels should explicitly indicate which curves correspond to axial versus polar contributions and which initial-data parameters are used, to make the large suite of simulations easier to interpret.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which we believe will improve the clarity and robustness of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [waveform extraction / methods] The prescription that maps the axial and polar master functions to the gravitational polarizations h+ and h× (described in the section on waveform extraction and used for all reported time series) is load-bearing for the headline result. Because the cubic EFT modifies the linearized equations, the standard GR relation between master functions and metric perturbations receives corrections at the same order; the paper adopts one particular truncation of this relation. A sensitivity test to plausible alternative truncations (or an explicit derivation showing the corrections are negligible) is needed to confirm that the reported difficulty of mode identification is not an artifact of this choice.

    Authors: We agree that the mapping between master functions and strain is central to the conclusions. In the manuscript we adopted the leading-order GR relation, consistent with the perturbative order of the cubic EFT corrections to the field equations. We will add an explicit discussion of the expected magnitude of the omitted corrections (which enter at higher order in the EFT expansion) together with a limited sensitivity test in which we vary the mapping coefficients within the range allowed by the truncation. This will be included in the revised version. revision: yes

  2. Referee: [results / time-domain analysis] The claim that identification is 'in general difficult' rests on the specific initial-data family and the cubic truncation. The abstract and results sections report only qualitative statements from the simulations; quantitative fits, error budgets, and a systematic scan over initial-data parameters would be required to make the 'generally difficult' statement robust rather than dependent on the chosen setup.

    Authors: We acknowledge that the current presentation relies on qualitative inspection of a broad suite of time-domain evolutions. In the revision we will supplement the results with quantitative mode-fitting analyses (including amplitude and phase residuals with error estimates) for representative cases and will extend the initial-data parameter scan to include a wider range of amplitudes and profiles. These additions will make the 'generally difficult' statement more quantitatively supported while remaining within the scope of the cubic truncation. revision: yes

Circularity Check

0 steps flagged

No circularity: central results from direct numerical evolution of EFT equations

full rationale

The paper derives its claims via time-domain numerical simulations of the cubic-curvature EFT equations for nonrotating black holes. The mapping from gauge-invariant master functions to gravitational polarizations is presented as an explicit modeling choice under stated assumptions, not as a fitted parameter or self-referential definition that is then relabeled as a prediction. No algebraic derivation reduces to its own inputs by construction, no uniqueness theorem is imported from self-citations to force the result, and no ansatz is smuggled via prior work. The findings are conditioned on the chosen prescription and initial data, but this conditioning does not create a circular loop; the numerical output remains independent of the input assumptions in the sense required by the circularity criteria. Self-citations to the authors' prior series are present but not load-bearing for the time-domain results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the validity of the chosen cubic-curvature EFT truncation and on the mapping between parity master functions and observable polarizations; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The effective-field-theory extension with cubic-in-curvature terms provides a controlled description of leading beyond-GR corrections to black-hole perturbations.
    This is the model adopted for the entire study.

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Forward citations

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  2. Gravitational electric-magnetic duality at the light ring and quasinormal mode isospectrality in effective field theories

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    Gravitational electric-magnetic duality at the light ring organizes and preserves quasinormal mode isospectrality in GR and selects duality-invariant higher-derivative corrections in effective field theories.

Reference graph

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