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Detection of a Higher Harmonic Quasi-normal Mode in the Ringdown Signal of GW231123

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The ringdown of GW231123 contains the fundamental 220 quasinormal mode and a statistically significant 200 mode, with log10 Bayes factor 5.3 at a start time 12 M after peak, and the pair passes a no-hair test.

desk verdict A clever new F-statistic evidence formalism wrapped around a ringdown detection whose headline claim looks physically doubtful because the 200 mode would have to be about a hundred times the fundamental at the merger peak. read the letter →

arxiv 2509.02047 v1 pith:5FMFDDN4 submitted 2025-09-02 gr-qc astro-ph.HE

Hai-Tian Wang , Shao-Peng Tang , Peng-Cheng Li , Yi-Zhong Fan This is my paper · ORCID
classification gr-qcastro-ph.HE PACS 04.30.-w04.70.-s
keywords blackholespectroscopyquasinormalmodesringdownGW231123F-statisticBayesfactorno-hairtheoremgravitationalwavedataanalysis
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 analyzes the ringdown of the binary black hole merger GW231123 and claims that, in addition to the expected fundamental 220 quasinormal mode, the signal contains a statistically significant 200 mode. The evidence peaks at a start time 12 M after the peak, with log10 Bayes factors of 5.3 (F-statistic) and 5.9 (traditional time-domain) over a fundamental-mode-only model; at the maximum-likelihood point the 200 amplitude is about an order of magnitude larger than the 220 amplitude. Using the two modes, the authors recover a redshifted remnant mass of 305.6^{+35.7}_{-47.3} solar masses and spin 0.84^{+0.07}_{-0.14}, consistent with full inspiral-merger-ringdown analyses with two independent waveform models, and a no-hair test returns no deviation from general relativity. If correct, this would be a rare secure detection of a subdominant ringdown mode inside the linear regime and a demonstration that the F-statistic can do full Bayesian model comparison.

What carries the argument

The load-bearing mechanism is the F-statistic extended to time-domain ringdown analysis. The quasinormal-mode waveform is split into linear amplitude/phase parameters and nonlinear parameters such as remnant mass, spin, inclination, sky location, and peak time; the linear parameters are maximized analytically, giving a profile likelihood in a low-dimensional nonlinear parameter space. The paper adds a Gaussian reconstruction of the linear-parameter posteriors plus an importance-sampling formula for the full Bayesian evidence, making model comparison possible. The central physical object is the (ell, |m|, n) = (2, 0, 0) Kerr quasinormal mode, the higher harmonic whose presence is claimed in t

What would settle it

Re-analyze the same data while marginalizing over sky location, polarization angle, and peak time rather than fixing them, and check whether the log10 Bayes factor for 220+200 over 220 stays above about 3 and the mass-spin posterior still overlaps the inspiral-merger-ringdown posteriors; alternatively, inject a 220-only signal into the same noise realization and ask how often the pipeline reports comparable 200-mode evidence.

Watch

Extended reading notes

Core claim

The central claim is that GW231123's ringdown contains at least two Kerr quasinormal modes: the fundamental 220 mode and the higher harmonic 200 mode, with the latter strongly preferred by the data at a start time 12 M after the polarization peak. The paper estimates the detection significance as log10 Bayes factor 5.3 with the F-statistic and 5.9 with the traditional time-domain sampler, and finds the 200-mode amplitude roughly ten times the 220-mode amplitude at the maximum likelihood. From the two-mode fit it derives remnant mass and spin, reports consistency with the NRSur7dq4 and SEOBNRv5PHM inspiral-merger-ringdown posteriors, and finds the frequency and damping time of the 200 mode co

Load-bearing premise

The detection rests on assuming that at 12 M after peak (about 18 ms) the signal is a linear Kerr ringdown, and that fixing sky location, polarization angle, and peak time to the NRSur7dq4 maximum-likelihood values does not bias the comparison.

Editorial extensions

If this is right

  • GW231123 becomes a two-tone black-hole spectroscopy source: the ringdown alone fixes the remnant's mass and spin, and those values agree with full inspiral-merger-ringdown analyses without the bimodal posteriors seen in fundamental-mode-only fits.
  • Start times as early as 12 M after peak can be used for ringdown model selection when subdominant modes are included, extending the usable ringdown data.
  • The no-hair test with the two detected modes places a direct observational constraint on the 200-mode frequency and damping time, and finds them compatible with Kerr predictions.
  • The F-statistic evidence-reconstruction formalism makes Bayesian model comparison between different quasinormal-mode combinations computationally practical, so scanning many mode combinations becomes feasible.
  • No three-mode combination, including 220+200+330, beats the two-mode model at safe start times, so the claimed spectrum is exactly two resolved tones.

Reading between the lines

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

  • Because the sky location, polarization angle, and peak time are fixed to the NRSur7dq4 maximum-likelihood values rather than marginalized, the 200-mode evidence could change under a fully marginalized analysis; re-running with those parameters varying would give the cleanest test.
  • The 200-to-220 amplitude ratio at 12 M is so large that it implies an even larger ratio at earlier times unless the 200 mode decays much faster; this points toward a strongly precessing or non-quasi-circular source, and comparing the measured mode amplitudes with precessing numerical-relativity templates would discriminate among the scenarios the paper lists.
  • The same F-statistic evidence and posterior reconstruction could be applied to other disputed ringdown claims, such as the earlier subdominant-mode detection in GW190521, where the conservative start time was a central issue.
  • If the 200-mode dominance is reproduced by independent pipelines, it would imply that some binary black holes ring down with mode energies quite different from quasi-circular aligned-spin expectations, informing ringdown templates for future detectors.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents a Bayesian time-domain ringdown analysis of GW231123 using two methods: a traditional stochastic sampler (TTD) and a newly extended F-statistic framework. It reports that the model containing the 220 and 200 quasinormal modes is strongly preferred over a 220-only model, with log10 Bayes factor 5.3 (F-statistic) at Δt=12M after the polarization peak, and that the two methods agree. The inferred remnant parameters are Mf=305.6^{+35.7}_{-47.3} Msun and χf=0.84^{+0.07}_{-0.14} (F-statistic), overlapping with full IMR results. A no-hair test allowing deviations in the 200 mode frequency and damping time finds no deviation at the preferred start time. A search for a third mode finds no compelling evidence.

Significance. If the detection is real, it would be an important result: a higher harmonic (200) mode in a ringdown, enabling a multimode no-hair test. The extension of the F-statistic to compute evidences and posteriors is a useful methodological contribution, and the cross-check between TTD and F-statistic is a strength. The authors are candid about the anomaly of the large 200 amplitude. However, the central claim depends on the assumption that at Δt=12M the signal is a linear superposition of Kerr QNMs; the paper does not yet provide the validation needed to exclude the alternative explanation that the fast-decaying 200 template absorbs the nonlinear merger transient.

major comments (4)
  1. [Sec. III; Fig. 4] The linear-regime assumption at Δt=12M is load-bearing and is not validated. The recovered amplitudes are A200=(2.9–3.9)e-20 and A220=(1.4–3.1)e-21, so A200/A220≈20. For the m=0 mode with τ≈6M, the corresponding peak amplitude would be far larger than the 220 mode, which is difficult to reconcile with a subdominant linear perturbation. The authors themselves write that 'The pronounced dominance of the 200 mode raises questions about its origin' and offer untested explanations. Moreover, Fig. 4 shows δf200 deviating from GR at Δt=8M and 10M, demonstrating sensitivity to non-linear contamination at nearby start times. I ask for an NR/IMR injection-recovery test showing that (i) a known 200 mode with moderate amplitude is recovered without bias at 12M, and (ii) a non-linear merger residual does not produce a spurious 200 detection with log10 BF≈5. Without such a test, the detection and the
  2. [Sec. II; Fig. 1] The headline log10 BF=5.3 is the maximum over a scan of start times (8 to 18M in 2M steps) and seven mode combinations. No trials-factor or look-elsewhere correction is applied. This inflates the statistical significance of the 200 detection. Please report the evidence integrated over the scan or a corrected Bayes factor. If the trials factor is applied, the evidence may remain large, but it must be quantified.
  3. [Sec. II] The analysis fixes RA, DEC, ψ, and the polarization peak time to the NRSur7dq4 maximum-likelihood values and then compares the resulting posteriors with NRSur7dq4 IMR posteriors. This makes the claimed consistency partially built in and can bias ringdown amplitudes and evidence if the extrinsic parameters are inaccurate. A marginalization over these nuisance parameters, or an injection test demonstrating that fixing them does not bias the 200 amplitude and evidence, is needed. Comparing with SEOBNRv5PHM is helpful but is not a substitute.
  4. [Appendix A, Eqs. (A13)–(A15)] The F-statistic evidence reconstruction is a new contribution, but no validation on simulated injections is shown. The cross-check with TTD is reassuring, but an explicit test that the importance-sampling evidence estimator is unbiased for the adopted priors and ACF would strengthen the methodological claim and the model comparison.
minor comments (5)
  1. [Fig. 5 caption] The caption says '220 + 201' in the bottom-right panel; it should be '220 + 200'.
  2. [Introduction] Typo: 'NRSurd7q4' should be 'NRSur7dq4'.
  3. [Sec. III] The statement that the 200 amplitude is 'approximately an order of magnitude larger' is imprecise; the maximum-likelihood ratio is about 20.
  4. [Sec. II] The sentence 'To isolate the ringdown signal and mitigate contamination from the pre-merger phase, we fix the sky location...' is confusing: fixing extrinsic parameters does not by itself remove pre-merger contamination. Please clarify.
  5. [Sec. II / Appendix A] The no-hair deviation parameters δf200 and δτ200 are introduced only in words; please include their definitions in the model equations and specify their priors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the ringdown detection is a likelihood-based model comparison using GR-derived QNM frequencies, not a reduction to fitted inputs or self-citations.

full rationale

The central claim—detection of the 200 mode—is obtained by comparing two time-domain likelihood models (220-only vs 220+200) on the same strain data, with QNM frequencies and damping times computed from Kerr perturbation theory for sampled (Mf, χf). The Bayes factor is a model-comparison statistic, not a refit of an input quantity; the 220+200 model is not derived from the 220-only fit. The fixed RA/DEC/ψ/tc values taken from the NRSur7dq4 IMR analysis are nuisance parameters used to isolate the ringdown segment, and the subsequent consistency check against IMR posteriors is a comparison, not an input to the detection. The no-hair test parameterizes deviations from GR and measures them, rather than assuming GR. The F-statistic evidence formalism is derived in Appendix A, and the methodological self-citations to earlier F-statistic work are not load-bearing for the physical result because the necessary likelihood and evidence expressions are presented in the paper. No equation reduces the claimed detection to a fitted value, to a self-citation chain, or to a uniqueness theorem. The physical concern that A200 ≫ A220 at Δt = 12M may indicate nonlinear contamination is a correctness and interpretation risk, not circularity: it does not make the Bayes factor equivalent to its input by construction. The paper also cross-checks against an independent TTD method and against the external SEOBNRv5PHM IMR model, so the analysis is not self-referential in a way that forces the conclusion.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim carries a modest ledger: fitted mass, spin, amplitudes, phases, inclination, and a chosen start time, plus standard Kerr-QNM and stationary-noise assumptions. The paper introduces no new physical entity; its exotic source interpretations reuse existing boson-star and Proca-star literature.

free parameters (8)
  • Redshifted final mass Mf = 305.6+35.7-47.3 Msun (F-stat, 90% CL)
    Posterior inferred from 220+200 ringdown; prior [100, 500] Msun; sets the QNM frequency scale.
  • Final spin chi_f = 0.84+0.07-0.14 (F-stat, 90% CL)
    Posterior inferred from the same two-mode model; prior [0, 0.99]; determines the QNM splittings.
  • Inclination cos iota = posterior not quoted; flat prior [-1,1]
    Sampled in both methods; modulates the spheroidal harmonic amplitudes and the relative mode excitation.
  • Amplitude A220 = 1.4e-21 (maximum likelihood, F-stat at dt=12M)
    Free linear parameter; analytically maximized in F-statistic, then reconstructed; anchors the fundamental mode.
  • Amplitude A200 = 2.9e-20 (maximum likelihood, F-stat at dt=12M)
    Free linear parameter; drives the reported Bayes factor for the 200 mode.
  • Phases phi220 and phi200 = not quoted; uniform prior [0, 2pi]
    Linear parameters in the F-statistic; analytically maximized and later reconstructed.
  • No-hair deviation parameters delta_f200 and delta_tau200 = 0.26+0.68-0.38 and -0.24+0.51-0.32 (90% CL)
    Introduced only in the no-hair test; fitted and consistent with zero, but constraints are broad.
  • Analysis start time dt/M = 12 M (selected from discrete scan over [8,18] M)
    Chosen as the Bayes-factor maximum over start times and mode combinations; not marginalized and no trials correction.
assumptions (7)
  • domain assumption Kerr no-hair theorem: QNM frequencies and damping times are functions of only Mf and chi_f.
    Used in Eq. (A1) and all template models; both the detection and the no-hair test assume these GR relations.
  • domain assumption The ringdown at dt >= 12 M is linear and well described by a finite sum of Kerr QNMs (here 220 and 200).
    The authors invoke the literature, but the recovered 200 amplitude is larger than the fundamental, and nonlinear contamination would make the detection spurious.
  • domain assumption Detector noise is stationary and Gaussian with covariance estimated from the same data segment.
    Used in the TD likelihood, Eq. (A2) to (A4); no injection-based calibration is shown.
  • ad hoc to paper Sky location, polarization angle, and peak time can be fixed to NRSur7dq4 maximum-likelihood values without marginalizing.
    Sec. II fixes (RA, DEC, psi) = (3.24, 0.25, 2.23); this imports IMR model information into the 'independent' ringdown measurement.
  • domain assumption Spin-weighted spheroidal harmonics can be replaced by spin-weighted spherical harmonics.
    Appendix A.1; standard for SNR-limited ringdown, but introduces a systematic approximation in the angular pattern.
  • domain assumption Analysis priors: amplitudes uniform in [0,5e-19], Mf flat in [100,500] Msun, chi flat in [0,0.99], phase uniform.
    Used for both TTD and as the F-stat target prior via importance resampling; the amplitude prior upper bound can affect evidence values.
  • standard math The F-statistic evidence reconstruction via importance sampling, Eq. (A13) to (A15), is unbiased with the finite number of draws used.
    Assumes 100 times N0-squared conditional draws and the retained posterior samples are sufficient; no convergence diagnostics are reported.

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Cite this review

Pith. "Pith review of Detection of a Higher Harmonic Quasi-normal Mode in the Ringdown Signal of GW231123." pith.science (2026). https://pith.science/paper/5FMFDDN4

@misc{pith2026250902047,
  author       = {Pith},
  title        = {Pith review of: Detection of a Higher Harmonic Quasi-normal Mode in the Ringdown Signal of GW231123},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FMFDDN4}},
  note         = {Machine review of arXiv:2509.02047}
}
abstract

The ringdown phase of a gravitational wave signal from a binary black hole merger offers a unique laboratory for testing general relativity in the strong-field regime and probing the properties of the final remnant black hole. In this study, we analyze the ringdown of GW231123 and find strong evidence for a multimode quasinormal spectrum. Our analysis employs two time-domain methodologies: a full Bayesian inference and an enhanced F-statistic framework, which we extend to enable the calculation of Bayesian evidence and the reconstruction of posterior distributions for all model parameters. We report a statistically significant detection of the $\ell|m|n=200$ mode, with a $\log_{10}$(Bayes factor) of $5.3$, commencing at $12\,M$ after the peak amplitude--a time well within the accepted linear regime. This two-mode analysis yields a redshifted final mass of $305.6^{+35.7}_{-47.3}M _{\odot}$ and a final spin of $0.84^{+0.07}_{-0.14}$ at $90\%$ credibility, from a ringdown signal with a network signal-to-noise ratio of approximately $14.5$. Furthermore, a test of the no-hair theorem performed using the two detected modes reveals no deviation from the predictions of general relativity. These results highlight the power of the F-statistic methodology to uncover nuanced features in gravitational wave signals, thereby providing novel insights into the fundamental properties of black holes.

Figures

Figures reproduced from arXiv: 2509.02047 by the authors.

Figure 1
Figure 1. FIG. 1: Bayesian evidence for various two-mode QNM com [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Posterior distributions for the redshifted final mass [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: A direct comparison of the analysis of the 220 + 200 mode combination using the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Tests of the no-hair theorem using the 220 + 200 mode combination in the ringdown signal of GW231123. The x [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The posterior distributions of the redshifted final mass [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Bayesian evidence for three QNMs combinations in the ringdown analysis of GW231123, quantified by the base-10 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Constraints on the redshifted final mass ( [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Foundations of Direct Waves in Schwarzschild Ringdown

    gr-qc 2026-07 accept novelty 7.0 of 10

    Direct waves in filtered Schwarzschild ringdown are the anti-causal filter-pole contribution sourced by near-horizon trajectory dynamics and do not vanish.

  2. Ensemble-Based Residual Tests of GW231123 across Waveform Models

    gr-qc 2026-08 conditional novelty 6.0 of 10

    GW231123's residuals are consistent with detector noise for all five waveform models, and the strain-level differences between models are too small to be detected.

  3. GW250114 reveals black hole horizon signatures

    gr-qc 2025-10 conditional novelty 6.0 of 10

    The merger signal of GW250114 contains a residual 'direct wave' component whose frequency and damping match the remnant horizon's rotation frequency and surface gravity.

  4. First Overtone Mode in the Ringdown Signal of GW231028

    gr-qc 2025-09 conditional novelty 6.0 of 10

    The ringdown of GW231028 contains evidence for the 221 first overtone mode alongside the dominant 220 mode, with a Bayes factor of 193 at 10M after the peak.

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Reviewed August 5, 2026 · model on record in the stance chip above.