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Testing general relativity with amplitudes of subdominant gravitational-wave modes

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Pith's one-line read An improved subdominant-mode amplitude test of general relativity, extended to the (4,4) and (3,2) modes, gives the strongest constraint yet on the hexadecapolar (4,4) mode amplitude, δA44 = −0.30^{+1.16}_{−3.45}, consistent with GR.

desk verdict Solid incremental extension of the SMA test to (4,4)/(3,2) with an honest benchmark, but the headline δA44 constraint is thinner and less calibrated than the abstract claims. read the letter →

arxiv 2511.11886 v2 pith:MSI6FDDI submitted 2025-11-14 gr-qc astro-ph.COhep-th

Ish Gupta , Purnima Narayan , Lionel London , Shubhanshu Tiwari , Bangalore Sathyaprakash This is my paper · ORCID
classification gr-qcastro-ph.COhep-th
keywords generalrelativitytestsgravitational-wavehigher-ordermodessubdominant-modeamplitudesbinaryblackholemergerswaveformsystematicsspinprecessionorbitaleccentricityBayesianparameterestimation
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

This paper extends a test of general relativity that rescales the amplitudes of subdominant gravitational-wave modes — the (2,1), (3,3), (4,4), and (3,2) harmonics — while holding the dominant quadrupole mode fixed. The authors show that the test recovers injected deviations, produces null results for GR-consistent signals in Gaussian noise for aligned and mildly precessing binaries, and yields the strongest constraint to date on the (4,4) mode amplitude from a real event, δA44 = −0.30^{+1.16}_{−3.45}, consistent with GR. They also show the test is sensitive to phase deviations and that strong spin precession or orbital eccentricity can mimic apparent GR violations when unmodeled. A sympathetic reader would care because it validates a complementary, amplitude-based null test of strong-field gravity and maps the regime where it can be trusted.

What carries the argument

The SMA modification of the waveform: h → dominant quadrupole terms + Σ_HOM (1+δAℓm) hℓm, with each subdominant mode amplitude modified independently while the (2,2) quadrupole mode is kept fixed. The test's statistical power is carried by the orthogonal mode SNR ρ⊥ℓm, which measures how much signal cannot be explained by the dominant quadrupole mode, and by the one-at-a-time Bayesian estimation of each δAℓm. A load-bearing degeneracy is between δA33, the inclination angle, and the reference orbital phase: when the (3,3) mode is weak, the reference phase becomes bimodal and δA33 develops a secondary peak near −2 that can mimic a deviation. The (3,2) mode is singled out because it contributes

What would settle it

Run the SMA test on hundreds of Gaussian noise realizations of a GW230814-like signal and require the fraction of runs with δA44=0 outside the 99% CI to be ≈1%; if the bimodality produces an inflated false-alarm rate, the quoted CI is too narrow. Separately, reanalyze GW230814 with an independent waveform model and check whether the δA44 posterior and the GR-consistency conclusion are unchanged.

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Extended reading notes

Core claim

The paper claims that an improved subdominant-mode amplitude (SMA) test, which lets only the amplitudes of the (2,1), (3,3), (4,4), and (3,2) modes float while fixing the quadrupole (2,2) mode, is a reliable null test of general relativity in the aligned-spin and mildly precessing binary black hole regime. Benchmarked on Gaussian noise injections and numerical-relativity waveforms, the test returns unbiased posteriors and GR-consistent Bayes factors for those systems. Applied to the events GW241011 and GW230814, it gives δA33 = 0.00^{+0.46}_{−1.82} and δA44 = −0.30^{+1.16}_{−3.45}, the latter the tightest published bound on the (4,4) mode amplitude deviation, both consistent with GR. The aut

Load-bearing premise

The event-level GR-consistency claims assume that the waveform model used for parameter estimation is an accurate GR template for those signals and that the deviation-parameter null distribution, calibrated from just 20 noise realizations, fully captures the degeneracies that can mimic deviations.

Editorial extensions

If this is right

  • The (3,2) mode extends the test to near-equal-mass and face-on binaries, where the (2,1) and (3,3) modes are weak, so amplitude tests can now cover a larger share of detected black hole mergers.
  • The constraint δA44 = −0.30^{+1.16}_{−3.45} for GW230814 is the strongest published bound on a (4,4) amplitude deviation and is consistent with GR.
  • The δA33 posterior for GW241011, 0.00^{+0.46}_{−1.82}, likewise does not reject GR and is among the tightest constraints on that mode.
  • For strongly precessing or eccentric high-mass systems, apparent deviations reported by the test (e.g., for GW231123) should be interpreted as waveform-modeling systematics rather than evidence against GR.
  • Because the test also responds to phase perturbations, a nonzero δAℓm is not proof of an amplitude anomaly; it can flag phase-level deviations that the model does not include.

Reading between the lines

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

  • A larger Gaussian-noise injection campaign (hundreds of realizations) would calibrate the bimodal δA33 and δA44 tails; the current 20-realization p-p plot leaves the quoted 99% false-alarm rate uncertain.
  • The test's phase sensitivity suggests a cheap diagnostic for catalog events: check the secondary spin posterior from the GR fit — near-extremal spins, as seen in the eccentric and phase-deformed injections, signal unmodeled physics rather than a real Kerr black hole.
  • A hierarchical combination of δAℓm posteriors across many events could turn the per-event null test into a population-level bound, with the (3,2) mode as the most phase-sensitive channel.
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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

3 major / 5 minor

Summary. The paper presents an extension of the subdominant-mode amplitude (SMA) test of general relativity to the (3,2) and (4,4) modes, in addition to the previously considered (2,1) and (3,3) modes. The test is benchmarked through Gaussian-noise injections, numerical-relativity simulations (SXS), injection-recovery of amplitude deviations, and responses to phase-modified waveforms. The authors then apply the test to several O4 events, reporting a constraint on the (4,4) amplitude deviation from GW230814, δA44 = -0.30^{+1.16}_{-3.45}, which they describe as the strongest to date, and a constraint on δA33 from GW241011. The paper also demonstrates that waveform systematics can mimic GR violations for strongly precessing or eccentric binaries, and that the test responds to phase perturbations as well as amplitude perturbations.

Significance. If the statistical calibration and event-level constraints hold, the SMA test would be a useful null test of GR that is complementary to standard phasing tests, and the extension to (3,2) and (4,4) modes broadens its applicability to more symmetric and face-on binaries. The paper's systematic benchmarking against SXS waveforms and its explicit demonstration of systematics-induced biases in high-mass precessing systems are valuable contributions. However, the headline robustness claim rests on a calibration with only 20 noise realizations and on event-level results for systems whose mode SNR is marginal by the paper's own selection criterion, so the empirical validation is weaker than the abstract suggests.

major comments (3)
  1. [Sec. III B, Fig. 3] The statistical calibration uses only N=20 Gaussian noise realizations for a single binary configuration. The p-p plot shows δA33 touching the 3σ contour and the 60% CI contains the injected value in >80% of runs, which is a notable deviation from expectation. Since the central claim that the SMA test is a validated null test rests on this calibration, 20 realizations is too small to establish the false-alarm rate, especially for δA33, whose bimodal degeneracy with reference phase and inclination (Appendix B) is shown to produce broad, over-covering intervals. The paper should either increase N substantially or present a quantitative uncertainty on the calibration curve and discuss how the δA33 behavior affects the interpretability of event-level δA33 constraints.
  2. [Sec. V A, Appendix A] The headline event-level result, δA44 for GW230814, is selected using the criterion that the 68% lower bound of ρ⊥44 exceeds 2.145 (the 90th percentile of a χ2 distribution). The paper reports ρ⊥44 = 3.39^{+0.41}_{-1.26}, whose 68% lower bound is 2.13, below the stated threshold. Thus GW230814 does not satisfy the paper's own selection criterion. Moreover, no injection-recovery study is performed for a GW230814-like configuration; the only weak-HOM injection study (Appendix B, Fig. 14) is for GW250114 and shows that such configurations yield broad, bimodal δA44 posteriors dominated by degeneracies and noise. Without event-specific calibration, the quoted interval δA44 = -0.30^{+1.16}_{-3.45} cannot be interpreted as a meaningful constraint on the (4,4) amplitude, and the claim that it is the 'strongest constraint' is not established.
  3. [Sec. IV A, Fig. 8] The injection-recovery tests for amplitude deviations use IMRPhenomXPHM both to inject and to recover the signals. This is a valid check of internal consistency, but it does not probe the ability of the test to recover deviations when the template family is imperfect, which is the relevant systematic for real events. The SXS injections in Sec. III C partially offset this, but they are performed only for GR-consistent signals and do not include nonzero δAℓm. The paper should either acknowledge this limitation explicitly in the interpretation of Fig. 8 or add at least one cross-family injection-recovery (e.g., an SXS waveform with a modified subdominant mode) to demonstrate that the recovery of amplitude deviations is not an artifact of template self-consistency.
minor comments (5)
  1. [Sec. VI] Typo: 'ampltiude' should be 'amplitude'.
  2. [Fig. 4 caption] Typo: 'feect' should be 'effect'.
  3. [Fig. 14 caption] Typo: 'thode' should be 'those'.
  4. [Fig. 3] The p-p plot would benefit from a legend or explicit statement that the shaded bands are 1σ, 2σ, and 3σ for N=20; the current shading is not self-explanatory.
  5. [Sec. II, Eq. (5)] The definition of ρ⊥ℓm as an orthogonal SNR is clear, but the dependence on the noise-weighted inner product should be stated explicitly, including the normalization convention, since the χ2 null distribution in Appendix A relies on this.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SMA test is a phenomenological consistency test validated by external NR and noise benchmarks, not by self-referential reduction.

full rationale

The paper's central operation is not a derivation that claims to predict δAℓm from first principles; it is a Bayesian parameter-estimation consistency test. Equation (2) defines δAℓm as a phenomenological rescaling of subdominant-mode amplitudes while the (2,2) mode is fixed, and the posteriors in Secs. III-V are estimated from data with flat priors. No quoted equation or fitted parameter reduces by construction to another quantity in the paper. The load-bearing validation is external rather than self-citational: the p-p plot uses 20 Gaussian noise realizations (Sec. III B, Fig. 3); the systematics checks use nine SXS numerical-relativity simulations plus two eccentric SXS simulations (Sec. III C, Figs. 5-7); the sensitivity checks inject explicit amplitude deviations and TIGER phase modifications (Secs. IV A-IV B, Figs. 8-9). These benchmarks are independent of the paper's own event-level claims. The event-level results for GW241011, GW230814, GW250114, and GW231123 are reported as posterior intervals from real detector data, not as predictions derived from the model's own inputs. Self-citations to the original SMA formalism [22,23] and to the authors' earlier GW241011 analysis [47] are contextual or comparative, and no load-bearing 'uniqueness theorem' or prior self-cited result is used to force the conclusions. The paper also explicitly documents its limitations: the calibration uses only N=20 noise realizations; δA33 bimodality and degeneracies with inclination and reference phase are identified in Sec. III B and Appendix B; waveform systematics are shown to mimic GR violations for strongly precessing and eccentric systems (Secs. III C and V C); and GW250114 yields an uninformative posterior due to limited mode SNR. These are robustness and calibration concerns, not circularity. The skeptical point that GW230814's ρ⊥44 selection is marginal and that no event-specific injection-recovery was performed is a statistical-strength concern, not a reduction of a predicted quantity to a fitted input. Overall, the paper is self-contained against external NR benchmarks and does not confuse fitted inputs with independent predictions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the δAℓm parameters being meaningful GR-violation handles, on the waveform model being faithful, and on the noise model being correct. No new physical entities are introduced. The SXS benchmarks partially externalize the waveform-model assumption, but the amplitude-deformation injections in Sec. IV A are closed-loop (same model for injection and recovery).

free parameters (4)
  • δA21 = GW231123: posterior piles up at prior boundary (|δA21| > 10 preferred); no well-defined central value
    The SMA non-GR model introduces this fractional subdominant-mode amplitude deviation; it is fitted to the data via Bayesian PE.
  • δA33 = 0.00^{+0.46}_{-1.82} (GW241011)
    Fitted deviation for the (3,3) mode; the GW241011 result is repeated from Ref. [47].
  • δA44 = -0.30^{+1.16}_{-3.45} (GW230814)
    The headline constraint on the hexadecapolar mode amplitude deviation.
  • δA32 = Not robustly constrained; broad posteriors and log B from -1.7 to 6.2 across SXS runs
    Fitted deviation for the (3,2) mode; in many runs it is weakly informative because of overlap with the (2,2) mode.
assumptions (5)
  • domain assumption The GW signal is accurately described by the multipole expansion Eq. (1) and by IMRPhenomXPHM's included modes (2,2), (2,1), (3,3), (4,4), (3,2).
    The entire SMA test compares data to this model; any missing mode or inaccurate amplitude maps into δAℓm. Invoked throughout Sections II-V.
  • domain assumption Detector noise is Gaussian and stationary with known PSD for likelihood evaluation.
    The Bayesian likelihood in Eqs. (6)-(7) assumes this; the p-p calibration only checks it in 20 Gaussian realizations.
  • standard math The orthogonal SNR definition ρ⊥ℓm in Eq. (5) correctly isolates the mode contribution.
    Used for event selection and interpretation; assumes the noise-weighted inner product and adequate subtraction of (2,2) overlap.
  • domain assumption The TIGER phase-perturbation parameterization δχi in Eq. (11) represents a plausible non-GR phasing.
    Used in Sec. IV B to argue SMA is sensitive to phase deviations; not independently justified as a physical theory.
  • domain assumption NR simulations from the SXS catalog are accurate GR ground truth.
    Used as injections to benchmark waveform systematics; assumed to be more accurate than IMRPhenomXPHM.

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Cite this review

Pith. "Pith review of Testing general relativity with amplitudes of subdominant gravitational-wave modes." pith.science (2026). https://pith.science/paper/MSI6FDDI

@misc{pith2026251111886,
  author       = {Pith},
  title        = {Pith review of: Testing general relativity with amplitudes of subdominant gravitational-wave modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSI6FDDI}},
  note         = {Machine review of arXiv:2511.11886}
}
abstract

We present an improved subdominant-mode amplitude (SMA) test of general relativity (GR), which probes amplitude-level deviations in the higher-order modes of gravitational-wave (GW) signals from binary black hole mergers while keeping the dominant quadrupole mode fixed. Using a comprehensive parameter-estimation campaign, we benchmark the test against Gaussian noise fluctuations, waveform modeling systematics, and physical effects such as spin precession and orbital eccentricity. When applied to numerical-relativity simulations, the SMA test performs reliably for aligned-spin and mildly precessing systems but exhibits measurable biases for strongly precessing or eccentric binaries. Although designed to detect amplitude deviations, the test also responds coherently to phase perturbations, yielding apparent GR violations when applied to phase-modified waveforms. Applied to recent GW detections, we report the strongest constraint on the hexadecapolar $(4,4)$ mode amplitude deviation, $\delta A_{44} = -0.30^{+1.16}_{-3.45}$, consistent with GR. With these results, this work establishes the SMA test as a robust and broadly sensitive null test of general relativity and demonstrates a systematic approach for assessing the robustness of GW tests of GR.

Figures

Figures reproduced from arXiv: 2511.11886 by the authors.

Figure 1
Figure 1. Absolute value of Y −2 ℓm as a function of θJN for different (ℓ, m) modes. Among the listed modes, only (2, 2) and (3, 2) contribute for face-on binaries, and other HOMs become relatively more important as inclination increases. Beyond the viewing angle, the detectability of HOMs also depends on the binary’s intrinsic parameters. In the inspiral regime, the phase of a given (ℓ, m) mode is related to the quadrupolar … view at source ↗
Figure 2
Figure 2. The ratio of the orthogonal , ρ ⊥ ℓm, in the (ℓ, m) modes compared to the (2, 2) mode, as a function of the total mass Mtot and the mass ratio q of the binary. dence enhances the contribution of HOMs in systems with higher masses. Furthermore, the amplitudes of odd￾m modes depend on q [29, 30]. These modes vanish in the equal-mass limit and grow in significance for asymmetric mass configurations. To quantify the imp… view at source ↗
Figure 3
Figure 3. P-P plot for the deviation parameters in a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (13 more)
Figure 5
Figure 5. Figure 5: In all four simulations, the posteriors for ev [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 4
Figure 4. Figure 4: Probability distributions for δA33 across 20 parameter estimation runs in different Gaussian noise realizations. Several runs show significant bimodality in the δA33 posteriors, which is attributed to the feect of Gaussian noise and parameter degeneracies (see Appendix…
Figure 5
Figure 5. Figure 5: Posterior distributions for δAℓm from the SMA test on SXS simulations listed in Tab. I. The black dashed line marks δAℓm = 0, while the dotted lines denote the 90% credible intervals. log B represents the logarithm of the Bayes factor in favor of the non-GR hypothesis.…
Figure 6
Figure 6. Figure 6: Same as Fig. 5 but for 200 M [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Posterior distributions for δAℓm when parameter estimation is performed on simulated eccentric signals using SXS simulations, SXS:BBH:1371 (e = 0.055) and SXS:BBH:1373 (e = 0.093). The quoted eccentricities are from Ref. [44], evaluated at a PN velocity squared of v 2 …
Figure 8
Figure 8. Figure 8: Recovered posteriors for δAℓm from simulated signals with injected deviations δAℓm = 3 (left, red) and δAℓm = 6 (right, blue) for each (ℓ, m) mode. The dotted horizontal lines indicate the 90% credible intervals. The re￾covered values agree well with the injections acr…
Figure 9
Figure 9. Figure 9: Results of applying the SMA test (one mode at a time) to simulated non-GR signals with phase deviations at different [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: summarizes the resulting posteriors on δA33 and δA44. For GW241011, we find δA33 = 0.00+0.46 −1.82, 2 with a corresponding log B = −1.3, indicating prefer￾ence for the GR hypothesis. The posterior distribution is mildly bimodal, featuring a dominant peak near δA33 = 0…
Figure 11
Figure 11. Figure 11: Posterior distribution of δA44 with GW250114. The dashed black line shows the applied prior, U[−10, 10], for the analyses. We infer an uninformative distribution, with significant support on prior boundaries. the maximum-likelihood parameters, into 20 independent Gaus…
Figure 13
Figure 13. Figure 13: The probability distribution for ρ ⊥ ℓm for a subset of events detected during O4a. The grey curve shows the χ distribution with 2 degrees of freedom, representing the null distribution for ρ ⊥ ℓm if the (ℓ, m) mode is not present in data. Appendix B: Bimodality in th…
Figure 14
Figure 14. Figure 14: Posterior distributions for δA44 for GW250114-like systems simulated in 20 Gaussian noise realizations. Several runs show bimodalities, similar to thode observed in [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: The posterior distributions for the reference phase [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: The posterior distributions for primary and secondary spin parameters, when inferred using the GR template applied [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.