Constraining the Quadratic-mode Amplitude Coupling in GW250114

Changfu Shi; Yi-Ming Hu; Yuxin Yang

REVIEW

Constraining the Quadratic-mode Amplitude Coupling in GW250114

Yuxin Yang, Changfu Shi, Yi-Ming Hu

General Relativity and Quantum Cosmology · Submitted 24 Sep 2026

Pith's verdict: First paper to try to directly measure the 220×220 amplitude-coupling coefficient as a free parameter, and it is honest about its limits; the consistency claim holds, but 440 leakage keeps it from being a clean direct measurement.

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arXiv ID2609.28953 (version 2)

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CitationChangfu Shi, Yi-Ming Hu, Yuxin Yang. "Constraining the Quadratic-mode Amplitude Coupling in GW250114." arXiv:2609.28953 (2026). https://pith.science/paper/OYUW5DDO

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Editor's note

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What you should know: this is the first paper I know of that tries to isolate the 220×220 amplitude-coupling coefficient R as a measured quantity, rather than just asking whether a quadratic mode is present via Bayes factors. That is a legitimate step forward. The analysis is careful and, unusually for this literature, the limitations are stated in the paper itself. My take: the consistency claim holds up, but the word “direct measurement” oversells what the data can currently support.

The new thing is the move from mode identification to coupling measurement. The model is simple: 220+221+220×220, with R free. They report R = 0.207 at Δt = 5tMz, broadly consistent with all the NR and perturbation-theory predictions, with no deviation beyond 1.3σ. Coming from this group, the analysis is competent. The injection tests in Fig. 5 are a real strength, and the start-time series, the fixed-DL/ι checks, and the explicit admission that the four-mode model only broadens the posterior (b129, b130) suggest the authors are not hiding from the systematics. Credit where due.

Now the soft spots. The first is the 440 mode. The paper's own injections show that including an unmodeled linear 440 mode biases the recovered R upward, and the measured R tends to sit above the theoretical curves (b96, b127). The four-mode check does not resolve this: the median barely moves while the interval widens by 52%. That is exactly what one expects from partial degeneracy between 220×220 and 440, which share the (4,4) angular dependence and sit close in frequency. The paper has not run a 440-only injection at the observed SNR to show that the R posterior cannot be produced without the quadratic mode. Until then, I would call this a measurement of the 44-multipole amplitude ratio under a three-mode model, not a clean direct measurement of the 220×220 coupling coefficient. This is an addressable gap, not a fatal flaw, and the authors flag the direction of the bias themselves (b127).

Second, the 1.3σ statement ignores uncertainty in the theoretical fitting formulas themselves, so the deviation is not fully defined. Minor, but worth fixing. Third, the headline R comes from a post-hoc chosen start time after earlier windows show model bias; the start-time dependence is shown and discussed, but it means the number is conditional in a way that is easy to understate.

The circularity worry from the reader report is overblown: R is a free parameter, and mapping the jointly-inferred χf through the theory curves is a reasonable way to compare. No issue there.

Overall: for ringdown and strong-field GR people, this is worth reading. It deserves a serious referee. I would recommend acceptance after a revision that adds a 440-only injection test and propagates theory uncertainty in the comparison. I would also want the abstract's “direct measurement” language softened or justified.

What would settle it

Run an injection-recovery test at the GW250114 start times: inject a signal that includes the 440 mode at the LVK-estimated amplitude alongside 220, 221, and 220×220, recover with the three-mode model, and compare the induced upward bias in $R$ against the observed excess of the median (≈0.21 at $\Delta t = 5\,t_{M_z}$) over the theoretical central predictions (≈0.16). If the 440-induced bias is comparable to or larger than that excess, the consistency claim rests on an unsubtracted systematic. A corroborating check: fit the real data with the four-mode model that jointly estimates $R$ and the 440 amplitude and verify that the $R$ median stays within 1.3σ of predictions; if jointly modeling 440 moves the central value materially, the three-mode agreement was an artifact.

Referee report

Significance

If established, this would be the first direct observational constraint on the amplitude-coupling coefficient of a quadratic quasi-normal mode, providing a genuinely new test of nonlinear gravitational dynamics in the strong-field regime. The analysis has real strengths: it scans ringdown start times, quantifies the information gain relative to priors, tests sensitivity to the fixed luminosity distance and inclination, and reports injection tests for mode-content mismatch. The central caveat is that the 220×220 mode and the linear 440 mode share the same angular structure and have nearby frequencies, so the inferred R can be biased by unmodeled 440 content. The paper's own injections show that a 220+221+440 signal shifts the recovered R upward, and the observed R tends to exceed the theoretical curves; the four-mode robustness check broadens the posterior without moving the median. Because of this degeneracy, the quantitative 1.3σ consistency claim is not yet fully supported, and the claim that R is a measurement of the 220×220 coupling coefficient needs additional evidence. The issues are specific and addressable within the manuscript's scope.

Major objections (3)

[Section 4, Fig. 5, b123-b130] The treatment of the 440-mode contamination is the central unresolved issue. The injection tests in Fig. 5 show that a signal containing 220, 221, and 440 but no 220×220 produces a recovered R that shifts markedly upward (b126), and b127 states that the observed R, which exceeds the theoretical predictions, may reflect a contribution from the 440 mode. Adding 440 to the recovery model (b129-b130) broadens the posterior by about 52% without moving the median, which is the expected signature of partial degeneracy rather than evidence that 440 is absent. The Discussion's claim that this systematic bias is 'subdominant to the statistical uncertainty' is not supported by any quantitative comparison in the paper. The authors should run a 440-only injection using the LVK-estimated 440 amplitude at the observed SNR, report the median and central 68% interval of the recovered R, and compare it directly with R = 0.207^{+0.086}_{-0.092}. If that posterior overlaps the observed value, the measured R cannot be uniquely attributed to the 220×220 coupling coefficient, and the abstract's implication that this coefficient has been measured should be correspondingly weakened. The authors acknowledge this: “The estimate ofRin Fig. 1, which exceeds the theoretical predictions, may therefore reflect not only noise fluctuations but also a possible contribution from the 440 mode in the data.”
[Section 4, b92, Fig. 1 lower panel] The 1.3σ consistency statement is obtained by taking the median of a theoretical distribution generated by mapping the ringdown-inferred χf posterior through each spin-dependent relation, then computing the percentile of that median within the marginal posterior of R. This procedure does not propagate the uncertainties of the theoretical fitting formulas themselves, and it does not account for the correlation between R and χf that arises because both parameters are inferred jointly from the same data. A theoretical curve can pass through the broad marginal R distribution even when the conditional posterior p(R|χf) at the relevant χf is inconsistent with the curve. The authors should compare the observed R with the prediction evaluated at the independent IMR reference value χf = 0.68, or report the conditional posterior p(R|χf) and evaluate the deviation at each χf, with theoretical uncertainties included. Without this, the quantitative statement 'no comparison shows a deviation exceeding 1.3σ' is not fully supported.
[Section 4, Fig. 1 upper panel, b78] The maximum deviation of approximately 1.3σ is quoted 'across all analysis start times and theoretical models considered' (b92), but this includes early start times where the authors themselves find the three-mode model inadequate: the upper panel of Fig. 1 and b78 show that for Δt ≲ 3 t_Mz the inferred χf is biased away from the IMR reference value, and the text states that this portion of the data is not adequately described by the model. The headline agreement claim should be restricted to start times for which the model is adequate (roughly Δt ≳ 3 t_Mz), or the paper should explicitly report the maximum deviation excluding the early start times and show that the conclusion is unchanged. As written, the most favorable quantitative statement includes times at which the model is known to fail.

Minor objections (4)

[Fig. 1 and Fig. 2 captions] The captions list 'Cheung et al. (2022)' and 'Cheung et al. (2023)' as separate curves, while the text refers to M. H.-Y. Cheung et al. (2023) for the spin-independent constant estimate and to M. H.-Y. Cheung et al. (2024) for the linear spin-dependent fit. The caption labels should be aligned with the reference list to avoid confusion.
[Fig. 3 and Fig. 4 captions] The start-time labels contain typographical artifacts: Fig. 3 uses '∆t>' and Fig. 4 uses 't > =t−t peak'. These should read Δt = t − t_peak.
[Section 3, Eq. (1)] The sign convention for the complex mode frequency should be stated explicitly. The text writes e^{iω(t−t0)} with complex ω, and the damping behavior depends on the sign of Im(ω); stating the convention (for example, Im(ω) > 0 for a decaying mode in this convention) would make the reference-time invariance argument in Section 2 easier to follow.
[Section 4, Fig. 3] The Kullback-Leibler information gain is computed by reweighting the same posterior samples to different prior ranges rather than by rerunning the inference. The presentation would be clearer if the text stated this explicitly and connected the analysis to the main conclusion, since the dependence on the prior range is the dominant effect visible in Fig. 3.

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The paper

The paper claims

This paper makes the first direct measurement of the amplitude-coupling coefficient of the quadratic 220×220 ringdown mode in GW250114, finding it consistent with general relativity's predictions — no comparison deviating more than 1.3σ —…

The reading

The paper attempts the first direct measurement of the amplitude-coupling coefficient that governs a nonlinear (quadratic) quasi-normal mode in a black hole ringdown. Second-order perturbation theory predicts that the 220×220 mode's amplitude equals the product of the parent 220-mode amplitudes times a coupling ratio $R$ that depends on remnant spin; that coefficient had never before been measured from gravitational-wave data. The authors perform Bayesian inference on the GW250114 ringdown with a 220+221+220×220 model and find $R$ consistent with predictions from numerical relativity fits and from perturbation theory, with no comparison deviating more than 1.3σ. The interest is that the frequency-sum relation alone is shared by modified-gravity frameworks, so the amplitude ratio is the more discriminating probe of nonlinear gravitational dynamics within general relativity.

What carries the argument

The central object is the complex amplitude ratio $\tilde R_{220\times220} = \tilde A_{220\times220}/(\tilde A_{220}\tilde A_{220})$, which is reference-time invariant: because the quadratic-mode frequency obeys the sum relations $\omega_{220\times220} = \omega_{220}+\omega_{220}$ and $\tau^{-1}_{220\times220} = \tau^{-1}_{220}+\tau^{-1}_{220}$, numerator and denominator acquire identical phase and damping factors under a shift of the ringdown start time. That invariance means analyses at different start times probe the same coupling, and it is what converts a mode-identification claim into a quantitative parameter measurement. The measurement itself is carried out by time-domain Bayesian inference (bilby with dynesty, 1000 live points) on an eight-parameter model $\{M_z, \chi_f, A_{220}, \phi_{220}, A_{221}, \phi_{221}, R, \phi_{220\times220}\}$, repeated over a grid of start times $\Delta t \in [0,10]\,t_{M_z}$. The comparison targets are the spin-dependent benchmark relations of Cheung et al. (2024), Redondo-Yuste et al. (2024), and Khera et al. (2025), mapped through the inferred $\chi_f$ posterior, against the spin-independent estimate of Cheung et al. (2023).

Extended reading notes

Core claim

The inferred amplitude ratio $R_{220\times220} = \tilde A_{220\times220}/(\tilde A_{220}\tilde A_{220})$ for GW250114, measured from post-peak data with a three-mode ringdown model, is consistent with the theoretical scale: across all analysis start times and the four benchmark predictions (a spin-independent numerical-relativity constant, linear and nonlinear spin-dependent numerical-relativity fits, and a higher-order perturbation-theory relation), the largest deviation is approximately 1.3σ, a two-sided Gaussian-equivalent tail probability of about 19%. The joint posterior in the ($\chi_f$, $R$) plane spans the region the theoretical curves cover, with a tendency for the inferred $R$ to sit above the central curves that the authors judge not statistically decisive. The measurement constrains $R$ relative to the prior — an information gain of roughly 4 bits under a $U(0,5)$ prior at early start times — and establishes the basis for more stringent tests, even though the current signal-to-noise ratio cannot distinguish among the competing spin-dependent models.

Load-bearing premise

The load-bearing premise is that the three-mode model (220, 221, and 220×220) adequately describes the analyzed data window with any unmodeled 440-mode content subdominant; the paper's own injection tests show that an injected 440 mode shifts the recovered $R$ upward, and the measured $R$ tends to sit above the theoretical curves, so a 440 contribution could be creating part of the observed agreement rather than confirming it.

Editorial extensions

If this is right

The first measured value of a quadratic-mode amplitude coupling in real data, even with broad uncertainty, turns a theoretical prediction of general relativity's nonlinear sector into an observable quantity that a single louder future event can tighten.
Because the frequency-sum relation is shared by modified-gravity and simplified nonlinear models, the now-measurable amplitude ratio $R$ is the quantity that discriminates general relativity's nonlinear predictions from alternatives.
The measurement pins $R$ to the theoretical scale at the 1.3σ level; if the predicted spin dependence were wrong, deviations larger than this would surface in better-SNR events.
The start-time dependence of the inferred spin (biased near the peak, consistent with the IMR reference beyond roughly $3t_{M_z}$) indicates the three-mode model is adequate only for the later portion of the ringdown, guiding where future analyses should begin.
Combining multiple high-signal-to-noise events would map $R$ against remnant spin over a range that this single event cannot probe, testing the predicted spin dependence directly.

Reading between the lines

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

The excess of the observed $R$ median over the theoretical central curves, combined with the paper's own injection tests showing that unmodeled 440 content biases $R$ upward, suggests the current agreement may be partly an artifact of model mismatch; a conservative reading treats the three-mode value as an upper bound on the true coupling until the 440 mode is jointly modeled.
The reference-time invariance of $R$ implies a built-in consistency check the paper does not exploit: posteriors from different start times should agree if the model is adequate, so the observed drift of $R$ and the spin bias near the peak are themselves diagnostics of model inadequacy.
A natural extension is to apply the same ratio measurement to historical lower-SNR ringdowns and to future loud events; agreement of $R$ across events would support the coupling's universality across remnant masses and spins, while disagreement would flag a modeling or theory problem.
If a catalog of events pins $R$ as a function of remnant spin, the comparison against the spin-dependent benchmark relations sharpens into a test of whether the ringdown's nonlinear sector is governed by general relativity's field equations rather than by a generic nonlinear mechanism.

Original abstract

Detecting quadratic quasi-normal modes in black hole ringdowns would provide evidence for nonlinear gravitational dynamics, while measuring their properties would enable tests of the corresponding predictions of general relativity. Specifically, second-order black hole perturbation theory predicts that their amplitudes scale with the product of the amplitudes of their parent linear modes, with a coupling coefficient that depends on the spin of the remnant black hole. This mode-specific coupling coefficient has not yet been directly measured from gravitational wave data. Here, we use Bayesian inference on the GW250114 ringdown, modeled with the $220$, $221$, and $220\times220$ modes, to infer the amplitude-coupling coefficient of the $220\times220$ mode. The inferred coefficient is consistent with predictions from numerical relativity fits and second-order perturbation theory; no comparison shows a deviation exceeding $1.3 \sigma$. Although the current data cannot distinguish among these spin-dependent predictions, this measurement establishes a basis for more stringent observational tests of nonlinear gravitational dynamics in black-hole ringdowns.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-25: 1 request, 5,098 input tokens.

What the abstract itself states, as probabilities: claims first 0.39 · states limitation 0.97 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.91 · compares with prior results 0.82 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Detecting quadratic quasi-normal modes in black hole ringdowns would provide evidence for nonlinear gravitational dynamics, while measuring their properties would enable tests of the corresponding predictions of general relativity. Specifically, second-order black hole perturbation theory predicts that their amplitudes scale with the product of the amplitudes of their parent linear modes, with a coupling coefficient that depends on the spin of the remnant black hole. This mode-specific coupling coefficient has not yet been directly measured from gravitational wave data. Here, we use Bayesian inference on the GW250114 ringdown, modeled with the $220$, $221$, and $220\times220$ modes, to infer the amplitude-coupling coefficient of the $220\times220$ mode. The inferred coefficient is consistent with predictions from numerical relativity fits and second-order perturbation theory; no comparison shows a deviation exceeding $1.3 \sigma$. Although the current data cannot distinguish among these spin-dependent predictions, this measurement establishes a basis for more stringent observational tests of nonlinear gravitational dynamics in black-hole ringdowns.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Detecting quadratic quasi-normal modes in black hole ringdowns would provide evidence for nonlinear gravitational dynamics, while measuring their properties would enable tests of the corresponding predictions of general relativity.	background 0.99	0.04	0.05	hedged possibility 0.68
s1	Specifically, second-order black hole perturbation theory predicts that their amplitudes scale with the product of the amplitudes of their parent linear modes, with a coupling coefficient that depends on the spin of the remnant black hole.	background 1.00	0.04	0.04	asserted 0.98
s2	This mode-specific coupling coefficient has not yet been directly measured from gravitational wave data.	gap 1.00	0.04	0.03	asserted 1.00
s3	Here, we use Bayesian inference on the GW250114 ringdown, modeled with the $220$, $221$, and $220\times220$ modes, to infer the amplitude-coupling coefficient of the $220\times220$ mode.	method 0.73	0.97	0.08	asserted 1.00
s4	The inferred coefficient is consistent with predictions from numerical relativity fits and second-order perturbation theory; no comparison shows a deviation exceeding $1.3 \sigma$.	result 0.96	0.93	0.61	hedged possibility 0.83
s5	Although the current data cannot distinguish among these spin-dependent predictions, this measurement establishes a basis for more stringent observational tests of nonlinear gravitational dynamics in black-hole ringdowns.	interpretation 0.66	0.77	0.45	asserted 0.90

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from analysis of existing data 0.88.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Time dependence of the inferred spin of the remnant BH and the quadratic-mode amplitude ratio for GW250114. The upper panel shows the posterior distribution of the remnant BH dimensionless spin χf vs. the analysis start time ∆t. The gray dashed line marks the reference maximum-likelihood value from the IMR waveform. The lower panel shows the corresponding amplitude ratio R for the 220 × 220 mode. At each start time, the right half of the blue violin gives the observed posterior of R, while the left half of the violin shows the theoretical predictions obtained by mapping the same χf…

**Figure 2.** Posterior constraints on the quadratic-mode amplitude ratio in the (χf , R) plane. The translucent blue areas show the 80% two-dimensional credible regions of the GW250114 posteriors for different analyses start times, with darker shades indicating earlier times and a thick contour highlighting ∆t = 5 tMz . The earlier start times are drawn with the darker contours and later start times with lighter contours. The orange point denotes the spin-independent prediction of M. H.-Y. Cheung et al. (2023), evaluated at χf = 0.68. The green, red, and purple curves show the spin-dependent…

**Figure 3.** Information gain for the inferred amplitude ratio R as a function of ringdown start time ∆t > and the upper bound Rmax of the uniform prior R ∼ U(0, Rmax). Colors and labeled contours show DKL in bits. The information gain decreases for narrower priors and shows a dip near ∆t = 8 tMz . distance DL. Equation 3 therefore implies that R also scales inversely with DL. Consequently, adopting a larger luminosity distance leads to a smaller inferred R, consistent with the trend shown in the figure. The inclination ι affects the inferred R through the spin-weighted spheroidal harmonics with…

**Figure 4.** Sensitivity of the inferred quadratic-mode amplitude ratio R to the fixed extrinsic parameters in the GW250114 ringdown analysis. The markers show the posterior median of R, and the error bars denote the central 68% credible interval, as a function of the ringdown start time t> = t − tpeak in units of tMz . The upper panel varies the fixed luminosity distance, DL = 367, 411, and 456 Mpc, while keeping the inclination fixed at ι = 0.76. The lower panel varies the fixed inclination, ι = 0.62, 0.76, and 0.88, while keeping DL = 411 Mpc. The similar start-time dependence across these…

**Figure 5.** Recovery of the quadratic-mode amplitude ratio R for four synthetic QNM injection configurations. All panels use the same recovery model containing the 220, 221, and 220 × 220 modes at t> = 5 tMz , while the injected signal content is varied as indicated by the panel titles. The thick blue curve shows the zero-noise posterior density, the thin gray curves show the posteriors from ten real-noise realizations, and the red vertical line marks the reference injected value R = 0.1637. The blue dashed and dotted vertical lines denote the zero-noise posterior median and central 68%…

Figures reproduced from arXiv:2609.28953 by the authors.

Discussion

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