Overtones and Mode Identifiability

An overtone is a quasinormal mode with the same angular labels as another mode but a larger damping rate. Adding overtones can improve an early-time waveform fit; that is not yet evidence that detector data identify each added mode. Identification requires the extra component to be distinguishable from noise, prompt response, start-time shifts, and other flexible corrections under declared priors. This page develops the reusable decision protocol; dated event and catalog verdicts belong to Research rather than to a parallel evidence table here.

Helpful background. Ringdown Likelihoods and Start-Time Dependence supplies the finite-window likelihood and reference-time transformation used here. Ringdown Overtones develops waveform construction, and Completeness, Excitation, and Ringdown Content explains the spectral interpretation.

What an overtone model says

For one angular multipole, write

$$h(t) = \operatorname{Re}\!\left[ \sum_{n=0}^{N} A_n(t_0)e^{-i\omega_n(t-t_0)} \right], \qquad \omega_n=\omega_n^R-\frac{i}{\tau_n}, \qquad t\geq t_0 .$$

With $e^{-i\omega t}$, higher $n$ generally means shorter $\tau_n$. An early reference time can therefore give an overtone a large amplitude even though it contributes little a few damping times later. The invariant comparison is not between raw amplitudes quoted at different $t_0$, but between waveforms or amplitudes transported to a common reference:

$$A_n(t_a)=A_n(t_0)e^{-i\omega_n(t_a-t_0)}.$$

The Kerr label $n$ has physical content only when the inferred complex frequency agrees with the predicted spectral branch. A free damped sinusoid that absorbs modeling error is not automatically a quasinormal overtone.

Fit quality, necessity, and identification

Three questions must be separated.

1. Reconstruction: Does a larger mode set reduce residuals or mismatch?
2. Predictive necessity: Does it improve out-of-sample or posterior-predictive performance after the additional parameter volume is counted?
3. Physical identification: Are its frequency, damping time, angular response, and start-time evolution consistent with one spectral mode and distinguishable from competitors?

A fixed-Kerr overtone model can win an evidence comparison because its frequencies are tied to a common $(M_f,\chi_f)$. That is evidence for the constrained waveform family relative to the alternatives actually compared. It is weaker than measuring the overtone frequency independently. Conversely, freeing every $\omega_n$ may make physical identification impossible at available signal-to-noise ratio even if a Kerr-tied component is useful.

Near-degenerate exponentials make the problem ill-conditioned. Sample the complex basis functions $q_n(t)=e^{-i\omega_n(t-t_0)}$ in the chosen window and define a Hermitian noise-weighted Gram matrix

$$G_{mn}=\langle q_m,q_n\rangle_C \equiv q_m^\dagger C^{-1}q_n.$$

Equivalently, first whiten the vectors by $C^{-1/2}$ and use their ordinary Hermitian inner product. The exponentials themselves are not already whitened in colored noise. For real detector data, the actual amplitude Fisher matrix uses the corresponding sine and cosine quadratures; the complex matrix is a useful conditioning diagnostic. If its smallest eigenvalue approaches zero, compensating amplitude changes leave the waveform nearly unchanged. Normalize the basis before quoting a condition number so that a rescaling of amplitudes is not mistaken for a loss of information.

An identifiability protocol

A credible overtone claim should survive all of the following:

• scan $t_0$, the end time, and peak-time uncertainty, transporting amplitudes to one reference time;
• compare Kerr-tied, freely damped, prompt-response, and reduced-mode models;
• vary noise estimation, calibration, gating, and the detector network;
• use amplitude priors whose transformation with $t_0$ is stated;
• demonstrate frequency–damping localization rather than only a nonzero constrained amplitude;
• recover injected overtones and measure the false-positive rate on fundamental-only injections and off-source data;
• test whether the same conclusion follows from independent implementations.

Residual compatibility with the assumed noise is necessary but not sufficient: a flexible wrong model can leave noise-like residuals, and an imperfect whitening filter can leave correlations even for a correct signal. Bayes factors are conditional on prior volume and the model list. When many start times and mode sets were examined, significance must include that search. Held-out predictions in colored noise must use the conditional noise distribution or a genuinely independent realization; simply dividing correlated samples does not make a validation set independent.

Reading an overtone claim

An overtone claim should identify which evidentiary rung it reaches and which diagnostic supports that rung. Moving upward requires new information; a larger Bayes factor for the same constrained model comparison does not automatically supply a frequency measurement or rule out an omitted competitor.

Evidentiary rung	Minimum diagnostic	What the diagnostic cannot establish by itself
Reconstruction	Residual, mismatch, or remnant recovery improves on a declared time window	Predictive necessity in noisy data
Predictive necessity	Held-out prediction, calibrated model comparison, or injection recovery favors the added component after parameter cost	A unique quasinormal-mode label
Physical identification	Frequency and damping localize on one Kerr branch and remain stable under admissible start-time, noise, and competitor changes	Universality across events or waveform families
Robustness	Overtone-present, overtone-absent, and off-source trials measure recovery, coverage, and false identification	Immunity to competitors that were never tested

Numerical-relativity reconstruction supplies a clean example of the first rung Giesler et al. 2019, Figs. 1–5 and remnant-recovery analysis. A detector-data reanalysis that varies start time, noise realization, and amplitude assumptions tests the higher rungs instead Cotesta et al. 2022, start-time scan and injection study. The finite-window likelihood explains why these tests answer different questions Isi and Farr 2021, §§V–VII. These papers are methodological case studies here, not a current evidence ranking.

Exercises

1. Let $\tau_0=10\ {\rm ms}$ and $\tau_1=3\ {\rm ms}$. At $t_0$, $\lvert A_1/A_0\rvert=4$. Find the ratio $8\ {\rm ms}$ later.

Solution

Each magnitude decays as $e^{-\Delta t/\tau_n}$, so

$$\left\lvert\frac{A_1(t_0+\Delta t)}{A_0(t_0+\Delta t)}\right\rvert = 4\exp\!\left[ -\Delta t\left(\frac{1}{\tau_1}-\frac{1}{\tau_0}\right) \right].$$

For $\Delta t=8\ {\rm ms}$ this is

$$4e^{-8(1/3-1/10)} = 4e^{-28/15} \simeq 0.62.$$

The apparently dominant overtone at the early reference is already subdominant, illustrating why raw amplitudes at different start times cannot be compared.

2. Two normalized basis functions have overlap $(q_0\mid q_1)=r$, with real $0\leq r<1$. Find the eigenvalues and condition number of their Gram matrix.

Solution

Here

$$\boldsymbol G= \begin{pmatrix} 1&r\\ r&1 \end{pmatrix},$$

whose eigenvalues are $1+r$ and $1-r$. The spectral condition number is

$$\kappa(G)=\frac{1+r}{1-r}.$$

It diverges as $r\to1$, so the two amplitudes become unidentifiable even though their sum may fit the waveform accurately.

References

• Cotesta, R., Carullo, G., Berti, E., and Cardoso, V. (2022). “Analysis of Ringdown Overtones in GW150914.” Physical Review Letters 129, 111102. DOI.
• Giesler, M., Isi, M., Scheel, M. A., and Teukolsky, S. A. (2019). “Black Hole Ringdown: The Importance of Overtones.” Physical Review X 9, 041060. DOI.
• Isi, M., and Farr, W. M. (2021). “Analyzing black-hole ringdowns.” arXiv:2107.05609.

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• Ringdown Overtones and Mode Identifiability