Overtones and Rapidly Damped Modes

An overtone is a QNM with radial index $n\gt 0$ in a fixed angular family. It is usually more strongly damped than the $n=0$ fundamental, so it matters most at early times and becomes hardest to distinguish precisely when the signal is most contaminated by merger and prompt response. Adding fixed-frequency overtones can improve a finite-window representation without proving that every coefficient is a uniquely projected or independently resolved physical mode. The status of early overtones is therefore a question about evidence and conditioning, not terminology alone.

Helpful background. Einstein Equations supplies perturbative order; Waves, Green Functions, and Geometric Optics supplies resonant response and nonorthogonal fitting; Black-Hole Perturbation Theory supplies the overtone spectrum; Gravitational Radiation and Waveforms supplies the asymptotic field and harmonic frame; and Black-Hole Dynamics and Collapse supplies the merger history that excites the remnant.

Radial index, damping, and amplitude bookkeeping

Use geometrized units $G=c=1$ in this remnant calculation. $M$ denotes the initial total mass as a time or length in quoted simulation intervals; $M_f$ denotes the final mass. A source-frame duration $bM_f$ restores to $bGM_{f,\rm phys}/c^3$, and a detector-frame duration has one further factor $1+z$.

Use a remnant-aligned inertial Bondi frame and

$$H=r(h_+-ih_\times) =\sum_{\ell m}H_{\ell m}\,{}_{-2}Y_{\ell m},$$

with unit-normalized spin-$-2$ harmonics. The inherited $e^{-i\omega u}$ convention places stable QNMs in the lower half-plane. For fixed spheroidal labels $(L,m)$, write

$$\omega_{Lmn}=\omega^R_{Lmn}-\frac{i}{\tau_{Lmn}}, \qquad n=0,1,2,\ldots,$$

where $n=0$ is the least-damped member. “Overtone” does not mean an integer multiple of the fundamental frequency; it labels successive radial resonances. A spherical mode contains spheroidal overlaps,

$$H_{\ell m}(u)= \sum_{L,n}\mu^m_{\ell L}(a_f\omega_{Lmn}) A_{Lmn}(u_*)e^{-i\omega_{Lmn}(u-u_*)}.$$

Thus an extra frequency in $H_{22}$ may be a $(2,2,n)$ overtone, leakage from a different spheroidal $L$ with the same $m$, a nonlinear combination, or an effective term absorbing omitted response. These possibilities must be tested rather than inferred from the spherical channel name.

This is a selected linear-QNM model, so the equality applies to its modeled contribution rather than to every part of the physical waveform. It also suppresses a frequency-branch label. Counterrotating or mirror branches, when retained, need their own frequencies, overlaps, and amplitudes as described in the QNM expansion.

Because the amplitude at the fit start changes by known evolution, compare the decay-compensated quantity

$$\bar A_n(u_0;u_*)=A_n(u_0)e^{+i\omega_n(u_0-u_*)}$$

at one common reference $u_*$. A stable physical component should have approximately constant $\bar A_n$ and phase across a range of windows in which it remains above the numerical floor.

Why more terms improve fits so easily

For fixed frequencies, fitting $N+1$ overtones is linear regression with columns

$$q_n(u)=e^{-i\omega_n(u-u_0)}.$$

Enlarging their span cannot worsen the minimum residual. The relevant question is whether the coefficients are identifiable. A useful pairwise warning is

$$\rho_{nm}= \frac{|\langle q_n,q_m\rangle|} {\sqrt{\langle q_n,q_n\rangle\langle q_m,q_m\rangle}}, \qquad 0\le\rho_{nm}\le1.$$

When $\rho_{nm}$ is close to one, small noise or an omitted component can move large, compensating amounts between $A_n$ and $A_m$ while barely changing the reconstruction. The full singular-value spectrum is more informative than pairwise correlations because many moderately correlated columns can still make the design matrix ill-conditioned.

Rapid damping creates a second difficulty. High-$n$ columns have support mainly near $u_0$; move the start slightly later and their information disappears, but move it earlier and prompt response, an evolving remnant, and nonlinear sourcing become more important. Consequently, the same overtone can be useful in a constrained model, unresolved in a free-frequency search, and unstable as an individual amplitude without contradiction.

Evidence status as of 2026-08-18

Whether a broad, unique early-time overtone decomposition exists remains unresolved; the evidence does not support a universal verdict. The matrix distinguishes representational success, mode-specific support, and limitations.

Evidence or result	Support	Qualification, scope, and limitation	Tension or failure mode
Fixed Kerr-frequency fits in 2019	For SXS:BBH:0305, including through $n=7$ in the spherical $(2,2)$ strain mode enabled low residuals and accurate final-mass/spin recovery from the strain peak; related behavior was reported across roughly 80 aligned-spin waveforms Giesler et al. 2019, §§ III–IV.A, PDF.	The main system had mass ratio $1.22$ and $\chi_f=0.6921$; amplitudes came from unweighted fits ending at peak plus $90M$, with frequencies constrained by one Kerr remnant.	The paper explicitly warns that mismatch can overfit numerical noise after fast modes decay; good remnant recovery is stronger than residual reduction but is not a unique projection of each overtone.
Frequency-perturbation tests in 2019	Kerr-consistent overtone frequencies recovered the remnant better than tested perturbed or randomized frequency sets Giesler et al. 2019, § IV.A, PDF.	This rejects those particular alternatives in that model and window; it does not search every prompt, nonlinear, or time-varying alternative.	Several correlated damped columns can encode background information collectively even when individual coefficients are not stable.
Synthetic noise and mode-dropping tests in 2025	In an analytic injection with known modes and Gaussian noise near $2\times10^{-8}$, the highest overtones became unstable first; dropping unresolved fast modes extended recovery of lower ones Giesler et al. 2025, § IV.B, PDF.	The injected content, amplitudes, noise model, and dropping rule were controlled and modeled on SXS:BBH:2423; this is a conditioning experiment, not a generic merger.	Modes can become unresolvable while their amplitude still exceeds the raw noise by orders of magnitude, and the selected dropping rule itself affects reported stability.
High-accuracy numerical amplitude stability in 2025	CCE waveforms in a remnant superrest frame supported stable amplitude and phase evolution for many overtones, including $n=5$ over roughly five decades in amplitude for the high-spin case Giesler et al. 2025, §§ IV–V, PDF.	The sample contained two equal-mass aligned-spin cases, SXS:BBH:2420 with $\chi_f=0.75$ and SXS:BBH:2423 with $\chi_f=0.92$; fitting used selected news content, variable projection, and mode dropping.	Stable decomposition began only about $4M$ or $8M$ after the $(2,2)$ strain peak, depending on spin; earlier unmodeled content remained, and generic precessing or unequal-mass transfer is open.
Green-function completeness limit	QNM poles are genuine parts of the retarded response Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF.	Prompt and branch-cut contributions accompany the poles, and QNM radial functions are not ordinary square-integrable normal modes.	A successful truncated overtone sum over one window is not proof that the merger waveform has a unique, complete discrete expansion.

Two emphases remain credible. One treats fixed Kerr overtones as physically informative components whose collective and increasingly mode-by-mode stability is supported in controlled simulations. The other stresses that early fitted columns may absorb prompt response, nonlinearities, background evolution, frame error, or noise and therefore demands stability and falsifiable alternatives before assigning individual content. Both agree that residual reduction alone is insufficient.

Reproducibility and selection protocol

A transferable overtone claim should state assumptions and expose selection choices:

1. Publish the waveform IDs, resolutions, extraction method, fitted field, Bondi/BMS frame, remnant mass scaling, peak definition, and exact interval grid. Use at least one system not employed to choose the mode set or thresholds.
2. Fit both fixed Kerr frequencies and free complex frequencies. Report priors or bounds, initialization, optimizer restarts, and the singular values of the fixed-frequency design matrix.
3. Inject known overtone sets into noise drawn at several amplitudes and with colored or correlated numerical structure. Run a no-overtone control to measure false components.
4. Transport amplitudes to one reference time; plot amplitude and phase stability with differences between at least two resolutions. Identify where each term crosses a mode-specific recovery floor, not only the raw waveform floor.
5. Repeat the start/end scan after dropping one mode, changing the maximum $n$, adding plausible spherical–spheroidal leakage, and adding a prompt or slowly varying nuisance term. Report conclusions that fail any of these changes.

The production simulations and convergence infrastructure belong to Computational Relativity. Individual detectability and empirical claims require a detector likelihood and belong to Gravitational-Wave Astronomy.

Exercise: diagnose rapid damping and correlation

Consider two real toy columns $q_0=e^{-t}$ and $q_1=e^{-2t}$ on $0\le t\lt \infty$. Compute their normalized correlation $\rho_{01}$. Separately, a fundamental has $M_f\operatorname{Im}\omega_0=-0.08$ and an overtone has $M_f\operatorname{Im}\omega_1=-0.50$. By what factors do their amplitudes fall over $10M_f$, and by what factor does the overtone-to-fundamental ratio change?

Solution

The inner products are

$$\langle q_0,q_0\rangle=\frac12, \qquad \langle q_1,q_1\rangle=\frac14, \qquad \langle q_0,q_1\rangle=\frac13.$$

Therefore

$$\rho_{01}=\frac{1/3}{\sqrt{(1/2)(1/4)}} =\frac{2\sqrt2}{3}\simeq0.943.$$

Even these visibly different decay rates produce strongly correlated fit columns. Over $10M_f$, the fundamental falls by $e^{-0.8}\simeq0.449$, while the overtone falls by $e^{-5}\simeq0.00674$. Their ratio is multiplied by $e^{-4.2}\simeq0.0150$. A slightly later start therefore erases most overtone leverage, while a short early window makes its column highly covariant with other fast and omitted contributions.

What could change the conclusion

The interpretation could change after new public high-accuracy waveforms broaden the binary parameter space, a reproducible opposing analysis changes amplitude-stability conclusions, mode-identification criteria are revised, or a cited overtone result is corrected. Preserve dated assumptions, injection settings, and negative recovery cases in the next comparison.

References

• Emanuele Berti, Vitor Cardoso, and Andrei O. Starinets, “Quasinormal Modes of Black Holes and Black Branes,” Classical and Quantum Gravity 26 (2009) 163001. DOI. Open PDF.
• Matthew Giesler, Maximiliano Isi, Mark A. Scheel, and Saul A. Teukolsky, “Black Hole Ringdown: The Importance of Overtones,” Physical Review X 9 (2019) 041060. DOI. Open PDF.
• Matthew Giesler, Sizheng Ma, Keefe Mitman, Naritaka Oshita, Saul A. Teukolsky, Michael Boyle, Nils Deppe, Lawrence E. Kidder, Jordan Moxon, Kyle C. Nelli, Harald P. Pfeiffer, Mark A. Scheel, William Throwe, and Nils L. Vu, “Overtones and Nonlinearities in Binary Black Hole Ringdowns,” arXiv:2411.11269v2 (2025), dated April 25, 2025. arXiv. Open PDF.