Parameterized Ringdown Spectra

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Parameterized ringdown tests replace one or more Kerr quasinormal frequencies and damping times by freely shifted values. They are valuable diagnostics of whether a remnant signal has the damped-mode structure predicted by linear perturbations of a Kerr black hole. They do not, by themselves, define a nonlinear merger, a stable alternative remnant, or a consistent relation among shifted modes.

Required background. Weak Metric Perturbations and Gauge Freedom supplies linear perturbations, radiative degrees of freedom, and gauge control. Probability and Inference for Gravity supplies likelihoods, priors, evidence, and correlated parameter inference.

Helpful background. Relativistic Two-Body Dynamics supplies the binary evolution and remnant parameters that connect inspiral and merger to ringdown.

Damped modes and spectral deformations

In a time interval where linear ringdown dominates, the response to a perturbed Kerr black hole can be approximated by damped quasinormal modes,

h(t)=\sum_{\ell mn} A_{\ell mn} e^{-(t-t_0)/\tau_{\ell mn}} \cos\!\left[\omega_{\ell mn}(t-t_0)+\phi_{\ell mn}\right], \qquad t\ge t_0.

This is an approximation over a chosen interval, not a complete all-time expansion: nonlinear merger can matter earlier, and power-law tails can dominate asymptotically late. Here $h$ denotes one real projected strain component, $\omega_{\ell mn}$ its real angular oscillation frequency, and $\tau_{\ell mn}\gt0$ its damping time. With time dependence $e^{-i\Omega t}$ , the complex frequency is $\Omega=\omega-i/\tau$ , in the lower half-plane. In GR the spectrum is fixed by the remnant mass $M_f$ and dimensionless spin $a_f=J_f/(GM_f^2)$ once the mode labels $(\ell,m,n)$ are chosen. A common parameterization writes

\omega_{\ell mn} =\omega^{\rm Kerr}_{\ell mn}(M_f,a_f) \left(1+\delta\hat\omega_{\ell mn}\right), \qquad \tau_{\ell mn} =\tau^{\rm Kerr}_{\ell mn}(M_f,a_f) \left(1+\delta\hat\tau_{\ell mn}\right).

The Kerr point has every shift equal to zero. Amplitudes and phases depend on the preceding merger, viewing geometry, mode mixing, and the basis used to represent the signal; treating them as free nuisance parameters avoids some excitation assumptions but sacrifices physical correlations.

A measured real frequency alone gives a curve in the $(M_f,a_f)$ plane. A single identified mode with both frequency and damping time measured supplies two real observables and can determine both remnant parameters, where the Kerr map is invertible. To see the counting, write

GM_f\omega=\widehat\omega_{\ell mn}(a_f), \qquad \frac{\tau}{GM_f}=\widehat\tau_{\ell mn}(a_f).

The product $\omega\tau=\widehat\omega\widehat\tau$ first constrains spin; the frequency then fixes mass. Unknown mode labels can leave discrete alternatives, and a poorly measured damping time leaves a broad degeneracy. A second identified complex frequency can overdetermine the same Kerr parameters—the operational core of black-hole spectroscopy Berti, Cardoso, and Will 2006, §§II, IV–V, PDF. “Two modes in a fit” is not enough: the data must distinguish them and the result must survive start-time and noise checks. For a source at cosmological redshift $z$ , detector-frame frequencies and damping times measure $M_{f,z}=(1+z)M_f$ ; spin and $\omega\tau$ are unchanged by that redshift.

Start time, overtones, and identifiability

The ringdown start time $t_0$ controls a genuine tradeoff. Starting early retains signal-to-noise ratio but exposes nonlinear merger dynamics and time-dependent mode content that a fixed linear sum may not describe. Starting late improves the perturbative interpretation while discarding data. Higher overtones decay rapidly and can approximate early-time structure; their inferred amplitudes are consequently correlated with $t_0$ , the peak definition, calibration, and unmodeled components.

Analyses of GW150914 illustrate the live methodological contrast. Isi and collaborators reported evidence for an overtone and a spectroscopy-style consistency test, while later reanalyses emphasized start-time uncertainty, noise realization, and prior sensitivity Isi et al. 2019, main analysis and Supplemental Material; Cotesta et al. 2022, main analysis. These are not merely competing tastes. They ask whether a second component is identified as a physical quasinormal mode rather than serving as a flexible fit to merger or noise.

Detector data are neither stationary nor perfectly Gaussian near every event. The likelihood must state the strain release, calibration, sampling, analysis segment, power spectral density or time-domain noise model, windowing, and treatment of glitches. Remnant mass and spin inferred from the inspiral cannot be reused as an independent ringdown prior unless the shared data and covariance are modeled.

Results and limitations through 2026-08-19

At the 2026-08-19 cutoff, ringdown observations are consistent with the Kerr spectrum within their declared models, but decisive, routinely repeatable multi-mode spectroscopy remains unresolved. The current LVK GWTC-5 tests document is a submitted draft rather than a peer-reviewed replacement for the GWTC-3 methods baseline LVK 2026, P2500781-v17, Table 1 and §§2–7.

Evidence or result	Status at the cutoff	Limitation or scope boundary
Dominant post-merger damped component	Compatible with a Kerr fundamental-mode interpretation in analyzed events	Remnant parameters, start time, waveform transition, calibration, and noise remain correlated
Additional overtone or subdominant-mode claims	Some analyses find support; conservative reanalyses weaken identification	Prior volume, peak-time uncertainty, merger leakage, mode mixing, and noise realization can control the result
Catalog-level parameterized shifts	No reproducible population inconsistency with the declared GR baseline	Event selection and heterogeneous signal strength make simple posterior stacking invalid
Complete alternative-remnant interpretation	Requires a separate remnant, excitation, and detector prediction for the chosen theory	Free frequency shifts alone establish neither a well-posed theory nor its comparative evidence

The strongest conservative position is that present ringdowns test consistency of a composite Kerr-plus-waveform-plus-noise model and have not falsified it. The stronger deviation interpretation would attribute an identified spectral shift to non-Kerr gravity. Evidence against that leap is the sensitivity of weak secondary components to analysis choices and the absence, so far, of a correlated shift pattern reproduced across independent events and modes.

From a spectral shift to new gravity

A complete theory must predict the remnant solution, its stability, its perturbation spectrum, the excitation amplitudes produced by merger, and the detector waveform. It may also shift inspiral generation or wave propagation. Varying one $\delta\hat\omega_{\ell mn}$ while fixing all other stages to GR is a useful null direction, but it is not generally a consistent theory slice.

An adversarial discriminator is a blinded multi-pipeline injection study. Inject numerical-relativity GR signals and signals from a specified stable alternative into real or representative noise; vary $t_0$ over a predeclared range; compare time- and frequency-domain likelihoods; and require the same correlated spectral shifts in at least two identifiable modes or independent events. A flexible overtone that absorbs early merger should fail the late-time and cross-event parts of this check.

The result should also converge as sampling rate, data segment, noise estimation, and waveform basis change within justified ranges. A counterexample in which a claimed shift disappears under an equally predictive noise or merger model is evidence against a gravity interpretation, even if the original posterior excluded zero.

What could change the conclusion. The interpretation could change when a superseding calibrated catalog or peer-reviewed tests release appears, when an independent event supports multi-mode shifts under frozen start-time rules, or when a viable alternative-remnant calculation predicts and passes the correlated injection test above.

Check your understanding

Why does fitting two damped sinusoids not automatically constitute a two-mode test of the Kerr spectrum?

Solution

Two basis functions can improve a fit even when the second absorbs nonlinear merger, spectral leakage, or a particular noise fluctuation. Spectroscopy requires that both complex frequencies be identifiable, that their mode labels and common remnant parameters are consistent, and that the result survive justified changes in start time and noise model. Therefore one should compare against GR merger injections, examine posterior correlations and predictive residuals, and demand a correlated Kerr or alternative-theory relation. If the second component changes freely with $t_0$ or disappears in independent data, the fit has not measured a second physical mode.

References

Berti, E., Cardoso, V., and Will, C. M. (2006), “Gravitational-wave spectroscopy of massive black holes with the space interferometer LISA,” Physical Review D 73, 064030. DOI; Open PDF.
Cotesta, R., Carullo, G., Berti, E., and Cardoso, V. (2022), “Analysis of ringdown overtones in GW150914,” Physical Review Letters 129, 111102. Cotesta et al. 2022, main analysis.
Isi, M., Giesler, M., Farr, W. M., Scheel, M. A., and Teukolsky, S. A. (2019), “Testing the no-hair theorem with GW150914,” Physical Review Letters 123, 111102. Isi et al. 2019, main analysis and Supplemental Material.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026), “GWTC-5.0: Tests of General Relativity,” submitted manuscript, arXiv:2607.19293, LIGO DCC P2500781-v17. LVK 2026, Table 1 and §§2–7.