Ringdown and Remnant Tests

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Ringdown tests ask whether a merger remnant responds with the damped-mode spectrum of the general-relativistic Kerr solution. The clean ideal is powerful: in GR, a mode’s complex frequency is fixed by the remnant mass and spin. Real data do not arrive with a unique boundary between merger and linear response, however. Start time, included modes, detector noise, progenitor waveform, and selection all influence the inferred spectrum, so an apparent extra mode or frequency shift is credible only if it survives those choices.

Helpful background. Black-Hole Perturbation Theory supplies quasinormal boundary conditions and Kerr spectra; Weak-Field Gravity supplies linear response; Probability and Statistical Inference supplies model comparison under uncertain start time; Gravitational-Wave Data Analysis and Inference supplies calibrated strain likelihoods; and Tests of Gravity supplies the bounded empirical conclusion.

The damped-mode hypothesis

Using the site’s time dependence $e^{-i\omega t}$ , a linear ringdown model beginning at $t_0$ is

h(t)=\sum_{\ell mn} A_{\ell mn} e^{-i\omega_{\ell mn}(t-t_0)} \,{}_{-2}S_{\ell mn}(\iota,\varphi), \qquad t\geq t_0,

where $h=h_+-ih_\times$ is the complex strain, $A_{\ell mn}$ is a complex excitation amplitude, and ${}_{-2}S_{\ell mn}$ is a spin-weight minus-two spheroidal angular function. The indices $\ell,m$ label angular structure and $n=0,1,\ldots$ labels the fundamental and its overtones. The displayed expression is a mode ansatz in a fixed remnant frame; a complete model must include the required partner modes and project $h_+,h_\times$ through each detector’s response. A stable retarded mode has $\operatorname{Im}\omega_{\ell mn}\lt0$ . Equivalently,

\omega_{\ell mn}=2\pi f_{\ell mn}-\frac{i}{\tau_{\ell mn}}.

For a four-dimensional, asymptotically flat Kerr remnant in GR, $GM_f\omega_{\ell mn}^{\rm source}$ is a function only of the dimensionless spin $\chi_f=J_f/(GM_f^2)$ . Observed time and frequency are redshifted: $M_{fz}=(1+z)M_f$ , $f^{\rm obs}=f^{\rm source}/(1+z)$ , and $\tau^{\rm obs}=(1+z)\tau^{\rm source}$ . Thus a detector-frame fit uses $GM_{fz}\omega^{\rm obs}$ , not source mass multiplied by observed frequency. Ringdown alone measures the redshifted mass scale.

One resolved complex frequency supplies two real numbers, $f$ and $\tau$ . With an assigned mode label it can generally determine mass and spin within the Kerr model; that alone need not overdetermine the model. A second identified frequency, or an independent remnant prediction, supplies extra conditions. A useful mass-independent quantity is the quality factor $Q=\pi f\tau$ , which depends only on spin and mode label in Kerr and is unchanged by cosmological redshift. A parameterized test may write $f_{\ell mn}=f_{\ell mn}^{\rm Kerr}(1+\delta\hat f_{\ell mn})$ and likewise for $\tau$ . These phenomenological shifts are not independent theory couplings, and a complete alternative can alter excitation, tails, nonlinear response, or progenitor evolution as well as the spectrum.

The mode sum describes an intermediate interval in which linear quasinormal response dominates, not an exact representation at every time or a guarantee of dominance at arbitrarily late times. Prompt response, nonlinear merger dynamics, mode mixing, branch-cut tails, recoil, precession, and detector projection can all matter. A good fit by several exponentials does not by itself establish that each fitted term is a quasinormal pole. The likelihood issues associated with truncation are developed by Isi and Farr 2021, §§II–III.

Start time, overtones, and noise

Choosing $t_0$ creates a bias–variance tradeoff. Starting near the strain peak retains signal-to-noise ratio but risks nonlinear contamination and waveform-model error. Starting later improves the linear approximation while exponentially discarding information. Treating $t_0$ as fixed after examining the data understates trials; marginalizing it changes the prior measure because amplitudes at different start times are related by exponential decay.

Overtones decay faster than the fundamental. Including them can extend a Kerr fit toward the peak, but their amplitudes become covariant with start time, sky-projected phase, higher harmonics, and nonstationary noise. A spectroscopic claim requires posterior support away from zero, stability across start times and noise realizations, and a false-alarm calculation for selecting the favored mode. Conversely, failing to resolve an overtone does not show it was not physically excited.

Truncating a time series changes its frequency-domain covariance; applying an ordinary stationary diagonal-noise likelihood after a sharp cut can bias results. Time-domain analyses must use a justified covariance or conditioning procedure. Full-signal parameterized waveform analyses reduce the arbitrary cut but inherit progenitor and merger model assumptions. Rational-filter methods remove hypothesized modes and compare remnant consistency, trading amplitude inference for a different noise and selection problem. These methods are complementary, not interchangeable.

Yunes, Siemens, and Yagi derive the no-hair logic and review the assumptions behind inspiral and ringdown implementations Yunes, Siemens, and Yagi 2025, §4.4.3, pp. 120–124. The peer-reviewed GWTC-3 analysis separately reports time-domain and full-waveform parameterized approaches LIGO–Virgo–KAGRA 2023, §VIII.A and Tables XI–XIII.

Release, likelihood, and controls

At the evidence cutoff 2026-08-19, the current strain and posterior baseline is GWTC-5/GWOSC stable parameter-estimation release 9 GWOSC 2026, official parameter-estimation and strain documentation, while P2600410-v1 supplies the ringdown result products LIGO–Virgo–KAGRA 2026, official tests data release. A ringdown likelihood must pin the calibrated strain file, detector set, event time, sample rate, conditioning, covariance or spectrum estimate, window, waveform family, remnant prior, mode-amplitude prior, and sampler convergence. The 16 kHz source-matched products are distinct from 4 kHz decimated products; frequencies below 10 Hz are not calibrated.

The submitted GWTC-5 tests manuscript reports three distinct current ringdown strategies: a time-domain damped-mode analysis, a frequency-domain full-signal parameterized model, and quasinormal-mode rational filters LIGO–Virgo–KAGRA 2026, submitted P2500781-v17, §§7.1–7.3 and Tables 1 and 8–10. It reports that spurious high-frequency content prevents a spectroscopic constraint for GW240621_195059 and that the GR point lies in the tails of the combined full-signal result, with limited catalog size identified as a possible cause. These are current method-specific qualifications, not an established non-Kerr detection. Event selections differ by method. A hierarchy over events must retain those selections and the interim event priors; it cannot multiply posteriors as though every event measured one universal, source-independent deformation.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or counterevidence
Kerr perturbation spectrum	Supplies a sharply overdetermined GR null for a settled linear remnant	Mapping assumes Kerr, linear response, declared mode labels, and retarded boundary conditions
Peer-reviewed GWTC-3 ringdown analyses	Found no established inconsistency in selected events under their models	Limited post-merger signal, start-time sensitivity, mode covariance, and O3 noise
Submitted GWTC-5 multi-method study	Cross-checks current remnants with three complementary constructions	Submitted status; high-frequency noise and method-dependent selections limit spectroscopy

The conservative position is that present selected ringdowns have not established a non-Kerr spectrum. A competing claim based on an overtone, subdominant mode, or shifted damping time depends on the chosen start time and noise/model treatment. Negative results also do not establish that every remnant is exactly Kerr; many deviations remain below resolution or outside the chosen parameterization.

An adversarial independent check injects numerical-relativity GR signals and controlled shifted-mode signals into many off-source noise segments. It repeats inference across start times, mode sets, covariance constructions, sample rates, and independently implemented methods. A true deformation should recover its injected scaling without appearing in GR controls; a noise feature should not track a common remnant spectrum. Until this discriminator passes for the relevant population, strong spectroscopy claims remain unresolved and model-dependent.

What could change the conclusion. The interpretation could change after a peer-reviewed GWTC-5 analysis, an independently reproduced multi-mode detection, a revised calibration or noise-covariance treatment, or a simulation result that changes the justified onset of linear ringdown.

Exercises

For a mode with damping time $\tau$ , compare the amplitude available at start times $t_0$ and $t_0+\tau$ . Explain why later is not always statistically safer even if linear theory is more accurate.

Solution

The amplitude evolves as $e^{-(t-t_0)/\tau}$ , so delaying the start by one damping time reduces it by $e^{-1}\simeq0.368$ and the corresponding squared signal scale by $e^{-2}\simeq0.135$ . Later data may therefore have much less information, making mode amplitudes and frequencies prior-dominated and more sensitive to a noise fluctuation. Although nonlinear contamination should decrease, the variance grows; hence a trustworthy result must demonstrate stability across a predeclared range rather than choosing the latest or earliest time automatically.

References

Gravitational Wave Open Science Center (2026), “GWTC-5.0 Data Release Documentation,” stable release 9 parameter-estimation, strain, and data-quality products, official release, DOI.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Data Release for GWTC-5.0: Tests of General Relativity,” version P2600410-v1 (2026), official release.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “GWTC-5.0: Tests of General Relativity,” submitted manuscript P2500781-v17 (2026), official DCC record.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Tests of General Relativity with GWTC-3,” Physical Review D 107, 022005 (2023), DOI.
Maximiliano Isi and Will M. Farr, “Analyzing Black-Hole Ringdowns,” Physical Review D 104, 084016 (2021), DOI.
Nicolás Yunes, Xavier Siemens, and Kent Yagi, “Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar-Timing Arrays,” Living Reviews in Relativity 28, article 3 (2025), DOI.