No-Hair and Parametrized Ringdown Tests

Parametrized ringdown tests ask whether measured postmerger frequencies and damping times follow the Kerr spectrum of one mass and spin. They test a modeled response under explicit assumptions; they do not prove the full mathematical no-hair theorem, nor do they identify the cause of a deviation. A discrepancy could arise from new compact-object physics, an inadequate excitation model, nonlinear dynamics, mode mixing, calibration error, or nonstationary noise.

Helpful background. Multimode Black-Hole Spectroscopy defines common-remnant mode resolution, and Ringdown Likelihoods and Start-Time Dependence defines the windowed likelihood. The Kerr spectral calculation is developed in Quasinormal Boundary Conditions and Resonances, not repeated here.

Two complementary consistency tests

An inspiral–merger–ringdown consistency test estimates remnant parameters from two portions of the signal and compares them. If $(M_f^{\rm IM},\chi_f^{\rm IM})$ and $(M_f^{\rm RD},\chi_f^{\rm RD})$ are inferred with their shared calibration and noise correlations accounted for, useful fractional coordinates are

$$\Delta_M = \frac{2(M_f^{\rm IM}-M_f^{\rm RD})} {M_f^{\rm IM}+M_f^{\rm RD}}, \qquad \Delta_\chi = \frac{2(\chi_f^{\rm IM}-\chi_f^{\rm RD})} {\chi_f^{\rm IM}+\chi_f^{\rm RD}}.$$

Kerr consistency predicts the origin, but posterior width and coverage depend on the data split, waveform families, remnant fit, and selection. The spin ratio is poorly conditioned when the sum of inferred spin magnitudes is close to zero; use an absolute spin difference or another nonsingular coordinate in that regime.

A spectral-deformation test instead writes, for mode $k=(\ell,m,n)$,

$$\omega_k^R = \omega_{k,{\rm Kerr}}^R(M_f,\chi_f) \left(1+\delta\widehat\omega_k\right), \qquad \tau_k = \tau_{k,{\rm Kerr}}(M_f,\chi_f) \left(1+\delta\widehat\tau_k\right).$$

The notation is dimensionless, and $\delta\widehat\omega_k=\delta\widehat\tau_k=0$ is the Kerr submodel. A decaying template requires $\delta\widehat\tau_k\gt-1$; a positive-real-frequency branch also requires $\delta\widehat\omega_k\gt-1$. A fractional change in damping time is not the same parameter as a fractional change in damping rate: if $\alpha=1/\tau$, then $\delta\widehat\alpha=-\delta\widehat\tau/(1+\delta\widehat\tau)$, approximately $-\delta\widehat\tau$ for small departures.

Because detector-frame $M_f$ sets the overall time scale, freeing the dominant-mode frequency while also freeing $M_f$ creates a near-degeneracy. One must specify which mode anchors the remnant, which deviations vary, and whether external inspiral–merger information is used.

What a deviation parameter means

A phenomenological $\delta\widehat\omega_k$ is not a theory coupling. Different theories can alter the background spacetime, perturbation equations, polarization content, excitation, and propagation simultaneously. Conversely, a flexible deviation can absorb ordinary waveform error. The safe interpretation is conditional:

Within the stated detector response, noise model, finite mode set, start-time range, priors, and Kerr-deformation parameterization, the data constrain these departures.

Mode labels are part of that statement. Allowing two sinusoids to exchange labels can create mirrored posteriors; fixing labels by ordered frequency can conflict with physical branches when their order changes. State the labeling rule and test alternatives.

Prior scale directly controls evidence for the Kerr submodel. If a likelihood for one deviation is approximately Gaussian with width $\sigma$ inside a uniform prior of width $W\gg\sigma$, the evidence for the free model carries an Occam factor of order $\sigma/W$. Changing $W$ changes the Bayes factor even when the likelihood is unchanged. Publish the likelihood or posterior samples and the prior, not a Bayes factor alone.

Trials, combination, and systematics

Searching many modes, start times, events, and deviation parameterizations creates a look-elsewhere effect. A local tail probability cannot be quoted as the probability of “non-Kerr” without calibrating the complete search on null injections or off-source data. Catalog combination also requires a population hypothesis. Assuming one universal deviation, independent deviations per event, or a distribution with unknown mean and scatter gives different answers.

Minimum robustness checks include:

• scans over start time, end time, mode set, angular basis, and peak-time uncertainty;
• alternative calibration and noise realizations, including glitch treatments;
• injections with prompt response, overtones, nonlinear mode coupling, precession, and mode mixing;
• coverage tests for the complete selection and combination pipeline;
• posterior-predictive residual checks and a comparison with a flexible non-Kerr-agnostic waveform error model.

Results and limitations through 2026-08-18

Evidence	Supported conclusion	Limitation
GW250114 spectroscopy and TGR analysis	The released analyses find the measured remnant response compatible with their Kerr predictions while placing event-level constraints on allowed departures LVK 2025, main text pp. 1–5 and Figs. 1–3.	Compatibility is conditional on model, priors, data quality, start time, and mode identification; it is not a proof of uniqueness.
GWTC-5.0 tests of general relativity	The official catalog selected at this cutoff combines multiple generation, polarization, residual, and ringdown tests and reports no stable catalog-level requirement for the tested deviations LVK 2026, §§4–8.	Different tests constrain different phenomenological directions and cannot be collapsed into one theory-independent number.
GW150914 constrained spectroscopy	A Kerr-tied fundamental-plus-overtone construction produced an early direct spectral consistency test Isi et al. 2019, pp. 1–6.	The inference is sensitive to start time and overtone identifiability; a competing start-time and injection analysis weakens a claim of an independently measured overtone Cotesta et al. 2022, main text and supplemental start-time scans.
General time-domain frameworks	Free-frequency and Kerr-deviation likelihoods can be formulated with detector response and truncated covariance explicit Isi and Farr 2021, §§VI–VII.	A statistically correct likelihood does not eliminate waveform incompleteness or identify the physical origin of a deviation.

This matrix is pinned to the cutoff. Updated strain, calibration, TGR products, independent high-information-event analyses, or a material change in injection-calibrated false-positive rates requires revision.

Exercises

1. A free frequency is modeled as $f=f_{\rm Kerr}(1+\delta)$, with $f_{\rm Kerr}=250\ {\rm Hz}$. A posterior gives $\delta=0.04\pm0.03$ under a locally Gaussian approximation. Find the corresponding mean frequency and standard deviation. Does this alone establish a departure from Kerr?

Solution

The mean and standard deviation are

$$\bar f=250(1.04)=260\ {\rm Hz}, \qquad \sigma_f=250(0.03)=7.5\ {\rm Hz}.$$

The nominal displacement is only $0.04/0.03\simeq1.33$ standard deviations. Even that local statement omits prior choice, trials, waveform and noise systematics, and coverage. It does not establish a departure.

2. Suppose $N$ events share a truly universal small deviation $\delta$, and independent Gaussian likelihoods have means $\mu_i$ and variances $\sigma_i^2$. Derive the combined mean and variance under a broad prior.

Solution

Multiplying the likelihoods gives a Gaussian with precision and mean

$$\sigma_{\rm comb}^{-2} = \sum_{i=1}^{N}\sigma_i^{-2}, \qquad \mu_{\rm comb} = \sigma_{\rm comb}^{2} \sum_{i=1}^{N}\frac{\mu_i}{\sigma_i^2}.$$

This improvement is valid only for the universal-deviation hypothesis. If deviations vary across sources, the product is misspecified and can hide population scatter.

References

• Cotesta, R., Carullo, G., Berti, E., and Cardoso, V. (2022). “Analysis of Ringdown Overtones in GW150914.” Physical Review Letters 129, 111102. DOI.
• Isi, M., and Farr, W. M. (2021). “Analyzing black-hole ringdowns.” arXiv:2107.05609.
• 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. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “Black Hole Spectroscopy and Tests of General Relativity with GW250114.” arXiv:2509.08099.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “GWTC-5.0: Tests of General Relativity.” arXiv:2607.19293.