Multimode Black-Hole Spectroscopy

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Black-hole spectroscopy asks whether two or more independently informative damped modes are compatible with one Kerr remnant. Seeing extra postmerger structure is not enough: the components must be resolved, assigned defensible mode labels, and shown to share a common mass and spin after start-time, excitation, angular-response, calibration, and noise uncertainties are propagated.

Helpful background. Predicting the Remnant from the Full Signal defines the inspiral–merger prediction, and Ringdown Likelihoods and Start-Time Dependence defines the windowed likelihood. The spectral map is developed in Quasinormal Boundary Conditions and Resonances; Spectroscopy Model Requirements explains the waveform requirements used here.

One remnant, several complex frequencies

For a Kerr black hole with detector-frame mass $M_f$ and dimensionless spin $\chi_f$ ,

\omega_k = \frac{c^3}{G M_f}\widehat\omega_k(\chi_f), \qquad k=(\ell,m,n), \qquad \omega_k=\omega_k^R-\frac{i}{\tau_k},

using $e^{-i\omega t}$ . Two complex frequencies therefore provide four real observables constrained by two remnant parameters, before amplitudes and orientation are considered. A useful consistency coordinate compares each measured mode with the common-remnant manifold:

p(M_f,\chi_f\mid d,k_1,k_2) \propto p(M_f,\chi_f) \int p\!\left[ d\ \middle|\ \sum_{a=1}^{2} h_{k_a}(M_f,\chi_f,\mathcal A_a),\eta \right] p(\mathcal A_1,\mathcal A_2,\eta)\, d\mathcal A_1\,d\mathcal A_2\,d\eta .

Here each complex amplitude $\mathcal A_a$ carries excitation and phase information, while $\eta$ collects shared noise, calibration, start-time, and response nuisances. There is one likelihood for the one data set: multiplying separate single-mode likelihoods for the same $d$ would count the data twice and would discard interference between the modes. An alternative “agnostic” fit first measures free $(\omega_a^R,\tau_a)$ and only then asks whether their joint posterior intersects a Kerr spectral pair. The constrained fit is usually more sensitive; the agnostic fit makes independent spectral information easier to diagnose. They answer different questions and should not be presented as interchangeable.

Resolution is more than peak separation

A Rayleigh-like rule such as $\lvert f_1-f_2\rvert\gtrsim1/(2\pi\tau)$ is a useful scale estimate, not a detection criterion. Unequal damping, amplitude, phase, detector antenna response, and colored noise alter resolvability. The operational requirement is that the likelihood contain information in directions that separate the modes.

For whitened complex templates $q_a$ , define the Hermitian Gram matrix $G_{ab}=\langle q_a,q_b\rangle=q_a^\dagger q_b$ . Its small eigenvalues diagnose nearly indistinguishable template combinations. The actual real-data amplitude Fisher matrix is built from the sine and cosine quadratures; nuisance parameters require an additional projection or marginalization. If frequencies are also free, inspect the full posterior for:

localization away from a single-mode boundary;
stable mode labels rather than label switching;
frequency and damping information not imposed almost entirely by a Kerr prior;
predictive improvement on held-out or posterior-predictive data;
recovery and false-identification rates in injections at the observed signal-to-noise ratio.

Signal-to-noise ratio alone cannot certify spectroscopy. A loud fundamental plus a tiny secondary mode may remain one-mode data, while a moderately loud signal with favorably separated and excited modes can be more informative.

For a simple illustration, in continuous white noise over $t\geq0$ let $q_a=e^{-\alpha_a t-i\omega_a^Rt}$ , with $\alpha_a=1/\tau_a$ . Direct integration gives

\langle q_1,q_2\rangle =\frac{1}{\alpha_1+\alpha_2+i(\omega_2^R-\omega_1^R)}, \qquad r^2=\frac{4\alpha_1\alpha_2} {(\alpha_1+\alpha_2)^2+(\omega_2^R-\omega_1^R)^2}.

Here $r$ is the normalized overlap magnitude. Coincident frequencies and damping rates give $r=1$ : two amplitudes then describe the same basis function exactly. Different damping rates can reduce degeneracy even when real frequencies coincide. Finite windows and colored noise change the integral, so this calculation explains the mechanism rather than supplying a universal detection threshold.

Excitation and angular systematics

The detector measures

h_I(t) = F_I^+h_+(t)+F_I^\times h_\times(t),

not an intrinsic mode catalog. Source inclination, polarization, sky position, and the network geometry govern which angular components are visible. Numerical waveforms are often decomposed in spin-weighted spherical harmonics, whereas Kerr angular eigenfunctions are spheroidal; their mixing can move power among apparent $(\ell,m)$ labels. Precession, recoil, nonlinear combination tones, and an evolving background can create additional structure without constituting an independent linear Kerr mode.

Consequently, a mode claim must state the spherical or spheroidal convention and include plausible mixing. Excitation amplitudes should be nuisance parameters or informed by calibrated merger physics with uncertainty. Forcing an absent mode to the amplitude predicted by a narrow source model can turn a waveform assumption into “spectroscopic evidence.”

Results and limitations through 2026-08-18

Evidence stream	Present support	Boundary of the claim
GW250114 spectroscopy	The LVK analysis reports that at least two damped components are required within its adopted postmerger model and performs common-Kerr-remnant tests with the released event data LVK 2025, main text pp. 1–5 and Figs. 1–3.	“Required by this model comparison” is stronger than fit improvement but remains conditional on event selection, calibration, noise treatment, start time, mode labels, priors, and waveform family; it is not model-independent identification of two physical poles.
GWTC-5.0 TGR catalog	The official catalog analysis combines three ringdown pipelines with different event selections and parameterizations LVK 2026, §7, pp. 16–25.	Catalog combination does not convert prior-dominated or unresolved event-level components into independently measured modes.
GW150914 analyses	A fundamental-plus-overtone interpretation enabled an early no-hair consistency test under a constrained Kerr spectrum Isi et al. 2019, pp. 1–6.	Competing analyses show sensitivity to start time, noise, and overtone priors; this event alone did not establish a universally robust free-frequency two-mode measurement.
Spectroscopy forecasts	Mode-separation and signal-to-noise calculations identify regimes in which a common-remnant test becomes possible Berti, Cardoso, and Will 2006, §§II–IV.	Forecast thresholds depend on excitation, detector spectrum, parameter correlations, and the adopted decision rule.

The strongest present evidence should neither be diluted into “nothing has been learned” nor inflated into a model-independent theorem. Refresh the assessment when official strain or calibration changes, GWTC TGR products are superseded, or an independent analysis alters mode-resolution or false-positive conclusions.

Exercises

Two measured cyclic frequencies satisfy $f_1=K_1(\chi_f)/M_f$ and $f_2=K_2(\chi_f)/M_f$ . Show why their ratio is a spin diagnostic independent of detector-frame mass.

Solution

Taking the ratio cancels the common scale:

\frac{f_1}{f_2} = \frac{K_1(\chi_f)}{K_2(\chi_f)}.

If the right-hand side is locally one-to-one, the measured ratio constrains $\chi_f$ . Substituting that spin into either absolute frequency then determines $M_f$ . In practice, correlated uncertainty and possible mode-label ambiguity must be propagated jointly.

Suppose normalized mode templates have overlap magnitude $0.98$ . What fraction of the squared norm of $q_2$ remains after projecting out $q_1$ ?

Solution

For normalized templates, the orthogonal residual is

q_{2\perp}=q_2-\langle q_1,q_2\rangle q_1, \qquad \lVert q_{2\perp}\rVert^2 = 1-\lvert\langle q_1,q_2\rangle\rvert^2.

Thus only $1-0.98^2=0.0396$ , about $4\%$ , remains. Most of the second component can be absorbed by changing the first amplitude, so a large total signal-to-noise ratio need not imply two-mode resolution.

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.
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.