Requirements for a Spectroscopy Waveform Model

A spectroscopy-ready waveform model must do more than return two complex frequencies. It must define the radiative field and frame, show that its modal columns are distinguishable on the chosen interval, identify their angular and radial labels, control prompt, mixing, nonlinear, and numerical alternatives, and test whether all assigned frequencies belong to one remnant. These are waveform-theory requirements. Detector calibration, noise likelihoods, event-level inference, and empirical spectroscopy verdicts belong to Gravitational-Wave Astronomy.

Helpful background. Einstein Equations supplies the stationary Kerr hypothesis; Waves, Green Functions, and Geometric Optics supplies resonance and inverse problems; Black-Hole Perturbation Theory supplies labeled QNM spectra; Gravitational Radiation and Waveforms supplies the asymptotic waveform object; and Black-Hole Dynamics and Collapse supplies remnant formation and relaxation.

The radiative field and mode conventions

Declare an inertial Bondi frame, preferably remnant aligned, and model

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

in unit-normalized spin-$-2$ harmonics. State any conversion from news or a tetrad-defined curvature scalar rather than identifying those fields with strain. Use $e^{-i\omega u}$, for which $\operatorname{Im}\omega\lt 0$ is damping, and quote frequencies in a declared source-frame or detector-frame time coordinate. A redshifted mass rescales all frequencies; it is not a change in the dimensionless Kerr spectrum.

For each fitted interval $[u_0,u_1]$, the model contract must include:

• the field, spherical mode set, Bondi/BMS frame, retarded-time origin, remnant mass scale, and angular normalization;
• the candidate spheroidal labels $(L,m,n)$ and spherical–spheroidal overlaps;
• whether frequencies are fixed to Kerr, tied through common $(M_f,\chi_f)$, or free;
• the amplitude reference time, start and end times, norm, preprocessing, and numerical-error estimate;
• included prompt, tail, overtone, nonlinear, memory, and slowly varying nuisance terms, plus declared omissions.

Without this metadata, two numerically similar complex frequencies can represent different physical hypotheses.

Parameter counting and the one-remnant constraint

Consider $K$ candidate modes in one or more spherical channels,

$$H_{\rm mod}(u)= \sum_{q=1}^{K}A_q e^{-i\omega_q(M_f,\chi_f)(u-u_0)}S_q.$$

If labels, frame, window, and $u_0$ are fixed, a Kerr-constrained model has $2K$ real amplitude parameters plus the two real background parameters $(M_f,\chi_f)$: $2K+2$ in total. An agnostic model with an independent complex amplitude and complex frequency for every term has $4K$ real parameters. Treating $u_0$ merely as a new reference does not add physical freedom when all complex amplitudes are free, because its change is absorbed by

$$A_q(u_0')=A_q(u_0)e^{-i\omega_q(u_0'-u_0)}.$$

For fixed mode labels, $K$ measured complex frequencies contain $2K$ real numbers. The Kerr family spans only two real parameters, so common-background consistency imposes $2K-2$ real restrictions. One complex mode can, in principle, determine $M_f$ and $\chi_f$ when the local spectrum map is invertible; a second mode supplies two additional real consistency conditions. This is why resolving two modes is necessary but not sufficient for a no-hair-style test Berti, Cardoso, and Starinets 2009, §§ 9.6–9.7, pp. 79–83, PDF.

The word “consistent” requires one label assignment and one $(M_f,\chi_f)$ that reproduce both oscillation frequencies and damping times within declared waveform and numerical uncertainty. Inferring a separate mass and spin from each term and then averaging them discards the test. The counting assumes that mass and spin are inferred from ringdown alone. If independent information constrains both background parameters, even one complex mode can provide a consistency test; it is a different information budget from a self-contained two-mode spectroscopy test.

Changing a coefficient’s reference time also differs from changing which data enter the fit. Moving the window start removes samples and alters the available information. The amplitude reparameterization above proves only that changing the reference inside a fixed model and fixed data interval adds no independent degree of freedom.

Four distinct gates

Gate	Question	Minimum waveform evidence	What it does not establish
Mathematical resolvability	Are the finite-window damped columns linearly independent at useful precision?	Singular values or condition numbers under window and mode-order changes; recovery of injected components.	That the columns correspond to physical modes rather than nuisance flexibility.
Mode identifiability	Is a term uniquely favored over prompt, leakage, overtone, nonlinear, and drifting-background alternatives?	Stable frequency, decay-compensated amplitude and phase; angular selection; matched-complexity controls; held-out performance.	Statistical identifiability in a particular detector dataset.
Model adequacy	Does the complete waveform family describe the target signal without material structured error?	Residuals compared with numerical error, start/end scans, resolution and extraction convergence, frame and preprocessing variants.	That calibration and noise assumptions in an observation are correct.
Detector inference	Do calibrated data and a stated noise likelihood constrain the claimed modes and common background?	This is a downstream requirement involving calibration uncertainty, nonstationary noise, priors, selection, and coverage.	It is not supplied by a numerical waveform fit or by this page.

The classic Rayleigh-style comparison of frequency or damping-time separation with measurement error illustrates resolvability, but its cited two-mode form assumed white noise and distinct angular labels; the review itself describes resulting thresholds as optimistic under simplified parameter knowledge Berti, Cardoso, and Starinets 2009, § 9.7, pp. 82–83, PDF. Such a criterion is not interchangeable with model adequacy or robust inference.

Calibration and contamination controls

Near peak, a many-overtone fixed-frequency model may recover the correct numerical remnant while individual fast-mode amplitudes remain unstable. Giesler and collaborators’ 2019 analysis supports the former statement under a documented spherical $(2,2)$, unweighted, finite-window setup Giesler et al. 2019, §§ III–IV, PDF. Their later high-accuracy study used CCE, remnant superrest-frame fixing, mass-unit correction, variable projection, injection tests, and amplitude stability; it also found unmodeled early content and strong noise sensitivity for highly damped terms Giesler et al. 2025, §§ IV–V, PDF. A spectroscopy model must preserve both lessons.

Before exporting a model downstream:

1. Repeat start and end scans with at least two numerical resolutions and state the extraction and BMS-frame procedure.
2. Compare fixed common-Kerr, free-frequency, reduced-mode, prompt-nuisance, and nonlinear/mixing alternatives at matched effective complexity.
3. Use synthetic injections to measure false-mode and missed-mode rates across the numerical floor and relevant conditioning range.
4. Hold out time intervals or waveforms used neither to select the mode set nor to tune stability thresholds.
5. Supply a waveform-error representation that a detector analysis can propagate; do not hide numerical truncation, calibration convention, or preprocessing uncertainty inside free QNM amplitudes.

Failure is informative. A mode that changes label under a frame correction, loses its amplitude under a one-step window shift, falls below its injection recovery floor, or forces incompatible background parameters must not be promoted to a spectroscopy degree of freedom. Likewise, agreement of two real frequencies without their damping times can be a useful partial check but does not satisfy the full complex-spectrum constraint.

Exercise: count the constraints

A three-mode model has fixed labels and window. Compare (a) a common-Kerr model with three free complex amplitudes and free $(M_f,\chi_f)$ and (b) an agnostic model with three independent complex amplitudes and frequencies. Count their real parameters. How many real common-background restrictions do three complex frequencies obey? Does fitting $u_0$ add an independent parameter if all amplitudes remain free?

Solution

Model (a) has $3\times2=6$ real amplitude parameters plus two background parameters, for $8$. Model (b) has three complex amplitudes and three complex frequencies, for $3\times2+3\times2=12$. Three complex frequencies provide six real spectral quantities, while the Kerr background has two real parameters, leaving $6-2=4$ real restrictions once the labels are fixed.

Changing $u_0$ only multiplies each free complex amplitude by its known phase and damping factor. It therefore does not add an independent physical parameter unless the amplitudes are constrained by an attachment prescription or another shared model. Parameter counting is only a first gate: the three columns must still be resolvable, identifiable, adequate, and passed to a valid downstream likelihood before any empirical spectroscopy conclusion.

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.