Ringdown Likelihoods and Start-Time Dependence

A ringdown likelihood is inseparable from its analysis window. Moving the start time changes the physical regime being modeled, the measurable mode amplitudes, the amount of signal-to-noise ratio, and—unless handled exactly—the noise covariance. A defensible result is therefore a stability statement over specified start times and noise treatments, not a frequency estimate with the window hidden.

Helpful background. Predicting the Remnant from the Full Signal provides the pre-ringdown Kerr prediction. Ringdown Start Time explains the waveform transition to a linear description, while Completeness, Excitation, and Ringdown Content explains why a finite quasinormal sum is not automatically a complete description of arbitrary postmerger data.

A finite-window signal model

Choose a detector-frame reference time $t_{\rm peak}$ and analyze samples $t\in[t_0,t_1]$. For detector $I$,

$$h_I(t;\theta,t_0) = \sum_{\ell mn} \operatorname{Re}\!\left[ \mathcal A_{\ell mn}^{I}(t_0) e^{-i\omega_{\ell mn}(t-t_0)} \right], \qquad \omega_{\ell mn}=\omega_{\ell mn}^{R}-\frac{i}{\tau_{\ell mn}} .$$

This uses the convention $e^{-i\omega t}$, so $\tau_{\ell mn}>0$ corresponds to decay. The complex detector amplitude contains source orientation, polarization response, intrinsic excitation, and the spherical–spheroidal angular projection. Those ingredients can be degenerate; a network does not measure a source-frame mode amplitude merely by fitting one real damped sinusoid.

Changing the reference from $t_0$ to $t_a>t_0$ leaves an exact linear mode unchanged only if

$$\mathcal A_{\ell mn}^{I}(t_a) = \mathcal A_{\ell mn}^{I}(t_0) e^{-i\omega_{\ell mn}(t_a-t_0)}.$$

Thus the magnitude decays by $e^{-(t_a-t_0)/\tau_{\ell mn}}$ while its phase advances by $-\omega_{\ell mn}^{R}(t_a-t_0)$. Priors that are uniform in amplitude at every start time are not the push-forward of one another; they can create an apparent start-time dependence even for identical data.

Likelihoods after truncation

For a sample vector $\boldsymbol d$ and residual $\boldsymbol r=\boldsymbol d-\boldsymbol h$, a time-domain Gaussian likelihood is

$$\ln \mathcal L = -\frac12\boldsymbol r^{\mathsf T} \boldsymbol C_{[t_0,t_1]}^{-1} \boldsymbol r -\frac12\ln\det\boldsymbol C_{[t_0,t_1]} +\text{constant}.$$

The relevant covariance is the submatrix for the retained samples, not the inverse of the corresponding sub-block of an already inverted long-segment covariance. These operations do not commute. Estimating $\boldsymbol C$ from neighboring data also introduces finite-sample uncertainty that should be propagated or stress-tested.

For stationary, band-limited sampled noise with one-sided PSD $S_n(f)$ and sampling interval $\Delta t$, the unwindowed covariance is

$$C_{ij}=\int_0^{1/(2\Delta t)} S_n(f)\cos\!\left[2\pi f(t_i-t_j)\right]df.$$

For a flat sampled-band spectrum $S_n(f)=S_0$, this gives $C_{ii}=S_0/(2\Delta t)$ and vanishing off-diagonal values at distinct sample times. A colored spectrum gives nonzero off-diagonal correlations, explaining physically why discarding early samples and conditioning on their values are different operations. Any filters applied to the data must also be applied to the model and covariance.

Frequency-domain implementations require equal care. A hard time cut convolves the spectrum with a sinc-like kernel, while a taper correlates neighboring Fourier bins. Treating the tapered bins as independent with the stationary long-segment power spectral density can bias mode parameters. Time-domain conditioning, inpainting, gated likelihoods, or an explicitly window-convolved frequency model are legitimate alternatives, provided injections recover known parameters with the same pipeline Isi and Farr 2021, §§III–V.

Why the start-time curve matters

An early window has more signal power but is most exposed to prompt response, nonlinear dynamics, evolving remnant parameters, mode mixing, and waveform-model error. A late window better approaches linear perturbation theory but loses rapidly damped modes and eventually becomes noise dominated. There is no data-independent instant at which all contaminants vanish.

For a grid of $t_0-t_{\rm peak}$, report at least:

• the event, detector network, calibration version, conditioning, and peak-time definition;
• the fitted mode set, angular basis, amplitude and frequency priors, and end time;
• posterior shifts in $(\omega^R,\tau)$ and remnant parameters, together with evidence or predictive scores;
• residual coherence, off-source background behavior, and recovery on signals injected into real noise;
• repetitions with plausible noise spectra, glitch treatments, waveform responses, and mode sets.

A plateau is useful only if its width exceeds the timing uncertainty and it survives these variations. Adjacent start-time results are highly correlated because they reuse almost all the same samples; counting them as independent confirmations is incorrect.

Raw evidences at different start times also refer to different data vectors. Their ratios are not Bayes factors for a common experiment merely because each likelihood is normalized. To infer $t_0$ as a model parameter, define one common data record and model or marginalize the pre-ringdown portion consistently; otherwise use separate start-time fits as robustness diagnostics. Within each fixed window, competing mode models can be compared on the same retained samples.

Evidence through 2026-08-18

Evidence stream	What it establishes	What remains conditional
GW250114 ringdown analyses	A high-information event supports strong remnant and mode-consistency tests under the released network data, calibration, start-time, and waveform choices LVK 2025, main text pp. 1–5 and Figs. 1–3.	The stability region depends on noise realization, mode content, priors, and treatment of early postmerger data.
GWTC-5.0 tests of general relativity	Catalog methods expose event selection, start-time scans, three ringdown parameterizations, and their event-level conclusions as of the cutoff LVK 2026, §7, pp. 16–25.	A catalog posterior does not remove event-level start-time or waveform systematics.
Time-domain ringdown methodology	Finite-duration covariance and detector response can be formulated without pretending truncated Fourier bins are independent Isi and Farr 2021, §§III–VI.	Correct formulation does not guarantee that a chosen finite mode sum is adequate.
Numerical-relativity comparisons	Starting earlier can require more rapidly damped content to reproduce a simulated waveform Giesler et al. 2019, §§III–V and Figs. 1–5.	Better reconstruction of a simulation is not, by itself, identification of the same content in noisy detector data.

The evidence should be refreshed when calibration, strain, noise-subtraction products, event posteriors, or the official TGR release changes. A tail in one start-time posterior is not a stable departure from Kerr unless the full scan and its trials are included.

Exercises

1. A mode has $\tau=4\ {\rm ms}$. Its fitted magnitude at $t_0$ is $A_0$. What magnitude represents the same mode at $t_0+6\ {\rm ms}$?

Solution

The reference-time transformation gives

$$\lvert A_1\rvert = \lvert A_0\rvert e^{-6/4} \simeq 0.223\,\lvert A_0\rvert .$$

Keeping the same numerical amplitude prior at both times therefore assigns different prior probability to the same physical signal.

2. Partition a covariance matrix as

$$\boldsymbol C= \begin{pmatrix} a&b\\ b&c \end{pmatrix}.$$

Show that retaining only the second sample gives covariance $c$, whereas retaining the lower-right element of $\boldsymbol C^{-1}$ and inverting it gives $c-b^2/a$.

Solution

The marginal distribution of the second sample has variance $c$. Direct inversion gives

$$\boldsymbol C^{-1} = \frac{1}{ac-b^2} \begin{pmatrix} c&-b\\ -b&a \end{pmatrix}.$$

The inverse of its lower-right entry is $(ac-b^2)/a=c-b^2/a$, which is the conditional variance given the discarded sample, not the marginal variance. They agree only when $b=0$.

References

• Giesler, M., Isi, M., Scheel, M. A., and Teukolsky, S. A. (2019). “Black Hole Ringdown: The Importance of Overtones.” Physical Review X 9, 041060. DOI.
• Isi, M., and Farr, W. M. (2021). “Analyzing black-hole ringdowns.” arXiv:2107.05609.
• 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.