Ringdown Start Time

There is no known invariant instant at which a merger waveform switches from “nonlinear” to “linear.” A ringdown start time is an operational choice tied to a field, frame, peak convention, model family, norm, and scientific question. Near-peak QNM models can be useful before every fitted amplitude is demonstrably stable; a later start can support a more conservative pole interpretation while discarding signal. The evidence below is therefore a dated comparison, not a verdict.

Helpful background. Einstein Equations supplies perturbative order counting; Waves, Green Functions, and Geometric Optics supplies prompt, pole, and tail response; Black-Hole Perturbation Theory supplies QNM response domains; Gravitational Radiation and Waveforms supplies asymptotic strain and frame conventions; and Black-Hole Dynamics and Collapse supplies the nonlinear remnant-formation history.

Three questions hidden in one time

When quoting the numerical studies below, use their geometrized units $G=c=1$: $M$ is the initial total mass as a time or length and $M_f$ is the final mass. Thus a source-frame interval $bM$ restores to $bGM_{\rm phys}/c^3$, or $(1+z)bGM_{\rm phys}/c^3$ in detector time. Initial-mass and final-mass time units must not be interchanged.

Fix an inertial Bondi frame aligned with the final remnant, use

$$H=r(h_+-ih_\times),$$

expand in unit-normalized spin-$-2$ harmonics, and take $e^{-i\omega u}$ so that $\operatorname{Im}\omega\lt 0$ means damping. A candidate $u_0$ may answer three different questions.

1. Perturbative smallness: after $u_0$, is a field perturbation about one stationary Kerr background small in a declared gauge-invariant or asymptotic norm?
2. Waveform adequacy: on $[u_0,u_1]$, does a specified finite QNM model approximate the chosen waveform quantity to its numerical or modeling accuracy?
3. Parameter use: does that model stably recover the intended remnant or modal parameters under reasonable changes of window and model order?

The formation of a common horizon is geometrically important but does not, by itself, answer any of these waveform questions at future null infinity. Conversely, a flexible damped-sinusoid fit can answer the second question without establishing the first. The older review literature already described ringdown start time as intrinsically ill-defined and showed that different operational definitions change inferred efficiencies Berti, Cardoso, and Starinets 2009, § 9.1, pp. 61–62, PDF.

Diagnostics that do not collapse to one criterion

For a model $H_N(u;u_0)$, one may monitor a finite-window residual

$$\epsilon_N^2(u_0,u_1)= \frac{\int_{u_0}^{u_1}\lVert H-H_N\rVert_{S^2}^2du} {\int_{u_0}^{u_1}\lVert H\rVert_{S^2}^2du}.$$

But adding columns to a least-squares design matrix cannot increase its optimal residual; reduction alone therefore favors complexity. Stronger checks transport every fitted amplitude to one reference time,

$$\bar A_q(u_0;u_*)=A_q(u_0)e^{+i\omega_q(u_0-u_*)},$$

and ask whether $\bar A_q$, its phase, the fitted frequencies, and the common remnant parameters remain stable when $u_0$, $u_1$, resolution, and model order change. Stability must be compared with the numerical floor: an apparently constant coefficient below that floor is not resolved.

Even the origin of time must be named. For the outgoing tetrad $(\ell,n,m,\bar m)$ with $\ell\!\cdot n=-1$, $m\!\cdot\bar m=1$, and $\Psi_4=-C(n,\bar m,n,\bar m)$, the leading curvature waveform is $\Psi_4^0=\partial_u^2H$; differentiation need not preserve a peak time. In SXS:BBH:0305, the peaks of strain amplitude, strain luminosity, $\Psi_4$ amplitude, and $\Psi_4$ luminosity occur at offsets of roughly $0$, $7M$, $10M$, and $11M$, respectively Giesler et al. 2019, § II, p. 2, PDF. A phrase such as “ten mass units after peak” is incomplete without the observable and mode.

Evidence status as of 2026-08-18

The universal-onset question remains unresolved; no universal verdict follows from the present evidence. The table separates what each result supports from the qualification built into its scope and the tension or failure mode that prevents a universal onset claim.

Evidence or result	Support	Qualification, scope, and limitation	Tension or failure mode
Retarded-response structure	A stationary black hole has a pole contribution satisfying ingoing future-horizon and outgoing future-infinity conditions.	The complete response also contains prompt and branch-cut pieces Berti, Cardoso, and Starinets 2009, § 3.2, pp. 18–20, PDF.	Pole usefulness need not begin when the background first becomes approximately Kerr, and a finite pole sum is not complete.
Fixed-frequency near-peak fits	For SXS:BBH:0305, a spherical $(2,2)$ strain model with Kerr frequencies through $n=7$, unweighted time-domain fitting, $u_1=u_{\rm peak}+90M$, and $u_0=u_{\rm peak}$ produced a low mismatch and accurate remnant recovery; similar recovery was reported for about 80 aligned-spin simulations Giesler et al. 2019, §§ III–IV, PDF.	The peak was the complex-strain amplitude peak; the principal system had mass ratio $1.22$, $M_f=0.9520M$, and $\chi_f=0.6921$. Frequencies were tied to one remnant, while amplitudes were linear fit parameters.	The study itself notes that mismatch can overfit numerical noise after fast modes decay; early fit success does not uniquely decompose prompt, nonlinear, or evolving-background content.
Variable-projection and amplitude stability	In two high-accuracy equal-mass, aligned-spin simulations, stable decay-compensated amplitudes were demonstrated from about $4M$ after the $(2,2)$ strain peak for $\chi_f=0.75$ and about $8M$ after it for $\chi_f=0.92$ Giesler et al. 2025, §§ IV–V, PDF.	The waveforms used CCE, a remnant superrest BMS frame, selected news content, variable projection, mode dropping, and a numerical floor near $10^{-8}$ rather than the roughly $10^{-5}$ floor of earlier data.	Before those times, fixed-frequency models could still recover remnant properties, but some individual amplitudes were unstable; unmodeled content and modes still being sourced were plausible causes.
Peak, norm, and window choice	Later starts usually suppress prompt response and highly damped terms, making a fundamental-mode interpretation simpler.	“Later” depends on strain, news, curvature, luminosity, spherical mode, end time, and whether the norm is unweighted or application-weighted.	Waiting reduces both contamination and available information; different observables can rank candidate starts differently.
Resolution and free-frequency tests	Agreement across resolutions and recovery of Kerr-consistent frequencies can reject obvious numerical artifacts or an incorrect background.	Free frequencies add nonlinear parameters and can be poorly conditioned when damped columns overlap.	A flexible model may absorb truncation error, frame error, a changing background, or noise; parameter agreement at one window can be accidental.

Three positions remain credible within their stated scope. The near-peak usefulness position emphasizes accurate finite-window representation and remnant recovery with several fixed Kerr frequencies. The conservative onset position waits for decay-compensated amplitudes, phases, and free-frequency estimates to stabilize above the error floor. The analysis-dependent position treats $u_0$ as a model hyperparameter because adequacy depends on the waveform observable and intended claim. These are not logically exclusive: the same waveform can be useful near peak and admit a cleaner pole interpretation later.

Reproducible comparison protocol

A controlled start-time study should publish enough information to repeat the selection, not only the favored time.

1. Select at least two public waveforms spanning remnant spin or another relevant regime, and state simulation IDs, resolutions, extraction method, and any memory subtraction. Preserve a blind holdout waveform if model order or thresholds were tuned on the others.
2. Map every waveform to a declared inertial or remnant superrest Bondi frame. State whether the fitted object is $H$, a spherical $H_{\ell m}$, news, or curvature; do not compare their peak times as if identical.
3. Declare $u=0$ by observable, angular mode, and resolution. Report alternative peak definitions rather than silently shifting them.
4. Scan a two-dimensional grid of $(u_0,u_1)$ at two or more resolutions. Repeat with fixed Kerr frequencies, free frequencies, and at least one reduced model order, using the same stated norm.
5. Transport coefficients to a common $u_*$, compare amplitudes and phases with resolution differences and injected noise, and report the condition number or singular values of the design matrix.
6. Record failure cases: coefficients that grow after decay compensation, frequencies that wander between families, sensitivity to dropping one mode, results below the numerical floor, or conclusions reversed by the end time.

This is a documented rerun recipe, not a claim that the cited numerical results were independently reproduced here. Production evolutions are treated in Computational Relativity; detector-specific likelihoods and event-level onset choices in Gravitational-Wave Astronomy.

Exercise: choose what the evidence permits

A fixed-frequency model fitted from the strain peak gives $\epsilon=10^{-5}$, but its transported fastest-overtone amplitude changes by $80\%$ when $u_0$ advances by $2M$. From $u_0=u_{\rm peak}+6M$, the residual is $4\times10^{-5}$ and all transported amplitudes change by less than $3\%$ across start times and two resolutions. Which start is supported for (a) a compact waveform approximation and (b) an individually stable modal decomposition?

Solution

For (a), the peak start is supported within the tested window and norm: it gives the smaller residual, provided the numerical error is no larger and the model is not being selected on the same data without a complexity check. For (b), the evidence supports the later $u_{\rm peak}+6M$ start, because amplitude stability—not residual alone—is the relevant diagnostic. Neither result proves that the spacetime became linear at either instant. One must also quote the end time, mode set, peak definition, frame, resolution difference, and numerical floor before transferring either conclusion to another waveform.

What could change the conclusion

The interpretation could change after a materially new controlled onset criterion, a public high-accuracy waveform comparison across frames and resolutions, a reproducible contrary analysis, or a correction to the cited fits. New comparisons should retain the dates and assumptions of earlier results: differing criteria need not converge to a single preferred time.

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.