Predicting the Remnant from the Full Signal

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The full inspiral–merger signal predicts the black hole that should ring after coalescence. The useful object is not a single “best” final mass or spin, but a posterior predictive distribution for the remnant and its Kerr quasinormal frequencies, constructed before a ringdown-only fit. This makes the later comparison a quantitative consistency test rather than a circular identification of whatever modes happen to fit.

Helpful background. Binary Parameters and Observable Combinations supplies the source parameters carried through the calculation, and Waveform-Model Uncertainty in Inference explains why waveform families must be treated as a systematic choice. The spectral boundary-value problem itself belongs to Quasinormal Boundary Conditions and Resonances; this page uses, rather than rederives, that theory.

From binary parameters to a remnant distribution

Let $\vartheta$ denote source-frame component masses and spin vectors, and let $\lambda$ collect calibration, waveform, and population choices. Numerical-relativity fits or calibrated waveform models provide maps

\vartheta \longmapsto \left(M_f,\boldsymbol{\chi}_f,\boldsymbol{v}_{\rm kick}\right),

constrained by radiated energy, linear momentum, and angular momentum. In the initial center-of-momentum frame, let $E_i$ be the initial total energy, including binding energy, and let $(E_{\rm rad},\boldsymbol P_{\rm rad})$ be the energy and momentum radiated to infinity. Then

\begin{aligned} E_f&=E_i-E_{\rm rad}, &\boldsymbol P_f&=-\boldsymbol P_{\rm rad},\\ M_f^2c^4&=E_f^2-c^2|\boldsymbol P_f|^2, &\boldsymbol v_{\rm kick}&=\frac{c^2\boldsymbol P_f}{E_f}. \end{aligned}

Thus $M_f\simeq(E_i-E_{\rm rad})/c^2$ neglects the recoil kinetic energy; it is not an exact identity for a kicked remnant, and $E_i/c^2$ at finite separation is not simply $m_1+m_2$ . Angular-momentum balance uses a common asymptotic frame and origin. The remnant’s dimensionless spin is $\boldsymbol\chi_f=c\boldsymbol S_f/(GM_f^2)$ , where $\boldsymbol S_f$ is its intrinsic angular momentum in the remnant rest frame; translation and boost contributions must be removed before identifying total angular momentum with spin.

Applying the map sample by sample to the inspiral–merger posterior preserves correlations that a Gaussian error bar would erase. If $\mathcal R$ denotes a remnant fit with uncertain coefficients $\alpha$ , the predictive density is

p(M_f,\chi_f\mid d_{\rm IM}) = \int p(M_f,\chi_f\mid\vartheta,\alpha,\mathcal R)\, p(\vartheta,\alpha,\mathcal R\mid d_{\rm IM})\, d\vartheta\,d\alpha\,d\mathcal R .

The integral makes three uncertainties visible: measurement uncertainty in $\vartheta$ , finite calibration error in each remnant fit, and model discrepancy among waveform or fit families. Recoil additionally depends on spin directions and orbital phase near merger; it is usually much less tightly predicted than $M_f$ or $\chi_f$ .

NRSur7dq4Remnant is one concrete example of such a map: it predicts final mass, spin, and recoil from a surrogate trained on 1,528 quasicircular numerical-relativity simulations with mass ratio $q\leq4$ and spin magnitudes no larger than $0.8$ . Its cross-validation errors provide a model-error scale inside that training domain; extrapolation outside the domain is not validated merely because the code returns a value Varma et al. 2019, §§V–VI. A production analysis must state which remnant map and version it uses, check whether every posterior sample lies in the calibrated domain, and propagate or bracket the map’s error.

Detector-frame and source-frame masses must not be mixed. If $M_f^{\rm src}$ is the physical remnant mass, then the time scale seen at the detector is set by

M_f^{\rm det}=(1+z)M_f^{\rm src}.

Redshift therefore rescales every observed frequency and damping time, but leaves the dimensionless spin unchanged.

Predicting the Kerr spectrum

For a Kerr remnant, perturbation theory supplies dimensionless complex eigenfrequencies $\widehat\omega_{\ell mn}(\chi_f)$ . With the time convention $e^{-i\omega t}$ ,

\omega_{\ell mn}^{\rm det} = \frac{c^3}{G M_f^{\rm det}}\, \widehat\omega_{\ell mn}(\chi_f), \qquad \omega_{\ell mn}=\omega_{\ell mn}^{R}-\frac{i}{\tau_{\ell mn}},

so $\operatorname{Im}\omega_{\ell mn}<0$ for a decaying mode. The waveform expansion, excitation amplitudes, and choice of a linear-ringdown start time are separate modeling questions developed in Quasinormal-Mode Waveform Expansion and Ringdown Start Time. A spectrum prediction says where Kerr poles lie; it does not say which poles are measurably excited.

The posterior predictive spectrum follows by pushing every $(M_f^{\rm det},\chi_f)$ sample through the Kerr map:

p(\omega_{\ell mn}\mid d_{\rm IM}) = \int \delta^{(2)}\!\left[ \omega_{\ell mn} -\frac{c^3\widehat\omega_{\ell mn}(\chi_f)} {G M_f^{\rm det}} \right] p(M_f^{\rm det},\chi_f\mid d_{\rm IM})\, dM_f^{\rm det}\,d\chi_f .

Here $\delta^{(2)}$ is the product of delta functions for the real and imaginary frequency parts. Frequency and damping-time predictions are correlated because the same mass and spin control both. Comparing separate one-dimensional credible intervals discards this geometry; a ringdown consistency test should compare the joint posterior or a calibrated posterior-predictive statistic.

Independence, leakage, and systematics

A nominal “inspiral–merger” prediction and a ringdown fit are not independent if they reuse the same time samples, calibration realization, or waveform assumptions. A clean split specifies the detector network, conditioning, peak-time convention, and disjoint or covariance-aware time windows. Tapering does not magically remove correlations: multiplication by a time window couples Fourier bins, and truncating colored noise changes its covariance.

Three checks are indispensable.

Repeat the push-forward with more than one remnant fit and waveform family; quote shifts relative to statistical width.
Verify energy balance, $0\leq\chi_f<1$ , and the $M_f^{-1}$ scaling of every frequency sample.
Run injections through the complete split analysis. The posterior-predictive rank should be calibrated when the injected signal obeys the assumed model, and failures should be exposed when precession, higher modes, eccentricity, or glitches are omitted.

The catalog-level evidence cutoff is 2026-08-18. The GWTC-5.0 tests-of-general-relativity analysis reports remnant and ringdown consistency within explicitly selected events, priors, waveform models, and data-quality choices; it is evidence about those analyses, not a theorem that every compact remnant is Kerr LVK 2026, §7.1, pp. 16–19.

Worked example: correlated scaling

Suppose a posterior sample has detector-frame remnant mass $M_f$ and spin $\chi_f$ , and consider a small perturbation. For a mode frequency

f_{\ell mn} = \frac{c^3}{2\pi G M_f}\, \widehat\omega_{\ell mn}^{R}(\chi_f),

logarithmic differentiation gives

\frac{\delta f_{\ell mn}}{f_{\ell mn}} = -\frac{\delta M_f}{M_f} +\frac{d\ln\widehat\omega_{\ell mn}^{R}}{d\chi_f}\,\delta\chi_f.

A $2\%$ upward mass shift therefore lowers the predicted frequency by $2\%$ at fixed spin. The spin term may reinforce or oppose that shift, so adding marginal mass and spin errors in quadrature is generally wrong unless their posterior covariance and the derivative of the spectral fit justify it.

Exercises

A source at redshift $z=0.2$ has $M_f^{\rm src}=60\,M_\odot$ . If a chosen dimensionless mode frequency is $\widehat\omega^R=0.50$ , express the observed cyclic frequency in terms of $G$ , $c$ , and $M_\odot$ , and compare it with the frequency obtained by mistakenly using the source-frame mass.

Solution

The detector-frame mass is $M_f^{\rm det}=72\,M_\odot$ , hence

f_{\rm det} = \frac{0.50}{2\pi}\frac{c^3}{72\,G M_\odot}.

Using $60\,M_\odot$ would give $f_{\rm wrong}=0.50c^3/(120\pi G M_\odot)$ . Therefore $f_{\rm wrong}/f_{\rm det}=72/60=1.2$ : neglecting redshift overpredicts the observed frequency by $20\%$ .

Explain why using the maximum-posterior binary parameters in a remnant fit does not generally produce the maximum of the remnant posterior.