Calibration, Extrapolation, and Uncertainty

Calibration uses numerical-relativity, black-hole-perturbation, and self-force information to constrain EOB terms that are unknown analytically or intentionally left flexible. Done carefully, it converts a broad physical framework into an accurate waveform model. Done opaquely, it can hide parameter degeneracies and replace unsupported extrapolation by a smooth-looking fit. The essential safeguards are a versioned model, a declared calibration set, held-out tests, a sector-by-sector error budget, and uncertainty statements that distinguish interpolation from extrapolation.

Evidence cutoff. The model and literature assessment on this page is checked through 2026-08-12. Later code releases, numerical-relativity catalogs, recalibrations, corrections, and independent reproductions are outside this evidence window.

Required background. Presymplectic Potentials and Currents supplies Hamiltonian structure and clarifies which fitted changes preserve canonical dynamics.

Helpful background. The Einstein Equation supplies the target dynamics; Weak Metric Perturbations and Gauge Freedom supplies analytic low-frequency constraints; Black-Hole Perturbation Theory supplies the large-mass-ratio anchor; and Relativistic Two-Body Dynamics supplies the regime variables needed to define a calibration domain.

Calibration as an inverse problem

Let $\Lambda$ denote physical parameters such as mass ratio and spins, and let $\boldsymbol\theta$ denote EOB flexibility parameters. A completed waveform is

$$h_{\rm EOB}=h(\Lambda;\boldsymbol\theta).$$

The flexibility parameters may enter the conservative potentials, gyro functions, factorized modes and radiation reaction, nonquasicircular corrections, or merger–ringdown attachment. They should represent missing terms at specified formal orders, not arbitrary deformations of every sector.

For detector-weighted comparisons define

$$(a\mid b) =4\,{\rm Re}\int_{f_{\rm low}}^{f_{\rm high}} \frac{\widetilde a(f)\widetilde b^*(f)} {S_n(f)}\,df$$

and the mismatch

$$\mathcal M =1-\max_{\Delta t,\Delta\phi} \frac{(h_{\rm NR}\mid h_{\rm EOB})} {\sqrt{(h_{\rm NR}\mid h_{\rm NR}) (h_{\rm EOB}\mid h_{\rm EOB})}}.$$

Here $S_n(f)$ is the one-sided detector noise power spectral density, $f$ is the observed gravitational-wave frequency, and tildes denote Fourier transforms of the real detector strain. The frequency bounds, detector spectrum, total mass, inclination, and retained modes belong to the comparison specification. They can change a mismatch without changing the underlying physical waveform error.

The maximization removes chosen time and phase offsets; it does not remove physical errors in phasing, mode content, or dynamics. A calibration likelihood may combine mismatch, merger-time difference, amplitude, and other diagnostics. In one hierarchical procedure, each numerical waveform first yields a posterior $p(\boldsymbol\theta_i\mid h_i^{\rm NR})$; selected summaries or the full posterior are then fitted across $\Lambda$. This is preferable to a single global least-squares step when it exposes multimodality and correlations, but the subsequent regression can still erase them if it retains only one point per configuration.

Identifiability is the first concern. If two parameters modify the waveform in nearly the same direction,

$$\frac{\partial h}{\partial\theta_1} \simeq C\frac{\partial h}{\partial\theta_2},$$

the data constrain only a combination of them. A narrow waveform mismatch does not imply that either coefficient is individually known. Alternative bases, priors, or sector assignments can produce similar calibrated waveforms and different extrapolations.

Results for a specified SEOBNRv5 version

A concrete snapshot prevents model-family names from becoming vague claims. Use $q=m_1/m_2\geq1$ and signed aligned spins $\chi_A=\mathbf S_A\cdot\hat{\mathbf L}/(Gm_A^2)$. The published SEOBNRv5HM model is for quasicircular, aligned-spin binary black holes and is specified in Pompili et al. Its calibration used 442 numerical-relativity waveforms—441 from SXS and one Einstein Toolkit configuration—plus 13 Teukolsky waveforms at $q=10^3$; its nonspinning radiation sector also incorporates second-order self-force information Pompili et al. 2023, § IV, Fig. 2 and pp. 10–15.

The numerical calibration coverage is highly nonuniform:

• for $1\leq q\leq3$, both spins are represented, reaching $\lvert\chi_{1,2}\rvert=0.998$ at equal mass and about $0.85$ at $q=3$;
• for $3\lt q\leq10$, the primary spin coverage is roughly $-0.9\leq\chi_1\leq0.85$;
• for $10\lt q\leq20$, the set is nonspinning or has only the heavier black hole spinning, with magnitude no greater than $0.5$;
• the 13 perturbative waveforms at $q=10^3$ cover background spins from $-0.99$ to $0.99$, but they are not dense finite-$q$ numerical simulations.

Those ranges are a calibration map, not a rectangular guarantee. Sparse corners, spin-difference directions, waveform length, eccentricity, numerical resolution, and mode availability also matter. The paper reports extensive internal sanity scans over $1\leq q\leq100$ and uses comparisons with NR and surrogate information, but comparisons on the 442 calibration waveforms remain in-sample tests. Its § V mismatch distributions must therefore be read together with the data-use statement in § IV, not as 442 independent predictions.

The precessing extension SEOBNRv5PHM supplies a useful holdout-style distinction: it reduces to the calibrated aligned-spin model but was not calibrated to precessing SXS waveforms. It was tested against 1,543 precessing numerical-relativity waveforms; over total masses $20$–$300M_\odot$, 99.8% of cases had maximum unfaithfulness below 3% and 84.4% below 1% Ramos-Buades et al. 2023, § V and Fig. 11. This supports the precession construction on that set, but it does not establish accuracy for arbitrary eccentricity, mass ratio, spin magnitude, starting frequency, or detector weighting.

The public software implementation is documented as pySEOBNR Mihaylov et al. 2025, §§ 2–5. Reproducibility requires a release or commit, the requested model name, mode set, sampling settings, and dependency versions; “SEOBNRv5” alone does not uniquely identify an executable result. As a dated comparator, the SEOBNRv5HM study used TEOBResumSv4.1.4-GIOTTO at the public commit printed in its § V footnote 6. That comparison describes that exact snapshot, not every later TEOBResumS release.

Two positions on calibrated reliability

Position A: robust interpolation within a declared domain. Analytic PN, PM, and self-force limits constrain the functional form, while a moderate number of numerical waveforms fixes a small number of remaining coefficients. If calibration is hierarchical, the fit is smooth, and withheld configurations agree in waveform and dynamical invariants, the model can be reliable inside the sampled physical domain.

Position B: degeneracy and extrapolation remain leading risks. Numerical waveforms have resolution, extrapolation, and residual-eccentricity errors; calibration posteriors can be correlated or multimodal; and a low mismatch can compensate conservative, dissipative, and merger errors against each other. Smooth regression outside sparse data may look stable while its error is uncontrolled, especially at high mass ratio, nearly extremal spin, large spin difference, precession, or eccentricity.

Both positions are credible. The first is supported by accurate in-domain and withheld comparisons of named models. The second is supported by known sparse corners, model-to-model differences, and the fact that many published validation sets overlap calibration data. The correct conclusion is conditional: reliable for which version, observable, parameter region, frequency band, and comparison set?

Building an uncertainty budget

Write the waveform error schematically as

$$\delta h =\delta h_{\rm NR} +\delta h_{\rm fit} +\delta h_{\rm form} +\delta h_{\rm missing}.$$

Here $\delta h_{\rm NR}$ includes numerical truncation, waveform extraction, finite length, and hybridization; $\delta h_{\rm fit}$ is uncertainty in fitted calibration functions; $\delta h_{\rm form}$ covers choices such as resummation and parameter basis; and $\delta h_{\rm missing}$ contains absent physics such as eccentric–precessing couplings or incomplete modes. These terms are correlated and should not automatically be added in quadrature.

If calibration parameters have covariance $C_\theta$, linear propagation gives a waveform covariance

$$C_h \simeq J_\theta C_\theta J_\theta^\dagger, \qquad (J_\theta)_a=\frac{\partial h}{\partial\theta_a}.$$

This describes uncertainty only along waveform directions spanned by the chosen calibration parameters. It cannot represent a missing effect orthogonal to that span. Nor does posterior width necessarily increase outside the training region: an ordinary regression can become confidently wrong.

Pompili, Buonanno, and Pürrer model the SEOBNRv5HM calibration-parameter posteriors with mixtures of Gaussians whose means, covariances, and weights depend on $\Lambda$. Marginalizing over those parameters reduced systematic biases in simulated loud-signal studies, usually with a modest increase in statistical uncertainty Pompili, Buonanno, and Pürrer 2025, §§ II–IV. Their tests included $q=3$, equal spins $\pm0.45$, and a wider grid with $q=1.5,3,6$ and equal spins $\pm0.45,\pm0.75$. The method did not eliminate every bias and, as their § V states, targets calibration uncertainty inside the numerical-relativity domain; it does not automatically cover extrapolation or missing physics.

Why mismatch is not the whole error

For a small waveform difference $\delta h=h_{\rm true}-h_{\rm model}$, the leading parameter bias is

$$\Delta\lambda_{\rm sys}^i \simeq (\Gamma^{-1})^{ij} \left(\partial_jh\mid\delta h\right),$$

where $\Gamma_{ij}=(\partial_i h\mid\partial_j h)$ is the Fisher matrix. Only the projection of model error onto parameter-derivative directions drives this leading bias. A small mismatch averaged over a mass interval can coexist with a significant bias for a loud signal, while a larger nearly orthogonal error can be less damaging for one parameter. The derivation and its linear-signal assumptions are given in Cutler and Vallisneri 2007, §§ II–III.

Consequently, validation should include unfaithfulness over relevant detector spectra, mode-by-mode phase and amplitude, binding energy, angular-momentum flux, merger time, and inference injections at representative signal strengths. No single scalar establishes universal adequacy.

Holdouts and extrapolation tests

A strong test plan separates three questions.

Interpolation. Withhold configurations inside the well-sampled region, including clusters rather than random individual waveforms. Random splits can leak nearly duplicate physical configurations into training and validation sets.

Boundary behavior. Withhold the largest mass ratios, largest positive and negative spins, and large spin-difference configurations. Refit with alternative polynomial or rational bases and reasonable priors. A prediction is robust only if physically equivalent fits agree within a declared tolerance.

Extrapolation. Test beyond the numerical convex hull using independent perturbation theory, self-force, long numerical simulations, or a different numerical code. A smooth waveform and monotonic frequency are necessary sanity checks but not accuracy evidence. The distance outside the data must include spin directions, eccentricity, waveform length, and mode content—not only $q$.

Cross-code recalibration is especially informative because one numerical catalog can carry correlated extraction or gauge systematics. Likewise, withheld scattering or periastron data can expose compensating errors that a waveform-only fit misses.

Scope and open questions

As of the cutoff, there is no generally accepted frequentist or Bayesian uncertainty band with demonstrated coverage for all EOB waveform sectors across generic binary parameter space. Calibration-posterior marginalization is a concrete advance, but its coverage is model- and direction-dependent. The 2025 LISA waveform review emphasizes that most EOB calibrations and tests remain concentrated around $q\lesssim20$, while LISA applications demand wider mass ratios, coupled precession and eccentricity, greater accuracy, and high efficiency LISA Consortium Waveform Working Group 2025, §§ 4.5.2 and 4.5.6.

These conclusions could change if any of the following occurs:

• a materially expanded numerical-relativity dataset fills present high-$q$, high-spin, precessing, eccentric, or long-inspiral gaps;
• an independent numerical code enables a cross-code recalibration and changes fitted coefficients or held-out residuals;
• a validated uncertainty method demonstrates calibrated coverage on withheld and extrapolation sets;
• a new analytic constraint removes a calibration direction or reveals that two fitted parameters were compensating errors;
• injection studies at higher signal strength change the balance between waveform systematic and statistical uncertainty.

Until then, report interpolation claims separately from extrapolation, and never turn a calibration-domain mismatch into a universal waveform-accuracy statement.

Common pitfalls

Calling the training set a validation set. In-sample mismatch measures fit quality. It does not estimate prediction error without additional holdouts.

Propagating only coefficient covariance. That covers uncertainty in the chosen parameter span, not missing modes, physics, or functional forms.

Defining the domain by mass ratio alone. Spin combinations, precession, eccentricity, waveform duration, and frequency weighting can move a case outside support even at familiar $q$.

Exercise

Consider a waveform linearized in two calibration parameters,

$$h=h_0+\theta_1 e_1+\theta_2 e_2,$$

with covariance

$$C_\theta= \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}.$$

Derive the variance of the waveform projection onto an arbitrary normalized direction $v$. Explain what happens if $e_2=C e_1$.

Solution

Define $a_i=(v\mid e_i)$. The projected perturbation is

$$(v\mid\delta h)=a_1\delta\theta_1+a_2\delta\theta_2.$$

Therefore

$${\rm Var}(v\mid\delta h) =a_1^2\sigma_1^2+a_2^2\sigma_2^2 +2\rho a_1a_2\sigma_1\sigma_2.$$

If $e_2=C e_1$, then $a_2=Ca_1$ for every $v$, and the waveform depends only on the combination $\theta_1+C\theta_2$. The orthogonal parameter combination is unidentifiable from waveform data even if a prior makes each marginal posterior look finite. Extrapolating $\theta_1$ and $\theta_2$ separately can then be unstable while their in-domain waveform combination remains well constrained.

References

• Cutler, Curt, and Michele Vallisneri. “LISA Detections of Massive Black Hole Inspirals: Parameter Extraction Errors Due to Inaccurate Template Waveforms.” Physical Review D 76, 104018 (2007). DOI. Open PDF.
• LISA Consortium Waveform Working Group. “Waveform Modelling for the Laser Interferometer Space Antenna.” Living Reviews in Relativity 28, 9 (2025). DOI.
• Mihaylov, Deyan P., et al. “pySEOBNR: A Software Package for the Next Generation of Effective-One-Body Multipolar Waveform Models.” SoftwareX 30, 102080 (2025). DOI. Open PDF.
• Pompili, Lorenzo, Alessandra Buonanno, and Michael Pürrer. “Accounting for Numerical-Relativity Calibration Uncertainty in Gravitational-Wave Modeling and Inference.” Physical Review D 112, 044062 (2025). DOI. Open PDF.
• Pompili, Lorenzo, et al. “Laying the Foundation of the Effective-One-Body Waveform Models SEOBNRv5: Improved Accuracy and Efficiency for Spinning Nonprecessing Binary Black Holes.” Physical Review D 108, 124035 (2023). DOI. Open PDF.
• Ramos-Buades, Antoni, et al. “Next Generation of Accurate and Efficient Multipolar Precessing-Spin Effective-One-Body Waveforms for Binary Black Holes.” Physical Review D 108, 124037 (2023). DOI. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Effective-One-Body Calibration and Extrapolation