Analytic Truncation and Calibration Error

Analytic waveform uncertainty has several origins that should not be collapsed into one “model error.” A formal expansion omits higher orders; a resummation reorganizes known information; calibration fits adjustable functions to finite reference data; the reference data have their own errors; and software interpolates or evaluates the result. Each component has a different diagnostic and domain. This page builds that partition without assigning current detector requirements or event-specific bias.

Required background. Probability and Statistical Inference supplies fitted-parameter uncertainty, validation, and the difference between a discrepancy and a probability distribution.

Helpful background. Einstein Equations supplies controlled expansions of relativistic dynamics. Waves, Green Functions, and Geometric Optics supplies asymptotic series and Fourier errors. Gravitational Radiation and Waveforms supplies approximants and modes. Computational Relativity supplies calibration and reference data.

Formal truncation error

Suppose a waveform quantity has an asymptotic expansion

$$h(x)=\sum_{k=0}^{N}c_kx^k+R_{N+1}(x), \qquad R_{N+1}=O(x^{N+1})$$

in a declared regime $x\ll1$. The big-$O$ statement controls scaling only under its hypotheses; it does not give the unknown coefficient, phase dependence, or probability distribution. The last retained term and the difference between adjacent orders are useful diagnostics when coefficients are regular and $x$ is small, but neither is a rigorous universal error bar.

A consistency test varies order and checks the expected scaling over more than one $x$. If convergence is nonmonotonic, the smallest adjacent difference can be accidentally optimistic. Near merger, a post-Newtonian parameter need not remain sufficiently small for the formal ordering to predict numerical accuracy.

Resummation ambiguity

Taylor, Padé, effective-one-body, and other reorganizations can share the same known series coefficients yet differ by formally higher-order terms. Their spread probes sensitivity to the reorganization; it does not sample all possible missing physics. Historical comparisons and the distinction among approximants are explicit in Damour, Iyer, and Sathyaprakash 1998, §§II–IV, PDF.

Two resummations can agree because they share the same omitted term or calibration choice. Conversely, a large spread can overstate error if one formulation violates a known physical constraint. Use limits, monotonicity, poles, test-mass behavior, and independent perturbative information to judge each variant before interpreting the envelope.

Calibration and reference-data error

Let a model depend on calibration functions $\alpha$ fitted to reference waveforms $d_i$:

$$\widehat\alpha =\arg\min_\alpha \sum_i\lVert h_m(\lambda_i;\alpha)-d_i\rVert_i^2.$$

Training residual measures agreement at the fitted points in the chosen norm. It is not prediction error. Validation requires held-out waveforms, alternative resolutions/extraction methods, and coverage of parameter-space boundaries. The calibrated SEOBNRv5 construction, fit objectives, numerical inputs, and stated domain provide a concrete example in Pompili et al. 2023, §§II–V and appendices, PDF.

Reference data are not exact: finite resolution, extraction, junk, frame correction, and hybridization contribute $\delta h_{\rm ref}$. A flexible calibration can absorb those artifacts. Fit uncertainty and reference uncertainty are correlated when the same simulations determine both model coefficients and validation summaries.

A structured error budget

For small errors, a useful bookkeeping equation is

$$\delta h_{\rm total} \simeq \delta h_{\rm trunc} +\delta h_{\rm resum} +J_\alpha\delta\alpha +\delta h_{\rm ref} +\delta h_{\rm interp} +\delta h_{\rm impl}.$$

$J_\alpha$ maps calibration-coefficient errors into waveform space. The sum is conceptual: components can be correlated, cancel in one norm, or be double counted. Scalar maxima are not statistical standard deviations; adding them in quadrature supplies no general error bound. For deterministic error waveforms, retain their cross-inner-products. A probabilistic quadrature budget instead needs a justified model with vanishing cross-covariances and controlled means. Report how each signed waveform difference or covariance model was constructed.

Calibration domain and extrapolation

A model domain includes mass ratio, spin magnitudes and directions, eccentricity, mode set, frequency range, total-mass scaling, and any training assumptions. Interpolation inside a sparse domain can still fail near sharp features; extrapolation outside it has no guarantee from a small training residual. Distance from the nearest calibration point is not sufficient because waveform curvature is anisotropic in parameter space.

Robust checks include leave-one-out or held-out validation, boundary tests, known analytic limits, alternative reference catalogs, and variation of fit complexity. A high-order fit that drives the training residual to zero can oscillate between points or violate a limit. Document model selection and regularization so that the calibration can be reproduced.

Counterexamples to shortcut uncertainty

Zero spread, nonzero error. Let two approximants both equal $h_0$, while the unknown truth is $h_0+\epsilon g$. Their spread is zero but their common error is $\epsilon g$.

Zero training residual, poor prediction. A sufficiently flexible interpolant passes through every noisy reference point. It can still fit numerical noise and deviate badly at a held-out parameter.

Small norm, wrong feature. An error localized in a low-weight band can have a small chosen norm yet corrupt a scientific feature targeted by another band or observable. Current noise curves and event-specific consequences remain outside this page.

Exercises

1. Test a controlled series. Approximate $e^x$ at $x=0.2$ with $S_1=1+x$ and $S_2=1+x+x^2/2$. Compare actual errors with the first omitted terms.

Solution

$e^{0.2}=1.22140276\ldots$, while $S_1=1.2$ and $S_2=1.22$. Therefore

$$\lvert e^{0.2}-S_1\rvert=0.0214028, \qquad \lvert e^{0.2}-S_2\rvert=0.00140276.$$

The first omitted terms are $x^2/2=0.02$ and $x^3/6=0.0013333$. They estimate the scale well here because the exponential series has regular positive coefficients at small $x$. This success is a controlled example, not proof that the same shortcut works for a resummed strong-field waveform.

2. Avoid double counting. Suppose calibration absorbs a reference phase offset $\delta\phi_{\rm ref}$ exactly. Should the calibrated-model difference and the reference offset be added as independent errors?

Solution

No. The fitted coefficient now carries the same reference artifact, so the two terms are correlated. Adding their magnitudes in quadrature would count one underlying error twice. Refit with an alternative reference treatment, or propagate the reference uncertainty through the calibration map $J_\alpha$, to estimate the combined effect coherently.

References

• Damour, Thibault, Bala R. Iyer, and B. S. Sathyaprakash. “Improved Filters for Gravitational Waves from Inspiralling Compact Binaries.” Physical Review D 57 (1998): 885–907. 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 (2023): 124035. DOI. Open PDF.