Accuracy, Faithfulness, and Waveform Uncertainty

Waveform accuracy is not a single percentage. A discrepancy is measured in a declared geometry; alignment projects out chosen transformations; faithfulness and effectualness answer different questions; analytic, calibration, numerical, and surrogate errors have different evidence; and scientific bias depends on the direction of error relative to model parameters. This chapter builds that chain while leaving current detector noise curves and event-specific parameter biases to Gravitational-Wave Astronomy.

Helpful background. Einstein Equations supplies the continuum waveform problem. Waves, Green Functions, and Geometric Optics supplies Fourier geometry. Gravitational Radiation and Waveforms supplies strain, modes, models, and frames. Probability and Statistical Inference supplies likelihood, priors, covariance, and coverage. Computational Relativity supplies reference waveforms and convergence evidence.

Enter through the scientific task

Before asking whether two waveforms agree, specify what their comparison is meant to preserve. A source-generation test may compare asymptotic modes in a source frame. A detector-facing recovery calculation first redshifts, rotates, projects, and weights them. A parameter-preserving test fixes intrinsic parameters; a recovery-oriented test may optimize them. A numerical validation may avoid a detector PSD entirely and examine phase, amplitude, constraints, and extraction convergence separately.

The required record is therefore: waveform identifiers and versions; source/detector frame map; modes and response; frequency band and one-sided PSD or time norm; alignment and maximized variables; parameter domain; and the error component being tested. Without this record, a mismatch is not reproducible and an “accuracy requirement” has no defined task.

Shared weighted geometry

The chapter uses

$$(a\mid b)=4\operatorname{Re} \int_{f_{\min}}^{f_{\max}} \frac{\widetilde a(f)\widetilde b(f)^*}{S_n(f)}\,df$$

for a declared one-sided positive PSD. This follows the compact-binary likelihood geometry of Cutler and Flanagan 1994, §§II–III, PDF. Norms are dimensionless for strain. Positivity and Cauchy–Schwarz hold where $S_n\gt0$; out-of-band components are quotiented from this task, not erased from the physical waveform.

For normalized waveforms, maximize only the declared time and phase shifts:

$$\mathcal M=\max_{\Delta t,\Delta\phi} (\widehat a\mid\widehat b_{\Delta t,\Delta\phi}), \qquad \mathfrak m=1-\mathcal M.$$

The small-error result

$$\mathfrak m\simeq \frac12\lVert\delta\widehat h_\perp\rVert^2$$

contains a projection: normalization and optimized tangent directions have already been removed. This is why preprocessing and alignment choices belong beside every result.

Six pages, one accuracy chain

1. Noise-Weighted Waveform Inner Products defines the positive-frequency geometry, units, whitening, multichannel extension, and unweighted quotient directions.
2. Match, Mismatch, and Alignment derives time-phase maximization and the quadratic projected-error limit, following the template-space construction of Owen 1996, §§II–III, PDF.
3. Faithfulness and Effectualness compares fixed-intrinsic-parameter agreement with optimization across a model domain and proves that effectualness cannot be lower when the faithful point is included.
4. Analytic Truncation and Calibration Error separates missing orders, resummation spread, fitted coefficients, reference-data error, interpolation, and implementation.
5. Numerical Waveform Error Budgets reproduces a fourth-order synthetic control and then stress-tests resolution, extraction, alignment, frame, junk, and correlation against catalog-scale evidence from Boyle et al. 2019, §4, pp. 24–28, PDF.
6. Model Error and Scientific Bias projects equal-norm errors parallel and orthogonal to parameter tangents, compares norm, Fisher, coverage, calibration, and cross-validation evidence through 2026-08-18, and states when review is triggered.

Faithfulness, effectualness, and displacement

For reference $h_*(\lambda)$ and model $h_m(\lambda')$,

$$\mathcal F(\lambda) =\max_{\rm extrinsic} \mathcal O[h_*(\lambda),h_m(\lambda)],$$

while

$$\mathcal E(\lambda) =\max_{\lambda'\in\mathcal D,\,\rm extrinsic} \mathcal O[h_*(\lambda),h_m(\lambda')].$$

If $\lambda\in\mathcal D$, then $\mathcal E\ge\mathcal F$. High effectualness can coexist with low faithfulness because an intrinsic displacement absorbs model error. Report that displacement and any boundary hit; otherwise a recovery-oriented success can be mistaken for parameter-preserving physics.

Error components and evidence

Formal truncation states scaling in a controlled expansion, not a probability distribution. Approximant spread probes selected reorganizations, not all unknown terms. Calibration residual measures fitted points, not held-out prediction. Surrogate cross-validation tests interpolation against its reference, not the reference against nature. Numerical resolution differences mix evolution and extraction unless separately varied.

A structured small-error budget is

$$\delta h\simeq \delta h_{\rm trunc} +\delta h_{\rm resum} +J_\alpha\delta\alpha +\delta h_{\rm ref} +\delta h_{\rm disc} +\delta h_{\rm extract} +\delta h_{\rm interp}.$$

These terms can be correlated. Signed waveform differences and their cross-inner-products are preferable to automatic quadrature. When correlations are unknown, the triangle inequality is conservative; when a covariance is justified, propagate it explicitly.

From residual to parameter shift

For truth-minus-model error $\delta h$ and Fisher matrix $\Gamma_{ij}=(\partial_i h\mid\partial_j h)$,

$$\Delta\theta^i_{\mathrm{sys}} =(\Gamma^{-1})^{ij} (\partial_jh\mid\delta h).$$

Only the tangent projection drives this first-order displacement. The derivation and qualifications are given in Cutler and Vallisneri 2007, §§II–III, PDF. Orthogonal residual can instead reveal lack of fit. Priors, parameter boundaries, nonlinear degeneracies, noise, and calibration require injection, coverage, and posterior-predictive studies beyond the Fisher approximation.

Norm conditions such as $\lVert\delta h\rVert\lt1$ give useful sufficient control under declared assumptions Lindblom, Owen, and Brown 2008, §II.A, eqs. (4)–(9), p. 3, PDF. In the locally linear Gaussian problem, the standardized bias of any linear parameter combination cannot exceed this absolute norm. The bound controls the worst case; the actual shift still depends on direction. A small fractional error or an optimized mismatch does not supply the same bound by itself. At fixed mismatch, the absolute residual grows with signal norm.

Chapter diagnostic

Let $e_\parallel$ be a unit parameter-tangent direction with $\partial_\theta h=5e_\parallel$, and let $e_\perp$ be unit and orthogonal. Compare two equal-norm errors

$$\delta h_1=0.2e_\parallel, \qquad \delta h_2=0.2e_\perp.$$

Find the first-order parameter shift and orthogonal residual for each. What does the example say about a norm-only bias rule?

Solution

Here $\Gamma=(\partial_\theta h\mid\partial_\theta h)=25$. For the tangent error,

$$\Delta\theta_1 =\Gamma^{-1}(\partial_\theta h\mid\delta h_1) =\frac{1}{25}(5)(0.2)=0.04,$$

and the best-fit orthogonal residual is zero at first order. For the orthogonal error, $(\partial_\theta h\mid\delta h_2)=0$, so $\Delta\theta_2=0$ and the full residual norm remains $0.2$.

Both errors have the same norm, yet one is absorbed as parameter displacement and the other remains as lack of fit. The Fisher statistical width is $1/\sqrt\Gamma=0.2$, so the first shift is only $0.2$ statistical standard deviations, consistent with the error-norm bound; the second has no leading bias. The norm bounds the shift but does not specify its direction or value. PSD, prior, parameterization, signal strength, and nonlinear manifold curvature add further conditional dependence.

Boundaries and review

This chapter develops definitions, model/numerical error decomposition, and conditional accuracy criteria. Production convergence is treated in Computational Relativity; current detector curves, likelihoods, calibration products, and event-specific parameter biases in Gravitational-Wave Astronomy. The frontier bias page is reviewed after validated coverage studies, revised mission requirements, public controlled model comparisons, or counterexamples to its criteria. Mutable observational verdicts do not belong in evergreen mathematical prose here.

References

• Boyle, Michael, et al. “The SXS Collaboration Catalog of Binary Black Hole Simulations.” Classical and Quantum Gravity 36 (2019): 195006. DOI. Open PDF.
• Cutler, Curt, and Éanna É. Flanagan. “Gravitational Waves from Merging Compact Binaries: How Accurately Can One Extract the Binary’s Parameters from the Inspiral Waveform?” Physical Review D 49 (1994): 2658–2697. DOI. Open PDF.
• Cutler, Curt, and Michele Vallisneri. “LISA Detections of Massive Black Hole Inspirals: Parameter Extraction Errors Due to Inaccurate Template Waveforms.” Physical Review D 76 (2007): 104018. DOI. Open PDF.
• Lindblom, Lee, Benjamin J. Owen, and Duncan A. Brown. “Model Waveform Accuracy Standards for Gravitational Wave Data Analysis.” Physical Review D 78 (2008): 124020. DOI. Open PDF.
• Owen, Benjamin J. “Search Templates for Gravitational Waves from Inspiraling Binaries: Choice of Template Spacing.” Physical Review D 53 (1996): 6749–6761. DOI. Open PDF.