Cross-Regime Uncertainty Budgets

A defensible cross-regime uncertainty is a statement about one shared invariant over a declared domain, not the spread of whatever outputs several codes happen to provide. Every contribution must identify its source, correlation with other contributions, and validation evidence. Disagreement can reveal missing physics, but it becomes a probability interval only through additional statistical assumptions and coverage tests.

Helpful background. The Einstein Equation supplies the governing theory; Weak Metric Perturbations and Gauge Freedom supplies approximation and gauge boundaries; Presymplectic Potentials and Currents supplies canonical invariants; Computational Relativity supplies numerical uncertainty; Gravitational Radiation and Waveforms supplies flux and extraction conventions; and Relativistic Two-Body Dynamics supplies the PN, PM, self-force, EOB, amplitude, and numerical regimes being combined.

Sources of uncertainty in two-body predictions

Evidence cutoff: 2026-08-12. Claims and literature on this page were checked through that date. Later calculations, model and data releases, corrections, and independent reproductions lie outside the evidence window.

The evidence does not establish one universally accepted combination rule. It supports several component estimators, while common blind coverage across all regimes remains sparse:

Evidence class	Quantity it can estimate	What it does not establish
PN or PM order variation	Scale and shape of the first omitted terms inside a controlled common limit	Convergence of an asymptotic series or errors after leaving the overlap
Self-force resolution and regularization variation	Numerical and subtraction error in a fixed perturbative coefficient	Unknown $O(\nu^2)$ finite-ratio dynamics
EOB calibration posterior	Uncertainty of fitted coefficients conditional on a basis, likelihood, and data set	Missing analytic structure or out-of-domain discrepancy
Numerical-relativity resolution and extraction variation	Discretization and finite-radius sensitivity for a specified simulation	A continuum error if the sequence is nonconvergent
Cross-method overlap residual	Incompatibility of two complete predictions in shared invariant variables	A unique allocation among shared bias, truncation, and numerical error
Withheld-benchmark coverage	Whether a combined interval covers unseen observables at sampled points	Coverage in unsampled strong-field, spin, eccentric, or mass-ratio regions

Order-by-order differences can be converted into degree-of-belief intervals when coefficient-size and expansion-parameter priors are stated. Furnstahl and collaborators derive such intervals and emphasize prior sensitivity and model checking rather than treating the last term as an automatic error bar Furnstahl et al. 2015, §§II–IV, pp. 3–14, PDF. Correlations across energy or angle require a non-diagonal theory covariance; a Gaussian-process construction and diagnostics for that case are given by Melendez et al. 2019, §§II–IV, pp. 3–16, PDF. These are general effective-field-theory methods. Applying them to gravitational dynamics still requires a justified expansion variable and a common-limit check.

Numerical and calibration evidence has different scope. SXS resolution comparisons reveal a broad mismatch distribution and explicitly identify nonconvergent cases and mixed error sources Boyle et al. 2019, §4, pp. 24–28, Eqs. (22)–(24), PDF. Calibration-posterior sampling can expose parameter correlations and improve uncertainty propagation, but held-out surrogate comparisons show that calibration freedom does not span every waveform discrepancy Pompili, Buonanno, and Pürrer 2026, §§II–IV and Appendix B, PDF. These results support conditional budgets, not a universal coverage claim.

Two credible positions

Position A: modular covariance budget. Truncation, numerical, regularization, calibration, and matching errors arise from sufficiently distinct mechanisms that they can be represented by random variables with an explicit covariance matrix. Independent contributions add in quadrature; known shared inputs produce off-diagonal terms. This position is attractive because each component can shrink under a targeted calculation and can be propagated through derivatives. It is credible when the component definitions are nonoverlapping and withheld residuals have acceptable coverage.

Position B: discrepancy or envelope budget. The methods share equations, analytic coefficients, numerical catalogs, resummations, and even calibration targets. Their missing physics is therefore strongly correlated and may not be visible in pairwise differences. One should use an envelope of controlled variants, or learn a discrepancy process only from genuine overlap and blind data, rather than assign independent Gaussian errors. This is credible where perturbative convergence is nonuniform or the model family is known to be incomplete.

Neither position is automatically conservative. Quadrature can undercover when errors share a sign. An envelope can also undercover if every model shares the same bias, and it lacks probabilistic meaning unless calibrated. Bayesian model averaging gives weights conditional on its candidate set and priors; it cannot assign probability to a missing model family. A responsible budget states which position is used and shows the sensitivity to the alternative.

Constructing a budget on a shared invariant

Choose an invariant $Q(z)$, where $z$ denotes fixed physical inputs. Suitable examples are $\chi(E,J)$, binding energy $E_b(\Omega)$, periastron advance, redshift, or flux with its extraction convention. For a model prediction,

$$Q_{\rm pred}(z) =Q_{\rm true}(z) +\delta_{\rm tr} +\delta_{\rm reg} +\delta_{\rm num} +\delta_{\rm cal} +\delta_{\rm map},$$

where the terms denote analytic truncation, regularization, numerical solution or extraction, calibration, and cross-regime mapping discrepancy. The decomposition is useful only if each term has an operational variation or benchmark.

If $\boldsymbol\delta$ has covariance $C_\delta$ and the reported observable $F$ depends on several predicted quantities, linear propagation gives

$$\sigma_F^2 =J_F C_\delta J_F^{\mathsf T}, \qquad (J_F)_a=\frac{\partial F}{\partial\delta_a}.$$

For a single $Q$ this reduces to

$$\sigma_Q^2 =\sum_a\sigma_a^2 +2\sum_{a\lt b}\rho_{ab}\sigma_a\sigma_b.$$

The correlation coefficients cannot be selected merely to widen or narrow the interval. Shared PN coefficients, common numerical waveforms, common regularization data, and calibration reuse identify structural correlations. Repeating an analysis after removing shared inputs can estimate some of them.

Avoiding double counting

A cross-method residual is already composite. If

$$d_{AB}=Q_A-Q_B,$$

then

$$d_{AB} =\delta_{{\rm tr},A}-\delta_{{\rm tr},B} +\delta_{{\rm num},A}-\delta_{{\rm num},B} +\delta_{{\rm cal},A}-\delta_{{\rm cal},B} +\cdots.$$

Adding the full magnitude of $d_{AB}$ in quadrature with the numerical and calibration uncertainties counts those contributions twice. First propagate known component covariances into the expected difference. Only an excess residual, with its own model and uncertainty, may be assigned to a discrepancy term.

There is an opposite danger: a common bias cancels from $d_{AB}$. If both models inherit the same incorrect coefficient, their close agreement supplies no bound on it. Independent analytic derivations, different numerical methods, and exact limiting cases are therefore more valuable than many implementations sharing one input.

Calibration leakage is a special form of double use. Data that fitted $\boldsymbol\theta$ constrain the calibration posterior; their residuals cannot also calibrate the model-discrepancy process without accounting for that conditioning. The split discipline reserves blind points for combined coverage.

Coverage and withheld tests

At blind points $z_i$ with reference values $Q_i^{\rm ref}$ and combined standard deviations $\sigma_i$, form

$$r_i=\frac{Q_{\rm pred}(z_i)-Q_i^{\rm ref}}{\sigma_i}.$$

For a claimed central 68% interval, approximately 68% coverage is a long-run target under the stated sampling scheme, not a guarantee for each point. Inspect coverage by regime as well as globally: a model can achieve nominal average coverage by being too wide in the weak field and too narrow near merger. Probability integral transforms, standardized-residual autocorrelation, and leave-one-family-out tests expose miscalibration.

Waveform-domain studies illustrate the difference between a point estimate and uncertainty model. Read constructs frequency-dependent amplitude and phase errors by comparing analytic models with multiple-resolution numerical calculations Read 2023, §§2–4, especially Eqs. (5)–(13), PDF. Owen and collaborators instead introduce uncertain higher-PN phase coefficients and test injection–recovery behavior, showing that marginalizing missing terms can reduce bias while widening inference Owen et al. 2023, §§II–IV, PDF. These are credible but different discrepancy models; neither alone establishes cross-regime coverage for all binary dynamics.

Noncommuting limits and domain boundaries

An uncertainty model inherits the limits of its approximation. At low velocity, fixed-PM coefficients contain inverse powers of velocity, so PM order variation is not uniform as $v\to0$ at fixed impact parameter. In self-force theory, $\nu\to0$ at fixed orbital invariant does not justify approaching a light-ring singularity at fixed finite $\nu$. Numerical limits in grid spacing, extraction radius, and simulation length can contaminate one another; varying one while freezing unconverged others does not isolate its error.

These issues are not cured statistically. A Gaussian process trained inside an overlap cannot be extrapolated through a separatrix merely by returning a larger variance unless that behavior is built into and validated for the kernel. Likewise, taking the number of calibration points to infinity within one limited domain does not commute with enlarging the physical domain. Every interval therefore carries a regime statement and a breakdown trigger.

Unresolved uncertainties

As of the evidence cutoff, no standard blind benchmark simultaneously spans PN, PM, first- and second-order self-force, amplitude, EOB, and numerical-relativity predictions over a common multidimensional domain. Correlation lengths for missing gravitational terms are rarely learned from enough independent orders. Numerical data remain sparse at high mass ratio, large spin, eccentricity, and strong scattering. Radiative and hereditary sectors are not covered by every conservative dictionary. Calibration posteriors remain conditional on chosen bases, priors, and catalogs.

The practical consequence is modest: publish a conditional budget with sensitivity analyses, not a universal “theory error.” State whether an interval is a Bayesian credible interval, a frequentist confidence interval, an empirical envelope, or a deterministic bound. Report failures of coverage rather than inflating the interval after viewing the blind data.

Solved correlated-budget check

For a dimensionless scattering observable, suppose the post-processed one-sigma components are

$$\sigma_{\rm tr}=0.30,\quad \sigma_{\rm num}=0.10,\quad \sigma_{\rm cal}=0.20,\quad \sigma_{\rm map}=0.25.$$

The mapping term has already had identifiable numerical and calibration contributions removed. Assume only truncation and mapping discrepancy are correlated, with $\rho=0.8$. Compare the independent and correlated budgets, then assess a withheld residual of magnitude $0.55$.

Solution

Independence gives

$$\sigma_{\rm ind} =\sqrt{0.30^2+0.10^2+0.20^2+0.25^2} \simeq0.45.$$

Including the covariance adds

$$2\rho\sigma_{\rm tr}\sigma_{\rm map} =2(0.8)(0.30)(0.25)=0.12,$$

so

$$\sigma_{\rm corr} =\sqrt{0.45^2+0.12} \simeq0.57.$$

The residual $0.55$ lies outside the independent one-sigma interval but inside the correlated one. This single success does not validate 68% coverage; many predeclared blind points are required. If the original mapping residual had still contained numerical and calibration error, either calculation would double count, and the decomposition would have to be rebuilt first.

What could change the conclusions

Any of the following could change these conclusions:

• a common blind benchmark compares three or more regimes on the same invariant;
• evidence shows that a standard quadrature, envelope, or discrepancy rule systematically undercovers withheld observables;
• a joint PN/PM/self-force/EOB/numerical analysis publishes correlations rather than only separate errors;
• a named model release, numerical catalog revision, correction, or independent reproduction changes a component materially;
• a new complete perturbative order or validated second-order self-force observable changes order-variation evidence.

The failure-diagnosis workflow determines which component should change before the budget is recombined.

References

• Boyle, Michael, et al. “The SXS Collaboration Catalog of Binary Black Hole Simulations.” Classical and Quantum Gravity 36, 195006 (2019). DOI. Open PDF.
• Furnstahl, R. J., N. Klco, D. R. Phillips, and S. Wesolowski. “Quantifying Truncation Errors in Effective Field Theory.” Physical Review C 92, 024005 (2015). DOI. Open PDF.
• Melendez, J. A., R. J. Furnstahl, D. R. Phillips, M. T. Pratola, and S. Wesolowski. “Quantifying Correlated Truncation Errors in Effective Field Theory.” Physical Review C 100, 044001 (2019). DOI. Open PDF.
• Owen, Caroline B., Carl-Johan Haster, Scott Perkins, Neil J. Cornish, and Nicolás Yunes. “Waveform Accuracy and Systematic Uncertainties in Current Gravitational Wave Observations.” Physical Review D 108, 044018 (2023). DOI. Open PDF.
• Pompili, Lorenzo, Alessandra Buonanno, and Michael Pürrer. “Accounting for Numerical-Relativity Calibration Uncertainty in Gravitational-Wave Modeling and Inference.” arXiv:2410.16859v3 (2026). Open PDF.
• Read, Jocelyn S. “Waveform Uncertainty Quantification and Interpretation for Gravitational-Wave Astronomy.” Classical and Quantum Gravity 40, 135002 (2023). DOI. Open PDF.