Cross-Regime Matching, Calibration, and Consistency

Relativistic two-body methods become mutually constraining only after their native variables are translated to common invariants. This chapter builds that translation: PN and PM coefficients meet in a double expansion; self-force fixes derivatives at the test-mass boundary; amplitudes, Hamiltonians, and EOB models meet through actions and scattering angles; numerical data calibrate only unresolved structure; and uncertainties are tested on withheld overlap data. Production waveform fitting belongs to Gravitational Radiation and Waveforms. Here the purpose is narrower and more fundamental: establish dynamical compatibility and locate failures.

Helpful background. The Einstein Equation supplies the common field theory; Weak Metric Perturbations and Gauge Freedom supplies weak-field expansions and gauge reasoning; Presymplectic Potentials and Currents supplies canonical equivalence; Computational Relativity supplies convergence and benchmark practice; and Gravitational Radiation and Waveforms supplies flux and extraction conventions.

Enter this chapter

The six pages form one inference chain and are best read in order.

1. PN–PM Matching asks which monomials in $G$ and velocity must agree, derives a 1PM low-velocity check, and shows why weak coupling and slow motion are nonuniform limits.
2. Self-Force Information at Finite Mass Ratio converts $\epsilon=m_2/m_1$-expanded invariants to fixed-$x$, symmetric-mass-ratio variables and identifies exactly what first-order data leave undetermined.
3. Amplitude–Hamiltonian–EOB Dictionaries tracks external-state normalization, Born iterations, canonical transformations, radial actions, and the EOB energy map to the common scattering angle.
4. Numerical Calibration Without Overfitting specifies immutable data, grouped train/validation/test splits, a reproducible environment, synthetic and physical benchmarks, stability tests, and extrapolation failures.
5. Cross-Regime Uncertainty Budgets separates truncation, regularization, numerical, calibration, and mapping errors, declares their correlations, and tests combined intervals on blind points.
6. Consistency Tests and Failure Diagnosis orders dimensional, symmetry, exact-limit, first-law, balance, convergence, and residual-scaling checks so a disagreement becomes a localized hypothesis.

The recurring lesson is simple: compare observables before representations, and compare representations only through a complete dictionary.

The common comparison record

Before two results can be matched, write down a common record:

Field	Required content
Physical system	$m_1,m_2$, spins, finite-size coefficients, orbit class, and incoming or bound state
Invariant inputs	Total energy, angular momentum, frequencies, radial actions, or asymptotic impact parameter
Invariant output	Binding energy, redshift, periastron advance, scattering angle, impulse, flux, or radiated charges
Approximation	Powers retained in $v$, $G$, $\nu$, spin, eccentricity, or finite-size scale
Sector and boundary condition	Conservative or dissipative; time-symmetric, retarded, or advanced
Representation data	Coordinates, canonical gauge, EOB energy map, spin condition, state normalization, and subtraction
Error statement	Truncation, regularization, numerical, calibration, mapping, and correlations
Evidence role	Matched input, calibration datum, validation point, or extrapolation stress test

This is not bureaucracy. Each field blocks a known false comparison. The same symbol $E$ can mean total relativistic energy, nonrelativistic binding energy, or an EOB effective energy. The same “potential” can mean a coordinate interaction, an EOB radial function, or a Fourier kernel. A scattering angle is invariant only after incoming and outgoing energies and angular momenta are defined in the presence of radiation.

Shared invariants by regime

No single observable spans every physical question, but a small set covers most interfaces:

Interface	Common invariant	Controlled common limit
PN ↔ PM	$\chi(E,J)$ or radial action	Slow motion, weak deflection, and fixed double-scaling path
PN ↔ self-force	$E_b(\Omega)$, redshift, or periastron advance	Weak field and $\nu\to0$ at fixed invariant frequency
PM ↔ self-force	$\chi(E,J)$	Weak deflection and small mass ratio at fixed relativistic energy
Amplitude ↔ Hamiltonian	Impulse, $\chi(E,J)$, or radial action	Classical limit after restoring $\hbar$ and subtracting iterations
Hamiltonian ↔ EOB	Binding and scattering actions	Same real–effective energy map and canonical asymptotics
Local reaction ↔ radiation	Energy and angular-momentum balance	Same boundary condition, hereditary terms, and averaging
EOB ↔ numerical relativity	Binding curve, scattering angle, flux, or aligned waveform invariant	Documented numerical continuum and extraction limits

Damour’s PM–EOB construction uses $\chi(E,J)$ to establish the real/effective energy dictionary at 1PM Damour 2016, §§II–VIII, PDF. Kälin and Porto show how the same scattering data determine a radial action and, under conservative locality assumptions, bound invariants Kälin and Porto 2020, §§2–5, pp. 6–30, PDF. These examples explain why the common object is an action or observable, not a unique Hamiltonian coefficient.

Matching, calibration, and validation are different operations

These terms answer different questions:

Operation	Question	Data allowed	Result
Matching	What coefficient is forced by two descriptions of the same overlap?	Analytic or numerical information in the declared common limit	A coefficient or dictionary identity
Calibration	What unresolved parameter best represents data within a chosen model family?	Prespecified training data and priors	A conditional parameter estimate
Validation	Does the completed model predict information not used to build or choose it?	Sequestered configurations or independent calculations	A domain-specific performance statement

For example, the low-velocity expansion of a PM scattering angle must reproduce PN coefficients. That equality is matching; it cannot be improved by fitting. A first-order self-force redshift fixes the coefficient linear in $\nu$ after converting the frequency variable. Choosing a nonlinear completion in $\nu$ is modeling, and fitting its free coefficient to numerical relativity is calibration. Comparing the result with a simulation excluded from both basis selection and fitting is validation.

Buonanno and Damour’s original EOB construction distinguishes the exact form of the real/effective map from finite-PN input and resummation choices Buonanno and Damour 1999, §§II–VI, pp. 3–18, PDF. A modern calibrated EOB model makes the later layers visible by listing analytic ingredients, calibration parameters, numerical simulations, fit requirements, and performance comparisons Pompili et al. 2023, §§II, IV, and V, PDF. A small calibration residual is still not a truncation error or an out-of-sample guarantee.

A running invariant: the scattering angle

Consider conservative, nonspinning hyperbolic motion. Fix $m_1,m_2$, center-of-mass energy $E$, and angular momentum $J=b p_\infty$. Several routes lead to the same quantity:

$$\chi(E,J) =-\pi-\lim_{R\to\infty}\frac{\partial}{\partial J} \left[2\int_{r_{\min}}^R p_r(E,J;r)\,dr\right].$$

This uses the full unnormalized action with a large-radius cutoff: take the derivative before the limit. Subtracting the free action removes the $-\pi$; the long-range logarithm requires an additional infrared subtraction if the action itself, rather than only its angular derivative, is needed.

A PN Hamiltonian yields $p_r$ as a velocity expansion. A PM calculation organizes $\chi$ in powers of $G/J$ at arbitrary energy. An amplitude gives an impulse or a Fourier kernel after the classical limit and iteration subtraction. Self-force computes the coefficient of the small-mass-ratio expansion. EOB rewrites the radial dynamics through an effective Hamiltonian and energy map. Numerical relativity can measure incoming and outgoing trajectories and radiated losses, then infer an angle with numerical error.

The amplitude route is especially sensitive to incomplete dictionaries. The impulse is defined by an expectation-value difference involving $S^\dagger P^\mu S$, not by selecting one visually classical graph; unitarity and cut contributions are needed for the correct classical observable Kosower, Maybee, and O’Connell 2019, §§3–5, pp. 6–49, PDF. At the Hamiltonian stage, a canonical transformation can change coefficients but changes the action only by controlled endpoint terms. At the numerical stage, radiated energy means that incoming, outgoing, or averaged $(E,J)$ must be specified.

This one example exercises the whole chapter. A factor error is caught by the Newtonian limit. An incorrect $G^n v^k$ term is caught by PN–PM double expansion. A residual proportional to $\nu^2$ after first-order matching localizes the missing mass-ratio order, although several modeling or mapping errors can have that scaling. A residual that vanishes under grid refinement is numerical. A comparison made only on training data is not validation.

Common limits and why their order matters

The phrase “take both limits” is insufficient. In Newtonian scattering the deflection parameter is

$$\alpha=\frac{GM}{b v_\infty^2}.$$

Slow motion at fixed $G$ and $b$ makes $\alpha$ large and exits weak deflection; a fixed-PM truncation is then nonuniform. PM-to-PN matching requires a path with $v_\infty^2\ll1$ and $\alpha\ll1$. Similarly, self-force theory takes $\nu\to0$ at fixed invariant orbit. Approaching a light ring at fixed finite $\nu$ can make the nominal perturbation large, so exchanging those limits is unjustified.

Other orderings matter:

• restore $\hbar$ before selecting a classical contribution from a loop amplitude;
• subtract lower-order iterations before reading a new potential coefficient;
• identify the conservative branch before analytically continuing scattering data to a bound orbit;
• take numerical resolution and extraction-radius limits through demonstrably convergent sequences;
• choose calibration structure on validation data before opening a blind test;
• determine component overlap before adding errors to a combined budget.

Blanchet’s near-zone/radiation-zone matching shows why hereditary terms cannot be treated as instantaneous coefficients Blanchet 2024, §2, especially §§2.2–2.4. The binary first law likewise requires derivatives at specified fixed variables, so changing from $Gm_1\Omega$ to $GM\Omega$ after differentiation introduces a spurious self-force discrepancy when $m_1$ is the background mass Le Tiec, Blanchet, and Whiting 2012, §§II–IV.

From discrepancy to uncertainty

First localize a discrepancy; then assign uncertainty. If two methods differ by $d=Q_A-Q_B$, the residual contains both methods’ truncation, numerical, regularization, calibration, and mapping terms. Adding $d$ on top of all those full errors double counts. Conversely, a shared wrong coefficient cancels from $d$ and makes the methods look falsely precise.

The remedy is a covariance-aware component model tested on withheld points. Numerical resolution supplies one component only when it converges. Calibration posterior width supplies uncertainty conditional on the chosen model family. Order variation supplies truncation evidence only in a controlled expansion. Pairwise overlap supplies a discrepancy after already-accounted components are removed. A blind benchmark tests the resulting interval.

Numerical catalogs illustrate both the strength and the limit of such evidence. The SXS catalog compares multiple resolutions and extraction representations, while flagging configurations that do not converge monotonically and noting that resolution changes can alter physical parameters slightly Boyle et al. 2019, §4, pp. 24–28, PDF. These are inputs to an uncertainty budget, not a universal error bar for every downstream model.

Solved chapter diagnostic

A new finite-ratio EOB model is compared with four information sources. In this constructed diagnostic at one fixed relative Lorentz factor, define the dimensionless angular momentum

$$j=\frac{J}{GM\mu}=\frac{J}{Gm_1m_2}.$$

1. its 1PM scattering angle agrees with PN theory in all shared low-velocity coefficients;
2. at 2PM, the signed difference is $\chi_{\rm amp}-\chi_{\rm EOB}=4\nu/j^2=4\nu(GM\mu/J)^2$;
3. an independent first-order self-force calculation, converted to the same $(E,J,\nu)$ variables, predicts exactly the coefficient $4\nu$ in that 2PM term;
4. two additional strong-field coefficients were fitted to 30 numerical simulations, and a withheld 31st simulation agrees within its continuum-extrapolation error.

The amplitude team also supplies a Hamiltonian related to another published Hamiltonian by a canonical transformation. Classify each piece as matching, calibration, or validation; state the first checks to perform on the 2PM discrepancy; and explain how to enter the withheld agreement into an uncertainty budget.

Solution

Item 1 is analytic matching: the shared PN–PM coefficients are compulsory. Items 2 and 3 together show that the EOB model is missing a matched 2PM contribution proportional to $\nu/j^2$. Because both amplitude and self-force determine the same invariant coefficient independently, it should be inserted through the correct dictionary, not fitted to numerical data.

Before changing the EOB potential, verify that all calculations use the same $E$, $J$, masses, and real/effective energy map; that the amplitude has the same external-state normalization; and that the iterated 1PM contribution has been subtracted. Then compute $\chi(E,J)$ from both Hamiltonians. The canonical transformation is not a physical explanation if the invariant angles still differ; if it makes the Hamiltonian coefficients differ while $\chi$ agrees, that difference is representational.

The two strong-field coefficients are calibration because the 30 simulations estimate unresolved model freedom. The 31st simulation is validation only if it was excluded from coefficient fitting, basis choice, priors, stopping rules, and all model selection. Its agreement supports empirical coverage at one point; it does not prove a global uncertainty.

For the budget, propagate the 31st simulation’s continuum and extraction covariance together with analytic truncation, calibration covariance, and any residual mapping discrepancy. Do not add the raw model–simulation difference as a separate independent error when it already contains those components. Repeat on a family of withheld points before claiming coverage, and mark any point outside the calibration hull as extrapolation.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4 (2024). DOI.
• Boyle, Michael, et al. “The SXS Collaboration Catalog of Binary Black Hole Simulations.” Classical and Quantum Gravity 36, 195006 (2019). DOI. Open PDF.
• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59, 084006 (1999). DOI. Open PDF.
• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States.” Journal of High Energy Physics 2020, 072 (2020). DOI. Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell. “Amplitudes, Observables, and Classical Scattering.” Journal of High Energy Physics 2019, 137 (2019). DOI. Open PDF.
• Le Tiec, Alexandre, Luc Blanchet, and Bernard F. Whiting. “The First Law of Binary Black Hole Mechanics in General Relativity and Post-Newtonian Theory.” Physical Review D 85, 064039 (2012). DOI.
• 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.