Approximants, Hybridization, and Surrogate Waveforms

Practical compact-binary waveforms combine information that is exact only in different regimes. Post-Newtonian expansions organize weak-field inspiral; effective-one-body models resum analytic information and attach merger–ringdown; numerical relativity supplies strong-field waveform products; hybrids extend short simulations; reduced-order and surrogate models compress expensive families; and calibration fits selected model freedoms over finite parameter support. This chapter explains how those layers are assembled and tested without treating construction accuracy as a detector-specific inference verdict.

Helpful background. Einstein Equations supplies the common field theory; Waves, Green Functions, and Geometric Optics supplies outgoing waves, Fourier transforms, and interpolation; Relativistic Two-Body Dynamics supplies analytic binary dynamics; Computational Relativity supplies production NR evolutions and extraction; and Gravitational-Wave Astronomy treats detector likelihoods and inference adequacy.

Enter the construction chain

The pages are ordered by what each new layer consumes:

1. Post-Newtonian Approximants compares time- and frequency-domain organizations that share finite weak-field coefficients but differ beyond their common order.
2. Effective-One-Body Waveforms separates the real–effective energy map, resummed potentials, radiation reaction, factorized modes, merger attachment, and calibration.
3. Numerical-Relativity Waveform Products defines versioned strain, curvature, extrapolation, resolution, frame, and metadata products suitable for reuse.
4. Waveform Hybridization aligns compatible analytic and NR modes over a common-validity window and blends them with explicit continuity and sensitivity tests.
5. Reduced-Order and Surrogate Models distinguishes basis projection, empirical interpolation, parameter regression, and total surrogate error.
6. Calibration and Extrapolation Domains records fit support, held-out evidence, sparse boundaries, convex-hull warnings, and extrapolation risk.

Readers choosing a named waveform family should not jump directly to its final match histogram. First identify the analytic and numerical inputs, then the hybrid or attachment operation, then compression, and finally calibration and validation support.

One waveform, several epistemic layers

Layer	Information consumed	New choice introduced	Minimum independent check
PN approximant	Truncated energy, flux, and amplitudes	Ratio expansion, integration or inversion, termination	Re-expansion through the declared order; weak-field and test-mass limits
EOB waveform	PN, perturbative, and self-force information	Resummation, NQC factors, attachment, calibrated terms	Analytic limits; held-out NR and attachment checks
NR product	Discrete spacetime evolution and extraction	Resolution, worldtube or extrapolation order, frame fixing	Resolution/extraction variation and balance laws
Hybrid	Analytic and NR waveforms with an overlap interval	Alignment norm, matching window, blend	Shift recovery, endpoint continuity, window and approximant variation
Surrogate	A finite waveform training family	Basis, nodes, parameter fit, split	Separate projection, interpolation, regression, and held-out errors
Calibration fit	Simulations at finite parameter locations	Loss, weights, fit family, regularization	Subset and boundary validation plus explicit domain flags

No downstream layer removes upstream uncertainty. A surrogate can reproduce a biased hybrid with very small numerical error; a well-calibrated EOB fit can inherit NR extraction error; a smooth hybrid can join mutually inconsistent phases. Error labels should therefore identify their layer before any combined norm is formed.

Buonanno and Damour introduced the EOB mapping and radiation-driven plunge architecture in Buonanno and Damour 1999, §§II–VII, PDF. Field et al. give the reduced-basis, empirical-interpolation, and parameter-fit decomposition in Field et al. 2014, §§II–IV, PDF. MacDonald, Nissanke, and Pfeiffer analyze alignment, matching-window, PN, NR, and hybrid errors in MacDonald, Nissanke, and Pfeiffer 2011, §§II–V, PDF.

Common waveform and match definitions

Throughout the chapter the spacetime signature is $(-,+,+,+)$ and $G$ remains explicit in physical formulas. A waveform product uses

$$h=h_+-ih_\times =\sum_{\ell,m}h_{\ell m}\,{}_{-2}Y_{\ell m},$$

with unit-normalized spin-weight $-2$ harmonics and inverse Fourier time dependence $e^{-i\omega t}$. Every comparison names its Bondi, asymptotically inertial, source, or coprecessing frame; time and phase alignment does not substitute for a frame transformation. A source waveform tabulated against $t_{\rm src}/(GM_{\rm src})$ is mapped to detector time using $t_{\rm det}/(GM_z)$, where $M_z=(1+z)M_{\rm src}$; substituting the unredshifted source mass in a detector-weighted comparison is a physical scaling error.

When a detector-weighted match is used, the chapter adopts the one-sided-PSD inner product

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

A quoted match or mismatch is incomplete without PSD, frequency band, total-mass scaling, modes, orientation or averaging, taper, frame, and maximized time/phase freedoms. Pompili et al.’s SEOBNRv5HM comparisons, for example, are tied to their named 2023 model, NR set, mass intervals, modes, PSDs, and alignment prescriptions Pompili et al. 2023, §§IV–V and Appendices, PDF. They cannot be detached from those conditions and read as a universal accuracy certificate.

The displayed positive-frequency inner product applies to real projected strain channels. Comparisons of complex spherical modes or sky-integrated radiation need an explicitly specified two-polarization or complex-field norm, as in the computational examples. One cannot discard negative-frequency content of a generic complex mode merely because a real detector strain admits a one-sided PSD.

Validation should be hierarchical

Start with algebraic and convention checks: PN re-expansion, EOB test-mass limits, strain/curvature signs, harmonic normalization, and frame identity. Next vary numerical controls: NR resolution and extraction, hybrid sampling and window, basis rank and nodes, fit order and regularization. Then use genuinely held-out physical cases, including boundaries and sparse corners. Finally apply the intended PSD and detector band—but leave the interpretation of likelihood and source inference to Gravitational-Wave Data Analysis and Inference.

The 2019 SXS catalog shows why multiple controls are needed: resolution differences, strain-versus-curvature extraction, and extrapolation-order differences probe different error sources Boyle et al. 2019, §4, pp. 24–30, PDF. Varma et al. similarly separate hybrid construction, surrogate cross-validation, parameter support, orientation, and mass-dependent mismatches Varma et al. 2019, §§V–VII, PDF.

A useful release record therefore states both what passed and what remains untested. Outside calibration support, missing modes, eccentricity, precession, matter effects, and incomplete BMS control are not small-print qualifications; they change the object being modeled.

Review the chapter

A candidate aligned-spin construction supplies bounds on four successive differences in a chain of waveform versions: changing the analytic approximant, refining the NR input, changing the hybrid window, and replacing the hybrid by its surrogate. They are respectively $2.0\times10^{-3}$, $4.0\times10^{-4}$, $1.0\times10^{-3}$, and $3.0\times10^{-4}$ in the same $L^2$ norm, over the same modes, frame, and time interval, divided by one common reference norm. No correlations are known. The requested parameter point also lies outside the convex hull of the calibration locations. Bound the difference between the chain’s endpoints and decide whether it certifies the waveform at that point.

Solution

Without independence or orthogonality, adding in quadrature is unjustified. The triangle inequality gives the conservative norm bound

$$\epsilon_{\mathrm{chain}} \le 2.0\times10^{-3} +4.0\times10^{-4} +1.0\times10^{-3} +3.0\times10^{-4} =3.7\times10^{-3}.$$

This bounds the difference between the chain’s endpoints because its intermediate waveforms cancel in a telescoping sum. It does not bound either endpoint’s distance from the exact physical waveform. Arbitrary sensitivity diagnostics with unrelated reference waveforms or different denominators would not automatically satisfy even this chain bound. It is not a confidence interval, and shared biases could evade all four comparisons. More importantly, the point is outside calibration support, so the calibration-fit contribution has no held-out bound there. The value $3.7\times10^{-3}$ therefore does not certify the requested waveform. A proper answer must flag extrapolation and obtain targeted validation or restrict the domain.

References

• Boyle, Michael, et al. “The SXS Collaboration Catalog of Binary Black Hole Simulations.” Classical and Quantum Gravity 36 (2019): 195006. DOI. Open PDF.
• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59 (1999): 084006. DOI. Open PDF.
• Field, Scott E., Chad R. Galley, Jan S. Hesthaven, Jason Kaye, and Manuel Tiglio. “Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models.” Physical Review X 4 (2014): 031006. DOI. Open PDF.
• MacDonald, Ilana, Samaya Nissanke, and Harald P. Pfeiffer. “Suitability of Post-Newtonian/Numerical-Relativity Hybrid Waveforms for Gravitational Wave Detectors.” Classical and Quantum Gravity 28 (2011): 134002. 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.
• Varma, Vijay, Scott E. Field, Mark A. Scheel, Jonathan Blackman, Lawrence E. Kidder, and Harald P. Pfeiffer. “Surrogate Model of Hybridized Numerical Relativity Binary Black Hole Waveforms.” Physical Review D 99 (2019): 064045. DOI. Open PDF.