Spinning Effective-to-Backwards One Body (SEBOB)

• SEBOB is a hybrid waveform modeling framework that analytically bridges EOB inspiral evolution with BOB merger-ringdown prescriptions for aligned-spin binary black holes.
• It minimizes reliance on numerical relativity by enforcing analytic continuity at the inspiral–merger interface and matching computed dynamics to NR benchmarks.
• Implemented in the NRPy framework, SEBOB delivers computational efficiency and high waveform fidelity, making it ideal for parameter estimation and low-latency detection.

The Spinning Effective-to-Backwards One Body (SEBOB) formalism is a hybrid waveform modeling framework for aligned-spin binary black hole coalescences. SEBOB combines the predictive, analytically motivated inspiral modeling of the Effective-One-Body (EOB) approach with a merger-ringdown prescription based on first-principles perturbation theory—the Backwards One Body (BOB) method—designed to reduce dependence on direct numerical relativity (NR) calibration. Its goal is to produce transparent, computationally efficient, and highly faithful gravitational waveforms that can generalize well even in regions of parameter space where high-accuracy NR data is limited or unavailable (Mahesh et al., 28 Aug 2025).

1. Motivation and Conceptual Overview

Traditional high-fidelity models such as SEOBNRv5HM rely critically on NR-informed non-quasi-circular (NQC) corrections for the inspiral-plunge and require direct calibration to NR simulations to ensure accuracy in the late-inspiral and merger-ringdown phases. This heavy phenomenological tuning, while empirically successful, limits physical robustness and extrapolation, especially for high mass ratios and near-extremal spins where NR data is sparse.

SEBOB introduces an analytic bridge between the inspiral and merger-ringdown by integrating the BOB model with EOB: the EOB Hamiltonian is evolved through the inspiral and matched onto a BOB-based analytic waveform at the plunge, minimizing NR-dependent ingredients. This construction allows for a physically motivated, controlled transition between regimes while offering computational advantages through a transparent, symbolic-to-optimized-C implementation in the NRPy framework.

2. Hybridization of EOB and BOB Approaches

The EOB methodology governs the inspiral and plunge via a canonical Hamiltonian evolution, including spin effects, radiation reaction, and NQC corrections. At a carefully defined matching time,

$t_0 = t^{ISCO} - \Delta t^{ISCO},$

where $t^{ISCO}$ is the time at which the system crosses the EOB-predicted innermost stable circular orbit (ISCO), the waveform transition occurs. Two main SEBOB variants are defined:

• seobnrv5_nrnqc_bob: The standard NQC corrections (fit to NR data) are retained in the inspiral-plunge, and only the BOB merger-ringdown prescription is appended at $t_0$.
• seobnrv5_bob: The BOB formalism supplies both the merger-ringdown and—in a key extension—the NQC corrections themselves, enforcing analytic $\mathcal{C}^2$ (second-derivative) continuity at the transition between inspiral and merger. The NQC coefficients for amplitude and frequency derivatives are matched not to NR but to BOB’s analytic predictions.

The BOB model treats the merger-ringdown as a non-adiabatic nonlinear perturbation on the final remnant black hole. For each multipole $(\ell,m)$, the amplitude of the Newman–Penrose scalar is modeled as

$$A_{\ell m}(t) = A_{p,\ell m} \, \operatorname{sech}\left(\frac{t - t_{p,\ell m}}{\tau_{\ell m}}\right),$$

while the frequency evolution is given by

$$\omega(t) = \left[\omega_0^4 + k \left(\tanh\left(\frac{t - t_p}{\tau}\right) - \tanh\left(\frac{t_0 - t_p}{\tau}\right)\right)\right]^{1/4},$$

with $k$ algebraically determined to enforce matching to the least-damped quasinormal mode (QNM) frequency $\omega_{QNM}$.

This analytic merging eliminates the need for extensive NR-based tuning of NQC time-derivatives and QNM amplitude coefficients, and—crucially for $\texttt{seobnrv5_bob}$—enables fully analytic, order-unity mapping from inspiral-to-merger.

3. Technical Implementation and Computational Efficiency

The SEBOB models are realized using the NRPy framework, which features automatic translation of symbolic EOB/BOB expressions to highly optimized C code. Key features include:

• Symbolic representations ensure mathematical transparency and facilitate mathematical inspection and extension.
• Automated common subexpression elimination (CSE) is used to reduce computational overhead in evaluating complicated Hamiltonians and waveform prescriptions.
• Integration with the GNU Scientific Library guarantees robust ODE integration and accurate root-finding essential for waveform generation.
• Performance benchmarks indicate that SEBOB C-code achieves at least a $\sim$3x speedup over Python-based implementations (e.g., pySEOBNR), while remaining competitive with other C-based waveform libraries.

These choices support both rapid scientific prototyping and production-level parameter estimation in next-generation detector analyses.

4. Validation Against Numerical Relativity

Comprehensive validation is performed against a wide catalog of NR simulations (notably from the SXS database), focusing on noise-weighted waveform mismatches for the dominant $(2,2)$ multipole. The SEBOB models achieve:

• Typical median mismatches around $2 \times 10^{-4}$, indicating very high agreement with NR data.
• Accuracy that is competitive with the fully NR-calibrated $\texttt{SEOBNRv5HM}$, and similar performance relative to TEOBResumS.
• Diagnostic comparison shows the BOB-informed NQC model (seobnrv5_bob) reproduces the frequency evolution across merger extremely well (e.g., peak frequency derivative errors are generally $\sim$4%), while amplitude curvature mismatches near the merger remain modest.

Even in regions where the analytic NQC prescription leads to some local amplitude discrepancy, the net overlap with NR is robust. This demonstrates that the analytic BOB merger construction can replace direct NR calibration without major loss of fidelity.

5. Physical and Methodological Advantages

SEBOB’s hybridization delivers several key physical and practical benefits:

• By minimizing NR-based input to only the remnant mass, spin, and peak strain/frequency, SEBOB is less sensitive to the sparseness of NR data for high mass-ratio and high-spin regimes. It is thus well-poised for extrapolation into the "terra incognita" of the parameter space, relevant both for extreme-mass ratio inspirals and for highly spinning systems.
• The analytic matching at the inspiral–plunge/merger interface leads to controlled and physically interpretable continuity of the strain and its derivatives, aiding physical intuition about the transition regime.
• Open, symbolic implementation in NRPy supports straightforward extensions to new physics—such as modeling higher-order harmonics, relaxing the adiabatic amplitude approximation, or directly implementing analytic interpolations for the gravitational-wave news.
• High computational efficiency allows for real-time waveform generation needed in low-latency gravitational-wave detection and Bayesian parameter estimation.

6. Future Prospects and Applications

SEBOB establishes a pathway for next-generation waveform modeling:

• Reducing reliance on NR waveform calibration and shifting toward first-principles analytic control ensures that models remain extensible as detector sensitivities and scientific requirements increase (e.g., for LISA, ET, CE).
• The analytic structure of the merger-ringdown in SEBOB positions it as a robust baseline for further refinement, including the potential to handle precessing spins and more general deviations from adiabaticity.
• By maintaining high accuracy and efficiency, SEBOB is suited to support high-SNR detections and stringent tests of strong-field general relativity.

Future research will likely focus on building upon the analytic BOB prescription to further improve amplitude modeling near the peak, incorporating additional physical effects, and refining matching strategies for more general classes of binaries.

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In summary, Spinning Effective-to-Backwards One Body (SEBOB) provides a hybrid, physically motivated framework that combines the analytic strengths of EOB evolution with a first-principles, BOB-based analytic merger-ringdown, reducing empirical calibration while maintaining high waveform fidelity and computational efficiency. These properties make SEBOB a promising tool for the high-precision gravitational-wave astronomy program of the next decades (Mahesh et al., 28 Aug 2025).