Black-hole scattering with numerical relativity: Self-force extraction and post-Minkowskian validation

TL;DR

This work leverages highly accurate NR simulations of unbound BBH encounters to benchmark perturbative approaches. By extracting SF coefficients from unequal-mass NR runs, the authors show that up to $2SF$ ($\nu^2$ terms) reproduces the NR scattering angle across mass ratios, including equal mass. They also compare NR results to state-of-the-art PM predictions in the weak-field regime, finding strong agreement at large $b$ and uncovering that the dominant unknown higher-PM contribution appears to be the $6PM(0SF)$ term rather than the $5PM(2SF)$ term. The study demonstrates the complementary roles of SF and PM in modeling BBH scattering and highlights NR as a crucial tool for validating and refining perturbative theories, with implications for future extensions to radiative observables and spinning binaries.

Abstract

The asymptotic nature of unbound binary-black-hole encounters provides a clean method for comparing different approaches for modeling the two-body problem in general relativity. In this work, we use numerical relativity simulations of black-hole scattering, generated using the Spectral Einstein Code, to explore the self-force and post-Minkowskian expansions of the scattering angle. First, we use a set of unequal-mass simulations to extract the self-force contributions to the scattering angle. Our main result is that using information up to second-order in the symmetric mass ratio (2SF) reproduces numerical relativity within the error bars across the full range of mass-ratios, including equal mass. Next, we compare our numerical relativity results to state-of-the-art post-Minkowskian predictions at larger impact parameters than previously explored. We find good agreement in the weak-field regime and discuss the relative importance of higher-order terms.

Figures (5)

• Figure 1: Trajectories of equal-mass black-hole scattering scenarios with $\Gamma=1.02264$ and impact parameters $b=[37.4, 30.4, 23.4, 17.1, 12.8, 11.1]M$. Shown are the Cartesian components of the separation vector between the centers of the BHs, rotated such that the incoming path is aligned with the $x$-axis. The $b=17.1M$ (red) trajectory corresponds to the largest impact parameter previously studied at this energy Damour:2014afaRettegno:2023ghrSwain:2024ngsLong:2025nmj.
• Figure 2: Scattering angle $\theta$ as a function of symmetric mass ratio $\nu$ at fixed $\gamma=1.0200$ and $\ell=4.8000$. The points represent the SpEC data and the lines represent the best fits to the data using Eq. (\ref{['eq:SFAngle']}). The $n$SF fit contains all terms up to $n$th order in $\nu$ with the extracted values shown in Table \ref{['Table:SFAngle']}. The known 0SF term (dotted black line) and the zero mass limit (thin vertical line) are shown for reference. Left: Fits where all SF coefficients are free parameters (Fits 1--3 of Table \ref{['Table:SFAngle']}). Right: Fits where the 0SF term is fixed to the known geodesic value $\vartheta_{\rm 0SF}=188.211^\circ$ (Fits 1'-- 3' of Table \ref{['Table:SFAngle']}).
• Figure 3: Scattering angle $\theta$ as a function of symmetric mass ratio $\nu$ at fixed $\gamma=1.0200$ and $\ell=4.8000$. The points represent the SpEC data and the lines correspond to the extracted SF expansion of Eq. (\ref{['eq:SFAngle']}) using the best-estimate coefficients in Eqs. (\ref{['eq:1SFFit']})--(\ref{['eq:2SFFit']}) and the known 0SF term. The known 0SF term (dotted black line) and the zero mass limit (thin vertical line) are shown for reference. The lower panel shows the difference between the data and the best-estimate fits.
• Figure 4: Scattering angle $\theta$ as a function of impact parameter $b$ at fixed $\Gamma=1.02264$ and $q=1$. The points represent the SpEC data and the lines represent the analytic PM predictions. The vertical dotted lines show the first confirmed capture from SpEC. The bottom panel shows the difference between the data and PM predictions with the inset showing a zoomed-in view of the weak-field region. The SpEC errors are too small to be seen on the scale of the main plot, but can be resolved in the inset.
• Figure 5: Scattering angle $\theta$ as a function of impact parameter $b$ at fixed $\Gamma=1.02264$ and $q=1$. The points represent the SpEC data with various PM orders subtracted off, as indicated in the legend, and the dashed lines represent the analytic PM predictions at the corresponding order. For example, the orange points represent the SpEC data with the 1PM prediction subtracted off and the orange dashed line represents the 2PM prediction. $\vartheta_{5\text{PM}}^{1\text{SF}}$ represents the coefficient of $G^5$ containing terms up to order $\nu$.