Resummed energy loss in extreme-mass-ratio scattering using critical orbits

TL;DR

This work develops a PM-SF resummation framework augmented by PN terms to model the total gravitational-wave energy radiated in extreme-mass-ratio scattering around Schwarzschild black holes. By exploiting the universal logarithmic divergence of energy losses near the separatrix and linking the coefficient to fluxes from the limiting unstable circular geodesic, the authors construct resummed expressions for $E^{+}_{\rm GW}$ and $E^{-}_{\rm GW}$ that interpolate between weak-field PM/PN results and strong-field separatrix dynamics. They validate the approach against high-precision BH perturbation theory data and show that a mixed PM/3'PN resummation yields robust accuracy across parameter space, while horizon absorption requires attenuation of the singular term. The methodology offers a general pathway to uniformly valid radiative observables in black-hole scattering and could inform bound-orbit results via scattering-to-bound mappings, representing a unifying organizing principle for gravitational-wave modeling across regimes.

Abstract

Motivated by recent efforts to bridge between weak-field and strong-field descriptions of black-hole binary dynamics, we develop a resummation scheme for post-Minkowskian radiative observables in extreme-mass-ratio scattering, augmented with post-Newtonian terms. Specifically, we derive universal interpolation formulas for the total energy emitted in gravitational waves out to infinity and down the event horizon of the large black hole, valid to leading order in the small mass ratio. We test our formulas using numerical results from direct calculations in black hole perturbation theory. The central idea of our approach is to utilize as a strong-field diagnostic the known form of divergence in the radiated energy along geodesics near the parameter-space separatrix between scattering and plunge. The dominant, logarithmic term of this divergence can be expressed in terms of instantaneous energy fluxes calculated along the unstable circular geodesics that form the separatrix, fluxes that we obtain using interpolation of highly accurate numerical data. The same idea could be applied to bound-orbit radiative observables via either unbound-to-bound mapping or a direct resummation of bound-orbit post-Newtonian expressions.

Figures (8)

• Figure 1: Representation of the space of (non-plunging) geodesic orbits around a Schwarzschild black hole, using the squared inverse specific orbital energy, $1/E^2$, and inverse pericenter radius, $M/r_0$, as compact coordinates on parameter space. Non-plunging geodesics consist of periodic/bound orbits ($1/E^2>1$) and scattering orbits ($1/E^2\leq 1$). The boundary of the region is formed by circular geodesics; it is made up of two segments, green and orange curves in the diagram, corresponding, respectively, to stable and unstable circular geodesics. The two segments join smoothly at the Innermost Stable Circular Orbit (ISCO). The locus of unstable circular orbits (orange curve) is the separatrix, represented in the text by the function $r_0=R(E)$. Points along the separatrix also represent homoclinic orbits (for $1/E^2>1$, $4M<R<6M$) or heteroclinic orbits (for $1/E^2<1$, $3M<R\leq 4M$), which display infinite "zoom-whirl" behavior. The two orbits illustrated are examples of near-separatrix homoclinic and heteroclinic geodesics [their parameters are $(E=0.962253, j=3.67426)$ and $(E=1.26486, j=5.77319)$, respectively]. Our resummation technique takes advantage of the singular behavior of geodesics near the separatrix as a marker of strong-field dynamics.
• Figure 2: Energy flux (per $q^2$) from unstable circular geodesics, at null infinity (${\cal F}^+_{\circ}$) and down the event horizon (${\cal F}^-_{\circ}$), as a function of orbital radius $R$. Triangular markers show results from our high-precision numerical calculations (with error bars indiscernible on the scale of this plot). Our analytical models (\ref{['eq:circular_flux_fit']}), with the best-fit parameters from Table \ref{['tab:flux_fit_coeffs']}, are shown in solid lines. The corresponding PN expressions are shown in dashed line for comparison. The bottom panel displays the relative difference between each flux model and the numerical data.
• Figure 3: Total energy radiated to infinity in the weak-field approximations $E^+_{\rm 5PM}$, $E^+_{\rm 3'PN}$ and $E^+_{\rm 5PM/3'PN}$, as a function of orbital angular momentum $j$ (lower scale) and periapsis distance $r_0$ (upper scale) at a fixed value $v=0.2$ of the velocity at infinity. "Exact" benchmark results $E^+_{\rm GW}$ from the numerical code of Ref. Warburton:2025ymy are shown for reference. (Estimated error bars on the numerical values are too small to be visible on the scale of this plot.) The lower panel displays relative differences with respect to the numerical data. The location of the separatrix is marked by the vertical line. Here and in all subsequent plots, the displayed quantities $E^{\pm}$ are normalized by a factor $q^2M(=m^2/M)$.
• Figure 4: Same as Figure \ref{['fig:weak_field_0.2']}, this time for $v=0.35$. We see that the 5PM formula takes over from the $\rm 3'PN$ model as a better approximation at large $j$. The mixed 5PM/$\rm 3'PN$ model still outperforms both pure PM and pure PN models at all $j$.
• Figure 5: Resummed energy-loss models $\tilde{E}^+_{\rm PM}$ and $\tilde{E}^+_{\rm mixed}$ [Eq. (\ref{['resum']})] as functions of $\delta j:=j-j_c(v)$ (lower axis) and periastron distance $r_0$ (upper axis) for fixed $v=0.2$. The resummation models are plotted against reference numerical data for $E^+_{\rm GW}$ from the code of Warburton:2025ymy. (Estimated error bars on the numerical values are too small to be visible on the scale of this plot.). Also shown for reference are the plain 5PM and 5PM/$\rm 3'PN$ weak-field approximations (without resummation). The lower panel displays relative differences with respect to the numerical data, with filled (open) markers denoting underestimation (overestimation) by the model.