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

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.

💡 Research Summary

The paper addresses the longstanding challenge of accurately modeling gravitational‑wave energy loss in extreme‑mass‑ratio (EMR) scattering events across the full range from weak‑field, post‑Minkowskian (PM) regimes to the strong‑field region near the scattering‑plunge separatrix. While PM and post‑Newtonian (PN) expansions provide high‑order results for the total radiated energy to infinity (E⁺_GW) and horizon absorption (E⁻_GW) in the weak‑field limit, they fail to capture the characteristic logarithmic divergence that occurs when a scattering orbit approaches the critical angular momentum j_c(v). In this limit the particle executes an arbitrarily large number of “zoom‑whirl” cycles around an unstable circular orbit of radius R, and both the azimuthal phase and the emitted energy diverge as log δj, where δj = j – j_c(v) → 0.

The authors’ central insight is to use the instantaneous energy fluxes computed on the unstable circular orbit (the “critical orbit”) as a strong‑field diagnostic. By performing high‑precision black‑hole perturbation theory calculations they obtain accurate fits for the fluxes to infinity Φ_∞(R) and down the horizon Φ_H(R). The coefficient of the logarithmic divergence in the total radiated energy is then κ_±(R) = Φ_±(R) · A(R), where A(R) = –½ √(6M/R – 1) is the known coefficient governing the divergence of the scattering angle. This provides an exact analytic expression for the leading singular term in E_±_GW near the separatrix.

Armed with κ_±(R), the authors construct a resummation scheme that augments existing PM/PN series with a multiplicative correction factor designed to reproduce the logarithmic behavior. The correction takes the form  F(δj) = 1 + α δj, and the resummed energy loss is  E_±^resummed = E_±^PM/PN ·