Computing gravitational radiation from highly eccentric bound binaries in the small-mass-ratio limit
Precise strong-gravity models describing the evolution of bound compact binary systems provide important information about the gravitational radiation such binaries generate and how they evolve due to the radiation's backreaction. The limit of very high eccentricity, $e \to 1$, is challenging to model: frequency-domain techniques require many harmonics to accurately describe radiation in this limit, and time-domain techniques can be slow to converge due to the systems' long timescales. Many recent astrophysical analyses have nonetheless emphasized the importance of understanding gravitational-wave emission in this regime, highlighting the fact that many systems may evolve through $e \simeq 1$ en route to becoming detectable sources, and may enter the sensitive band of detectors with substantial eccentricity. In this paper, we show how to adapt frequency-domain methods for computing gravitational radiation in the small-mass-ratio limit to effectively handle high eccentricity. The key piece of this analysis is an integral over the source of the frequency-domain Teukolsky equation. We examine features of this integral that make it difficult to evaluate as $e \to 1$, and study methods to circumvent these difficulties. We find a simple fix which allows us to compute gravitational-wave amplitudes and the associated fluxes of energy and angular momentum with ease up to eccentricities $e = 0.99$; with patience (and stamina), we have pushed to $e = 0.999$, though numerical accuracy appears to degrade for such extreme cases. Interestingly, we find that the well-known and often used classic formulas of Peters and Mathews are subject to a systematic error which persists at quite wide separation. This has implications for models of compact binary formation which evolve through $e \simeq 1$.