Resumming Post-Minkowskian and Post-Newtonian gravitational waveform expansions

TL;DR

This paper develops a new resummation method for gravitational waveform expansions in Post-Minkowskian and Post-Newtonian regimes, providing high-order calculations and a novel mathematical approach to special functions.

Contribution

It introduces a resummation technique for gravitational waveforms at a given order, utilizing a hypergeometric representation of Heun equation solutions, and computes high-order waveforms for circular orbits.

Findings

Waveform and energy flux computed to 30PN order.

New hypergeometric representation of Heun solutions.

Validation against existing literature results.

Abstract

We derive formulae that resum, at a given order in the soft limit, the infinite series of Post-Minkowskian (small gravitational coupling) or Post-Newtonian (small velocities) corrections to the gravitational waveform produced by particles moving along a general (open or closed) trajectory in the Schwarzschild geometry in the probe limit. Specifying to the case of circular orbits, we compute the waveform and the energy flux to order 30PN, and compare it against the available results in the literature. Our results are based on a novel hypergeometric representation of the solutions of the Heun equation (and its confluence), that leads to a simple mathematical proof of the Heun connection formula.