Post-Newtonian expansion of gravitational waves from a particle in circular orbit around a Schwarzschild black hole

TL;DR

This work analytically derives the post-Newtonian expansion of gravitational waves from a test particle in a circular orbit around a Schwarzschild black hole, obtaining waveforms and luminosity up to $O(v^8)$ beyond Newtonian. It uses a Green's-function solution to the Teukolsky equation, mapped from the Regge-Wheeler equation, and performs a systematic $\epsilon$- (with $\epsilon = 2M\omega$) and small-$z$ expansion to extract the ingoing amplitudes and luminosity coefficients, including exact $\ln v$ terms at $v^6$ and $v^8$. The results are shown to be in excellent agreement with prior numerical analyses, and the authors provide a detailed assessment of the impact on orbital-phase evolution for coalescing binaries, indicating the PN order required for accurate template construction. Although restricted to the test-particle limit, these analytical benchmarks guide full PN calculations and improve our understanding of tail effects and logarithmic contributions in gravitational-wave emission.

Abstract

Based upon the formalism recently developed by one of us (MS), we analytically perform the post-Newtonian expansion of gravitational waves from a test particle in circular orbit of radius $r_0$ around a Schwarzschild black hole of mass $M$. We calculate gravitational wave forms and luminosity up to $v^8$ order beyond Newtonian, where $v=(M/r_0)^{1/2}$. In particular, we give the exact analytical values of the coefficients of $\ln v$ terms at $v^6$ and $v^8$ orders in the luminosity and confirm the numerical values obtained previously by the other of us (HT) and Nakamura. Our result is valid in the small mass limit of one body and gives an important guideline for the gravitational wave physics of coalescing compact binaries.