General Fourier expansion of post-Newtonian binary dynamics based on a quasi-Keplerian framework

TL;DR

This work develops a unified Fourier-domain framework for nonspinning binary dynamics in the post-Newtonian regime by introducing elliptic-integral generalizations that represent PN Fourier modes and by providing a Fourier-series solution to the PN-Kepler equation up to $4\mathrm{PN}$. It computes the full set of $3\mathrm{PN}$ dynamical quantities and gravitational-wave modes, including detailed multipole-moment Fourier coefficients and waveforms, with all results available in a public code (pyPNFourier). A key advance is the resummation of tail contributions to the energy and angular-momentum fluxes into compact expressions under the adiabatic approximation, enabling more efficient period-averaged waveform modeling in frequency space. The framework is validated against post-circular results and is designed to scale to high eccentricities, offering a practical route toward accurate, frequency-domain templates for upcoming low-frequency GW detectors. Limitations include focus on nonspinning binaries and the high computational cost for large eccentricities, with future work aimed at spins, higher hereditary terms, and performance optimization.

Abstract

We have introduced a new method for computing gravitational-wave emission from nonspinning binaries which systematically unifies the various integrals arising in the Fourier expansions of post-Newtonian dynamics, providing a simple, practical scheme for calculations at arbitrary precision. Using this approach, we derived the full set of 3PN dynamical quantities and gravitational-wave Fourier modes and have released the corresponding numerical code as open source. Furthermore, when radiation-reaction effects is not included, we found that the tail contribution to the energy and angular momentum fluxes can be resummed into an exceptionally compact expression with the help of the new method. These advances pave the way for more convenient and accurate frequency-domain waveform modeling in the future.

Figures (3)

• Figure 1: Comparison of the leading-mode waveforms $h_{\ell\ell}$ (first four multipoles up to 3PN) computed with the Fourier-mode expansion and the post-circular expansion. In this example the binary has mass ratio $\nu=0.22$, initial frequency $v_0=0.25$, and the evolution ends at $v=1/\sqrt{6}$. The initial eccentricity is $e_0=0.3$. The black curve corresponds to the Fourier-mode approach presented in this paper, obtained by summing $H_{\ell m}=\sum_p H_{\ell(-m)p}\mathrm{e}^{\mathrm{i} p l}$ until the required accuracy is reached; the red curve shows the post-circular waveform truncated at $\order{e^{10}}$.
• Figure 2: Comparison of the leading-mode waveforms $h_{\ell\ell}$ (first four multipoles up to 3PN) computed with the Fourier-mode expansion and the post-circular expansion. In this example the binary has mass ratio $\nu=0.22$, initial frequency $v_0=0.25$, and the evolution ends at $v=1/\sqrt{6}$. The initial eccentricity is $e_0=0.4$. The black curve corresponds to the Fourier-mode approach presented in this paper, obtained by summing $H_{\ell m}=\sum_p H_{\ell(-m)p}\mathrm{e}^{\mathrm{i} p l}$ until the required accuracy is reached; the red curve shows the post-circular waveform truncated at $\order{e^{10}}$.
• Figure 3: Comparison of the leading-mode waveforms $h_{\ell\ell}$ (first four multipoles up to 3PN) computed with the Fourier-mode expansion and the post-circular expansion. In this example the binary has mass ratio $\nu=0.22$, initial frequency $v_0=0.25$, and the evolution ends at $v=1/\sqrt{6}$. The initial eccentricity is $e_0=0.5$. The black curve corresponds to the Fourier-mode approach presented in this paper, obtained by summing $H_{\ell m}=\sum_p H_{\ell(-m)p}\mathrm{e}^{\mathrm{i} p l}$ until the required accuracy is reached; the red curve shows the post-circular waveform truncated at $\order{e^{10}}$.