Modeling Gravitational Wave Modes from the Inspiral of Binaries with Arbitrary Eccentricity

TL;DR

This work develops a $1$PN post-Newtonian framework to compute Fourier amplitudes of gravitational waves from eccentric binary inspirals across all $(l,m)$ modes, valid for arbitrary eccentricities. It provides closed-form expressions for the Fourier-mode coefficients $N_p^{lm}$ by leveraging a quasi-Keplerian parametrization and Bessel-function techniques, and analyzes each mode's contribution, mean frequency, and frequency spread. A key contribution is a practical, mode-truncation scheme that minimizes the number of Fourier modes needed to reconstruct the signal within a specified tolerance, improving computational efficiency for waveform generation. The authors also outline extensions to higher PN orders, including tail and spin effects, and discuss how their framework can be expanded to deliver more accurate eccentric waveform models for current and future gravitational-wave detectors.

Abstract

Eccentric binaries are key targets for current and future gravitational wave (GW) detectors, offering unique insights into the formation and environments of compact binaries. However, accurately and efficiently modeling eccentric waveforms remains challenging, in part due to their complex harmonic structure. In this work, we develop a post-Newtonian (PN) framework to compute the Fourier amplitudes of GWs from the inspiral of eccentric binaries, deriving simple expressions at 1PN order for all relevant $(l, m)$ multipoles, valid for arbitrary eccentricities. We then characterize the GW emission by analyzing the contribution of each $(l, m)$ mode to the strain, its mean frequency, frequency spread, and asymptotic behavior at high frequencies. Additionally, we introduce a method to determine the minimal set of Fourier modes needed to reconstruct the waveform to a given accuracy. Finally, we discuss how our framework can be extended to higher PN orders, obtaining closed-form expressions for the leading-order tail and spin contributions and outlining the steps required to include higher-order corrections. Our results provide both a deeper theoretical understanding of eccentric GW emission and practical tools for developing more accurate and efficient waveform models.

Figures (7)

• Figure 1: Absolute value of the 1PN Fourier mode amplitudes, $N^{l m}_{p}$, as a function of $p$. Each panel shows $N^{l m}_{p}$ for a different value of the eccentricity $e$, with fixed PN parameter $y=0.2$ and mass ratio $q=m_2/m_1=0.1$. To compute the plotted $N^{l m}_{p}$ we have used Eq. \ref{['eq:Np']}.
• Figure 2: Norm of each 1PN GW mode, $\Vert \hat{H}^{l m}\Vert^2$, as a function of eccentricity $e$. Each panel shows $\Vert \hat{H}^{l m}\Vert^2$ for specific values of the PN parameter $y$ and mass ratio $q = m_2/m_1$. To compute $\Vert \hat{H}^{l m}\Vert^2$ we have used Eq. \ref{['eq:ezNorms']}.
• Figure 3: Average, $\mu^{l m}$ (top panel), and standard deviation, $\sigma^{l m}$ (bottom panel), of $p$ for each GW mode as a function of eccentricity $e$, for a fixed value of the PN parameter ($y=0.2$) and mass ratio $(q = m_2/m_1 = 0.1)$. To compute $\mu^{l m}$ and $\sigma^{l m}$, we have used the 1PN moments of Eqs. (\ref{['eq:ezNorms']},\ref{['eq:ezM1']},\ref{['eq:ezM2']}) to evaluate Eq. \ref{['eq:mus_M']}.
• Figure 4: Guess for the number of Fourier modes that have to be included for each GW mode multiplied by the square root of the tolerance, $n_{l m}^\mathrm{guess} \sqrt{\epsilon_N}$, as a function of eccentricity $e$. Each panel shows $n_{l m}^\mathrm{guess}\sqrt{\epsilon_N}$ for values of the PN parameter $y$ and mass ratio $q = m_2/m_1$ matching the configuration in Fig. \ref{['fig:norms_Hlm']}. To compute $n_{l m}^\mathrm{guess} \sqrt{\epsilon_N}$, we have used the 1PN moments of Eqs. (\ref{['eq:ezNorms']},\ref{['eq:ezM1']},\ref{['eq:ezM2']}) to evaluate Eq. \ref{['eq:nlm_guess']}.
• Figure 5: Optimal number of Fourier modes $\mathrm{len}(\bm{p}_{l m}^\mathrm{sel})$ needed to represent each 1PN GW mode as a function of eccentricity $e$. Each panel shows $\mathrm{len}(\bm{p}_{l m}^\mathrm{sel})$ for values of the PN parameter $y$ and mass ratio $q = m_2/m_1$ matching the configuration in Fig. \ref{['fig:norms_Hlm']}. We use an amplitude tolerance of $\epsilon_N = 10^{-4}$, typical in data analysis applications Morras:2025nlp. To find $\bm{p}_{l m}^\mathrm{sel}$ as described around Eq. \ref{['eq:strain_error_ineq']}, we compute the Fourier mode amplitudes $N^{l m}_p$ with Eq. \ref{['eq:Np']} and the 1PN norms of the GW modes with Eq. \ref{['eq:ezNorms']}, except for the $(2,2)$ mode, for which we use Eq. \ref{['eq:normH22']}.