Tail effects in the third post-Newtonian gravitational wave energy flux of compact binaries in quasi-elliptical orbits

TL;DR

The paper develops a frequency-domain, semi-analytical method to compute hereditary tail contributions to the 3PN gravitational-wave energy flux for compact binaries on quasi-elliptical orbits, leveraging a 1PN quasi-Keplerian orbital representation and the doubly periodic structure of the motion. It provides a systematic Fourier framework for the multipole moments, derives Newtonian and 1PN tail contributions (including tail-of-tail and tail-squared terms), and introduces eccentricity-enhancement functions that are evaluated numerically. By combining these hereditary results with the instantaneous terms (from the companion paper), it yields a complete 3PN energy flux for eccentric binaries and establishes a path toward ready-to-use templates for detectors like LIGO/Virgo and LISA. The work also outlines a robust numerical procedure for computing the necessary Fourier coefficients and highlights future directions, including extensions to angular momentum and higher PN orders.

Abstract

The far-zone flux of energy contains hereditary (tail) contributions that depend on the entire past history of the source. Using the multipolar post-Minkowskian wave generation formalism, we propose and implement a semi-analytical method in the frequency domain to compute these contributions from the inspiral phase of a binary system of compact objects moving in quasi-elliptical orbits up to 3PN order. The method explicitly uses the quasi-Keplerian representation of elliptical orbits at 1PN order and exploits the doubly periodic nature of the motion to average the 3PN fluxes over the binary's orbit. Together with the instantaneous (non-tail) contributions evaluated in a companion paper, it provides crucial inputs for the construction of ready-to-use templates for compact binaries moving on quasi-elliptic orbits, an interesting class of sources for the ground based gravitational wave detectors such as LIGO and Virgo as well as space based detectors like LISA.

Figures (6)

• Figure 1: Variation of $\varphi(e)$ with the eccentricity $e$. The function $\varphi(e)$ agrees with the numerical calculation of Ref. BS93 modulo a trivial rescaling with $f(e)$. The inset graph is a zoom of the function (which looks like a straight horizontal line in the main graph) at a smaller scale. The dots represent the numerical computation and the solid line is a fit to the numerical points. In the circular orbit limit we have $\varphi(0)=1$.
• Figure 2: Variation of $\beta(e)$ (left panel) and $\gamma(e)$ (right panel) with the eccentricity $e$. In the circular orbit limit we have $\beta(0)=\gamma(0)=1$.
• Figure 3: Variation of $\chi(e)$ (left panel) and $F(e)$ (right panel) with the eccentricity $e$. In the right panel, the exact expression of $F(e)$ given by Eq. \ref{['Fe']} is used. In the circular orbit limit we have $\chi(0)=0$ and $F(0)=1$.
• Figure 4: Variation of $\alpha(e)$ (left panel) and $\theta(e)$ (right panel) with the eccentricity $e$. In the circular orbit limit we have $\alpha(0)=\theta(0)=1$.
• Figure 5: Variation of $\psi(e)$ (left panel) and $\zeta(e)$ (right panel) with the eccentricity $e$. The inset graph is a zoom of the function (which looks like a straight horizontal line in the main graph) at a smaller scale. The dots represent the numerical computation and the solid line a fit to the numerical points. In the circular orbit limit we have $\psi(0)=\zeta(0)=1$.