Second post-Newtonian gravitational wave polarizations for compact binaries in elliptical orbits

TL;DR

The paper derives the instantaneous second post-Newtonian ($2$PN) corrections to the gravitational-wave polarizations $h_+$ and $h_\times$ for non-spinning compact binaries in elliptical orbits using the $2$PN corrections to $h^{TT}_{ij}$ and the generalized quasi-Keplerian parametrization. It expresses the results in ADM coordinates and organizes the waveform with two angular variables via $\phi = \lambda + W(l)$, producing an expansion $(h_{+,\times})_{inst} = \frac{G m \eta}{c^2 R}\,\xi^{2/3}\,[H^{(0)} + \xi^{1/2}H^{(1/2)} + \xi H^{(1)} + \xi^{3/2}H^{(3/2)} + \xi^2 H^{(2)}]$ in terms of $\xi = \frac{G m n}{c^3}$ and eccentricity parameters. The circular limit agrees with earlier results (BIWW96) when $e_t\to0$, and the analysis shows how eccentricity, periastron precession, and inclination shape the spectral content of the Newtonian part, setting the groundwork for future inclusion of tail terms and radiation reaction to produce fully evolving, ready-to-use templates for eccentric binaries. The work provides a framework to generate and analyze gravitational-wave templates for quasi-elliptical inspirals, which is relevant for both ground-based detectors and LISA. It also establishes a method to transform harmonic-coordinate results to ADM coordinates consistently, enabling cross-comparisons with other PN formulations.

Abstract

The second post-Newtonian (2PN) contribution to the `plus' and `cross' gravitational wave polarizations associated with gravitational radiation from non-spinning, compact binaries moving in elliptic orbits is computed. The computation starts from our earlier results on 2PN generation, crucially employs the 2PN accurate generalized quasi-Keplerian parametrization of elliptic orbits by Damour, Schäfer and Wex and provides 2PN accurate expressions modulo the tail terms for gravitational wave polarizations incorporating effects of eccentricity and periastron precession.

Figures (10)

• Figure 1: The orientation of unit vectors, which defines $\times$ and $+$ polarization waveforms. The unit vectors p and q are the gravitational wave's principal axes with ${\bf q} =$N$\times$p. Note that N is a unit vector lying along the radial direction to the detector and p lies along the line of nodes. The Newtonian angular momentum vector ${\bf L } = \mu\, {\bf r} \,\times\,{\bf v}$ is normal to the orbital plane and helps to define orbital inclination angle $i$. In this paper, the origin for $\phi = \lambda +{\rm W}$ is $+$ve x-axis, hence it is related to $\phi_{\rm BIWW}$ by $\phi = \phi_{\rm BIWW} -{\pi \over 2}$.
• Figure 2: Plots for scaled GW polarization waveform, $H^{0}_{\times}$ as a function of the mean anomaly, $l$ and the corresponding normalized relative power spectrum using Newtonian orbital motion, for various values of eccentricity $e$. Note in $H^{0}_{\times}(l)$ a 'burst' of GW emission near periastron passage and a shift in the position of the dominant harmonic in the power spectrum as $e$ increases. In the Fourier domain, the former results in a broad frequency rich peak. In all panels, the (arbitrary) periastron precession constant and the orbital inclination angle take values $0.1$ and ${\pi \over 3}$ respectively.
• Figure 3: The configuration is similar to Fig.\ref{['fig:c_N_k_0.1']}, but in the panels, $k$, the (arbitrary) periastron precession constant is varied for fixed eccentricity $e=0.5$ and orbital inclination angle $i = {\pi \over 3}$. Note the splitting and shifting of spectral lines from integer multiple values of $f_r$ as $k$ is increased.
• Figure 4: Plots of scaled GW polarization waveform, $H^{0}_{+}$ as a function of mean anomaly, $l$ and its corresponding normalized relative power spectrum for Newtonian orbital motion, when the orbital inclination angle $i$ is varied. In all frames, eccentricity $e=0.5$ and periastron precession constant, $k=0.1$.
• Figure 5: The ratio of the total power measured in $\times$ and $+$ polarization for Newtonian motion as a function of orbital inclination angle $i$ for various values of periastron precession constant $k$ and eccentricity $e$. From the plots, it is clear that the ratio is independent of the orbital elements like $e$ and $k$.