Phasing of gravitational waves from inspiralling eccentric binaries at the third-and-a-half post-Newtonian order

TL;DR

This work addresses the challenge of accurately phasing gravitational waves from inspiralling eccentric binaries at $3.5$PN order. It combines a $3$PN conservative generalized quasi-Keplerian parametrization with $1$PN reactive corrections (i.e., $2.5$PN and $3.5$PN) using an improved variation-of-constants method and a two-scale decomposition to separate secular drift from fast orbital oscillations. The authors derive explicit expressions for the secular and periodic variations of key orbital elements, enabling nearly analytic templates for the GW polarizations $h_{+}$ and $h_{ imes}$ in harmonic coordinates, suitable for both ground- and space-based detectors. They also demonstrate the expected qualitative behavior—adiabatic chirp, periastron precession, and periodic modulations—in representative visualizations, and discuss extensions to include spins, amplitude corrections, and EMRI regimes for future work.

Abstract

We obtain an efficient description for the dynamics of nonspinning compact binaries moving in inspiralling eccentric orbits to implement the phasing of gravitational waves from such binaries at the 3.5 post-Newtonian (PN) order. Our computation heavily depends on the phasing formalism, presented in [T. Damour, A. Gopakumar, and B. R. Iyer, Phys. Rev. D \textbf{70}, 064028 (2004)], and the 3PN accurate generalized quasi-Keplerian parametric solution to the conservative dynamics of nonspinning compact binaries moving in eccentric orbits, available in [R.-M. Memmesheimer, A. Gopakumar, and G. Schäfer, Phys. Rev. D \textbf{70}, 104011 (2004)]. The gravitational-wave (GW) polarizations $h_{+}$ and $h_{\times}$ with 3.5PN accurate phasing should be useful for the earth-based GW interferometers, current and advanced, if they plan to search for gravitational waves from inspiralling eccentric binaries. Our results will be required to do \emph{astrophysics} with the proposed space-based GW interferometers like LISA, BBO, and DECIGO.

Figures (2)

• Figure 1: The plots for $\bar{n} / n_{i}$ and $\tilde{n} / n$ versus $l / (2 \pi)$, which gives the number of orbital revolutions. The adiabatic increase of $\bar{n}$ is clearly visible in panel 1, and the quasi-periodic nature of the variations in $\tilde{n}$ is portrayed in panels 2--6. These variations are governed by the reactive 2.5PN and 3.5PN equations of motion. In the second and third row, these contributions to $\tilde{n}$ are plotted individually and separated for the initial and final stages. The parameters $e_t^i$ and $e_t^f$ denote initial and final values of the time eccentricity $e_t$, while $\xi^i$ and $\xi^f$ stand for similar values of the adimensional mean motion $\xi = G M n / c^3$. The panels are plotted for $\eta = 0.25$ and the orbital evolution is terminated when $j = \sqrt{48}$.
• Figure 2: The plots for the scaled $h_{+}(t)$ and $h_{\times}(t)$ (Newtonian in amplitude and 3.5PN in orbital motion) as functions of $l / (2 \pi)$. The slow chirping and the amplitude modulation due to the periastron precession are clearly visible in the two upper panels. In the two bottom panels, we zoom into the initial stages of the orbital evolution in order to show the effect of the periodic orbital motion and the periastron advance on the scaled $h_{+}(t)$ and $h_{\times}(t)$. The initial and final values of the relevant orbital elements are marked on top of the plots. The panels are plotted for a binary consisting of equal masses, so that $\eta = 0.25$, and the orbital inclination angle is given by $i = \pi / 3$. The orbital evolution is terminated when $j = \sqrt{48}$.