Phasing of gravitational waves from inspiralling eccentric binaries

TL;DR

This work develops a formalism to generate high-precision gravitational-wave templates for inspiralling binaries on eccentric orbits by merging three time scales through a refined variation-of-constants approach, implemented at 2.5PN reactive order. It employs a generalized quasi-Keplerian representation in ADM coordinates, with a two-scale decomposition into secular and periodic variations of orbital elements, yielding both a secular phasing consistent with Peters-type results and smaller post-adiabatic periodic corrections to the waveforms $h_{+}$ and $h_{ imes}$. The method produces Newtonian-amplitude waveforms with 2.5PN-accurate phasing and provides a framework for extending to higher PN orders, spins, and EOB-based near-LSO dynamics, making it relevant for the LIGO/Virgo/GEO network and future detectors like LISA. Overall, the paper delivers a principled, analytically tractable approach to eccentric-binary GW phasing that captures essential secular and periodic effects beyond the traditional adiabatic approximation, with clear pathways for enhancements and applications.

Abstract

We provide a method for analytically constructing high-accuracy templates for the gravitational wave signals emitted by compact binaries moving in inspiralling eccentric orbits. By contrast to the simpler problem of modeling the gravitational wave signals emitted by inspiralling {\it circular} orbits, which contain only two different time scales, namely those associated with the orbital motion and the radiation reaction, the case of {\it inspiralling eccentric} orbits involves {\it three different time scales}: orbital period, periastron precession and radiation-reaction time scales. By using an improved `method of variation of constants', we show how to combine these three time scales, without making the usual approximation of treating the radiative time scale as an adiabatic process. We explicitly implement our method at the 2.5PN post-Newtonian accuracy. Our final results can be viewed as computing new `post-adiabatic' short period contributions to the orbital phasing, or equivalently, new short-period contributions to the gravitational wave polarizations, $h_{+,\times}$, that should be explicitly added to the `post-Newtonian' expansion for $h_{+,\times}$, if one treats radiative effects on the orbital phasing of the latter in the usual adiabatic approximation. Our results should be of importance both for the LIGO/VIRGO/GEO network of ground based interferometric gravitational wave detectors (especially if Kozai oscillations turn out to be significant in globular cluster triplets), and for the future space-based interferometer LISA.

Figures (6)

• Figure 1: The 2PN accurate effective radial potential $W_j (u)$ as a function of the dimensionless radial variable $\frac{1}{u} =\frac{c^2\,R}{G\,m}$, for various values of the dimensionless angular momentum $j$. The point, marked c, denotes a stable circular orbit, while the line, noted e, stands for a precessing elliptical orbit. The line with label p denotes an elliptical orbit which is about to plunge. Note that the left end of the line p is tangent to the effective potential, and corresponds to an unstable circular orbit. The plots are for $\eta =0.25$.
• Figure 2: The plots for $\bar{n}/n_i, \tilde{n} /n, \bar{e}_t$ and $\tilde{e}_t$ versus $\frac{l}{2\,\pi}$, which gives the number of orbital revolutions. These variations are governed by the reactive 2.5PN equations of motion. Periodic nature of the variations in $\tilde{n}$ and $\tilde{e}_t$ are clearly visible. $e_t^i$ and $e_t^f$ denote initial and final values for the time eccentricity $e_t$, while $\xi^i$ and $\xi^f$ stand for similar values of the adimensional mean motion $\frac{G\,m\, n}{c^3}$. The plots are for $\eta=0.25$ and the evolution is terminated when $j^2=48$.
• Figure 3: The plots $\bar{c}_l,\tilde{c}_l, \bar{c}_\lambda$ and $\tilde{c}_\lambda$ against orbital cycles, given by $\frac{l}{2\,\pi}$. Similar to Fig. \ref{['fig:n_e_BT']}, these variations are governed by the reactive 2.5PN equations of motion. Periodic nature of the variations in $\tilde{c}_l$ and $\tilde{c}_\lambda$ as well as the constancy of $\bar{c}_l$ and $\bar{c}_\lambda$ are clearly visible. The symbols have the same meaning as in Fig \ref{['fig:n_e_BT']}.
• Figure 4: The plots showing scaled time derivative of $l$ and $\lambda -l$ as function of orbital cycles, given by $\frac{l}{2\,\pi}$. The right panels show clearly both the secular drift and the periodic oscillations of the plotted quantities. Similar to earlier figures, these variations are governed by the reactive 2.5PN equations of motion. The initial and final values of the relevant orbital elements are marked on the plots. The plots are for $\eta=0.25$ and $n_i$ is the initial value of the mean motion $n$.
• Figure 5: The scaled $h_+$ ( Newtonian in amplitude and 2.5PN in orbital motion) plotted against orbital cycles, given by $\frac{l}{2\,\pi}$. The 'chirping', amplitude modulation due to periastron precession are clearly visible in the upper panel. In the bottom panel, we zoom into the initial stages of orbital evolution to show the effect of the periodic orbital motion and the periastron advance on the scaled $h_+(t)$. The initial and final values of the relevant orbital elements are marked on the plots. The plots are for $\eta=0.25$ and the orbital inclination is $i= \frac{\pi}{3}$.