Gravitational waves from compact binaries in post-Newtonian accurate hyperbolic orbits

TL;DR

The paper tackles gravitational-wave modeling from compact binaries in hyperbolic orbits within post-Newtonian gravity, delivering a $3PN$-accurate Keplerian-type parametrization for conservative dynamics in both ADM and modified harmonic coordinates. It validates the hyperbolic solution via analytic continuation from eccentric counterparts and constructs $3.5PN$-accurate GW polarization templates by incorporating radiation reaction and solving a PN Kepler equation with Mikkola’s method. The resulting framework provides ready-to-use time-domain waveforms $h_+(t)$ and $h_\times(t)$ for hyperbolic encounters, capturing relativistic effects such as periastron advance and memory. These templates enable searches for hyperbolic GW bursts in ground-based detectors and inform connections to possible electromagnetic counterparts in neutron-star encounters.

Abstract

We derive from first principles third post-Newtonian (3PN) accurate Keplerian-type parametric solution to describe PN-accurate dynamics of non-spinning compact binaries in hyperbolic orbits. Orbital elements and functions of the parametric solution are obtained in terms of the conserved orbital energy and angular momentum in both Arnowitt-Deser-Misner type and modified harmonic coordinates. Elegant checks are provided that include a modified analytic continuation prescription to obtain our independent hyperbolic parametric solution from its eccentric version. A prescription to model gravitational wave polarization states for hyperbolic compact binaries experiencing 3.5PN-accurate orbital motion is presented that employs our 3PN-accurate parametric solution.

Figures (2)

• Figure 1: Scaled $H_+|_Q(l)$ and $H_{\times}|_Q(l)$ plots for non-spinning compact binaries with total mass $m= 20\, M_{\odot}$ and mass ratio $q=1$. We let the eccentricity $e_{t}$ take three values $1.5$, $1.3$ and $1.2$, while choosing an impact parameter $b \sim 30 \,Gm/c^2$ and inclination angle $\theta=\frac{\pi}{4}$. We observe the expected linear memory effect in the cross polarization state.
• Figure 2: Trajectories and the associated scaled $H_{\times}|_Q(l)$ for hyperbolic compact binaries, with a choice of two different impact parameters $b$, eccentricity $e_t=1.1$, total mass $m=20 M_{\odot}$, mass ratio $q=1$, and inclination angle $\theta=\frac{\pi}{4}$. For the trajectories, we adopt the geometric unit system. Newtonian and 3.5PN-accurate hyperbolic orbits are denoted by black and red lines, respectively. The orbital trajectory of the relativistic system is clearly different, especially for hyperbolic passages with smaller $b$ values, which is attributed to the advance of periastron. Relativistic effects also change the nature of the waveforms, as evident from the associated $h_{\times}|_Q(l)$ plots.