Analytical approaches for rapid prediction of gravitational waveforms for relativistic binary systems

TL;DR

This work tackles rapid prediction of gravitational-waveforms for relativistic binary systems by deriving a fully analytical evolution for the early inspiral. By recasting the PN evolution as $dx/dt = -F/(dE/dx)$ and performing a separable, analytic treatment of $I_A(x)=\int_{x_{low}}^x d\zeta/A(\zeta)$, the authors obtain an implicit $x(t)$ and an accompanying analytic phase and amplitude template for the GW signal. They anchor the method with physically motivated bounds ($x_{low}$ from detector sensitivity and $x_{high}$ from ISCO) and show how to construct the complete analytical waveform $h(t)$ from $r(x)$, $\dot r(x)$, and $\omega(x)$ with a consistent PN framework up to 4PN/4.5PN. Validation against numerical results demonstrates sub-percent accuracy for BH-BH and BH-NS, and excellent accuracy ($<10^{-4}\%$) for NS-NS over most of the pre-merger interval, enabling early monitoring times of hundreds of seconds and facilitating rapid detection and multi-messenger alerts.

Abstract

We present a fast method for obtaining fully analytical approximations for gravitational waveforms produced by merging of neutron stars and/or black holes for the earliest stages of the merger process. The obtained analytical formula is compared with numerical calculations, its accuracy and limits of applicability are evaluated. Our results may be useful not only for the earliest evalution of properties of the nature of binary system in gravitational-wave detectors but also will give early alerts for gamma-ray, optical and neutrino observatories.

Figures (6)

• Figure 1: Evolution of the PN parameter over time for various BH systems
• Figure 2: Evolution of the PN parameter over time for various NS systems
• Figure 3: GW form for the BH system $(20+20) M_\odot$
• Figure 4: Comparison of the numerical and analytical form of the GW for the BH system $(20+20)M_\odot$ half a second before the merger
• Figure 5: Comparison of the numerical and analytical form of the GW in the last seconds before the merger for the system $(1.5+1.5)M\odot$ (for this system, the critical time is $t_{cr}=905.9$ s --- see Fig. \ref{['pn-ns']})