The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries

TL;DR

The paper tackles the high-precision modeling of the dominant gravitational-wave quadrupole mode for non-spinning compact binaries on quasi-circular orbits at 3.5PN order. Using the multipolar post-Newtonian framework, it constructs the radiative quadrupole $U_{ij}$ from instantaneous, tail, and memory pieces and expresses the waveform mode $h^{22}$ via $h^{22} = -rac{G}{ oot{2}{}} rac{U^{22}}{R}$, with the phase absorbed into a tail-canceling variable to manage logarithmic contributions. It provides explicit 3.5PN expressions for source moments, notably the quadrupole $I_{ij}$ including a new radiation-reaction term $ ext{C}_{RR}$, and combines them to yield the amplitude factor $H^{22}$ as a function of the invariant PN parameter $x$ and mass ratio $ u$, including tail and memory effects and logarithms. The results agree with black-hole perturbation theory in the test-mass limit and advance the construction of a complete 3.5PN waveform, enabling more accurate parameter estimation and sky localization for gravitational-wave detectors.

Abstract

We compute the quadrupole mode of the gravitational waveform of inspiralling compact binaries at the third and a half post-Newtonian (3.5PN) approximation of general relativity. The computation is performed using the multipolar post-Newtonian formalism, and restricted to binaries without spins moving on quasi-circular orbits. The new inputs mainly include the 3.5PN terms in the mass quadrupole moment of the source, and the control of required subdominant corrections to the contributions of hereditary integrals (tails and non-linear memory effect). The result is given in the form of the quadrupolar mode (2,2) in a spin-weighted spherical harmonic decomposition of the waveform, and may be used for comparison with the counterpart quantity computed in numerical relativity. It is a step towards the computation of the full 3.5PN waveform, whose knowledge is expected to reduce the errors on the location parameters of the source.

Figures (1)

• Figure 1: Definition of angles and polarization vectors. $\bm{N}$ points towards the observer, whose spherical coordinates are $(\Theta,\Phi)$, with $\Theta=\pi/2$ corresponding to the orbital plane of the binary; $\Theta$ is also the inclination of the orbit on the sky from the observer's viewpoint. We define then $\bm{P}=-\bm{e}_\Phi$ and $\bm{Q}=\bm{e}_\Theta$. The ($\bm{n}$, $\bm{\lambda}$, $\bm{\ell}$) triad is displayed, $\varphi$ being the orbital phase, and $\phi \equiv \varphi-\Phi+\pi/2$ being the orbital phase relative to the direction of the ascending node of the orbit, which is also the direction of $\bm{P}$.