Gravitational radiation from compact binary systems: gravitational waveforms and energy loss to second post-Newtonian order

TL;DR

<3-5 sentence high-level summary>The paper develops a finite, self-consistent Epstein-Wagoner framework to compute gravitational radiation from inspiralling compact binaries to second post-Newtonian order, including tail effects, by solving the relaxed Einstein equations with a split into near-zone multipole moments and a convergent radiation-zone integral. It shows how to combine near-zone EW moments with a carefully treated radiation-zone contribution to produce a fully finite waveform $h_{TT}^{ij}$ and energy flux $rac{dE}{dt}$ that agree with independent post-Minkowskian/post-Newtonian results and correctly incorporate tails that propagate along null directions of the curved spacetime. The authors derive explicit two-body, quasi-circular, and spinning-body waveform templates, including spin-orbit and spin-spin contributions, and provide ready-to-use expressions suitable for gravitational-wave data-analysis, including phase evolution and tail-induced phase shifts. This framework paves the way for higher PN orders and improved interfaces with numerical relativity for late-inspiral dynamics, ultimately aiding precise parameter estimation in detectors like LIGO/VIRGO.

Abstract

We derive the gravitational waveform and gravitational-wave energy flux generated by a binary star system of compact objects (neutron stars or black holes), accurate through second post-Newtonian order ($O[(v/c)^4] \sim O[(Gm/rc^2)^2]$) beyond the lowest-order quadrupole approximation. We cast the Einstein equations into the form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve it formally as a retarded integral over the past null cone of the chosen field point. The part of this integral that involves the matter sources and the near-zone gravitational field is evaluated in terms of multipole moments using standard techniques; the remainder of the retarded integral, extending over the radiation zone, is evaluated in a novel way. The result is a manifestly convergent and finite procedure for calculating gravitational radiation to arbitrary orders in a post-Newtonian expansion. Through second post-Newtonian order, the radiation is also shown to propagate toward the observer along true null rays of the asymptotically Schwarzschild spacetime, despite having been derived using flat spacetime wave equations. The method cures defects that plagued previous ``brute- force'' slow-motion approaches to the generation of gravitational radiation, and yields results that agree perfectly with those recently obtained by a mixed post-Minkowskian post-Newtonian method. We display explicit formulae for the gravitational waveform and the energy flux for two-body systems, both in arbitrary orbits and in circular orbits. In an appendix, we extend the formalism to bodies with finite spatial extent, and derive the spin corrections to the waveform and energy loss.

Figures (10)

• Figure 1: Past harmonic null cone $\cal C$ of the field point $(t,{\bf x})$ intersects the near zone $\cal D$ in the hypersurface $\cal N$.
• Figure 2: Same as Fig. 1, for field point inside the near zone.
• Figure 3: Taylor expansion of retarded time dependence on $\cal N$ results in multipole moments integrated over the spatial hypersurface $\cal M$.
• Figure 4: Two-dimensional hypersurfaces $\cal F$ formed by intersection of past null cone of field point with future null cones from the origin. Field point is in radiation zone. For $u^\prime$ from $-\infty$ to $u-2{\cal R}$, F covers full $4\pi$ solid angle around the origin. From $u-2{\cal R}$ to $u$, F terminates at boundary of the near zone $\cal N$.
• Figure 5: Same as Fig. 4, for field point in near zone. Integral over $u^\prime$ terminates at $u^\prime=u-2{\cal R}+2r$.