Octupolar bremsstrahlung waveform up to the two-loop level and the third-and-a-half post-Newtonian accuracy

Abstract: Extending our recent work (which focussed on the even-parity quadrupolar part of the waveform), we compute the even-parity octupolar contribution, $U_3$, to the gravitational waveform $$W \equiv \frac{c⁴ r}{4G} \bar m^{i} \bar m^{j } h_{i j}$$ emitted during the scattering of two masses. We work within the Multipolar Post-Minkowskian (MPM) formalism, and use the 3.5 Post-Newtonian (PN) accurate radiation-reacted quasi-Keplerian representation of the hyperbolic motion. We explicitly evaluate the frequency-domain value $\hat U_3(ω, θ,φ)$ of $U_3$ up to the 2-loop level, i.e. $O(G<sup>4)$ contributions to $h_{ij}(ω, θ,φ)$, corresponding to $O(G<sup>3)$ contributions to $\hat U_3(ω, θ,φ)$. As a crucial partial confirmation of our result, we find that the 1-loop truncation of our 3.5 PN frequency-domain MPM waveform agrees with corresponding existing Effective Field Theory (EFT) results when taking into account exactly the {\it same} (2.5PN-level) difference in the definitions of the center-of-mass origins within the two formalisms that was deduced from our previous quadrupolar comparison.