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

Extending our recent work (which focussed on the even-parity quadrupolar part of the waveform), we compute the even-parity octupolar contribution, U₃, to the gravitational waveform W ≡ c⁴ r/4G mⁱ mʲ hi 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 U₃(ω, θ,ϕ) of U₃ up to the 2-loop level, i.e. O(G⁴) contributions to hᵢⱼ(ω, θ,ϕ), corresponding to O(G³) contributions to U₃(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 { 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.