Radiated Energy Spectrum, Radiated Angular Distribution and Non-linear Memory from the One-loop Gravitational Bremsstrahlung Waveform

The frequency-domain gravitational waveform emitted by the scattering of two non-spinning massive particles has recently been derived at next-to-leading, $\mathit{\text{i.e.}}$ one-loop, post-Minkowskian order, $h(\omega ,\theta ,\phi )\sim G^2+G^3$. Building on this one-loop-accurate frequency-domain gravitational waveform, we successively derive the spectral gravitational-wave (GW) radiance, $d{E}^{\mathrm{g}\mathrm{w}}\mathrm{/}(d\omega d\mathrm{\Omega })$, the radiated GW energy spectrum, $d{E}^{\mathrm{g}\mathrm{w}}\mathrm{/}d\omega$, and the radiated GW angular distribution, $d{E}^{\mathrm{g}\mathrm{w}}\mathrm{/}d\mathrm{\Omega }$, up to order $G^4$ included. We deduce from the radiated angular distribution the multipole expansion of the non-linear memory up to order $G^5$ included, thereby extending previous results. We work in the center-of-mass frame, and our results reach the fractional 7.5PN accuracy. For completeness, we include the tree-level information (considered in the center-of-mass frame).