We study the [helicity-($-2$)] quadrupolar part, $U_2=\frac{1}{2!}{\overline{m}}^i{\overline{m}}^j{U}_{ij}$, of the gravitational waveform $W\equiv \frac{R}{4G}{\overline{m}}^i{\overline{m}}^j{h}_{ij}$ emitted during the scattering of two masses. Working within the multipolar post-Minkowskian (MPM) formalism, we compute the time-domain value of $U_2$ at the third-and-a-half post-Newtonian (3.5PN) accuracy by using the 3.5PN radiation-reacted quasi-Keplerian representation of the hyperbolic motion. We then explicitly evaluate the frequency-domain value of $U_2$ up to the two-loop level, i.e., $O(G^4)$ contributions to $h_{ij}(\omega ,\theta ,\varphi )$, corresponding to $O(G^3)$ contributions to ${\widehat{U}}_2(\omega ,\theta ,\varphi )$. The nonlinear memory contribution to the waveform in the center-of-mass frame is computed too and checked against the soft-limit of the waveform. The one-loop truncation of our 3.5PN frequency-domain MPM waveform is found to agree with corresponding existing effective field theory (EFT) results when subtracting the dipolar part of the Veneziano-Vilkovisky supertranslation connecting the MPM and EFT Bondi-Metzner-Sachs (BMS) frames.

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• General relativity