High post-Minkowskian gravitational waveform for hyperbolic encounters in the extreme-mass-ratio limit

The frequency-domain waveform emitted by a two-body scattering process is computed in the extreme-mass-ratio limit through the fifth post-Minkowskian (PM) order (i.e., $O(G^5)$) and the fractional sixth post-Newtonian (PN) order. The current accuracy of the scattering waveform obtained by quantum amplitude methods is the one-loop level corresponding to the 3PM order, whereas the 4PM waveform is known up to the 2PN order only as derived within the traditional multipolar-post-Minkowskian formalism. Direct comparison between these waveforms to the first order in the mass ratio shows that they differ at most by the effect of an angular-independent time shift, leading to a complete physical agreement at the same level of accuracy. The new results at the 4PM and 5PM orders thus provide a benchmark for future multiloop calculations. The soft limit of the waveform at the leading order in the small-frequency expansion gives the gravitational wave memory, which is evaluated at the 4PM and 5PM orders. The waveform is also used to obtain the radiated energy at $O(G^6)$, improving its knowledge from 3PN to 6PN level.