Comparing One-loop Gravitational Bremsstrahlung Amplitudes to the Multipolar-Post-Minkowskian Waveform

We compare recent one-loop-level, scattering-amplitude-based, computations of the classical partof the gravitational bremsstrahlung waveform to the frequency-domain version of the corresponding Multipolar-Post-Minkowskianwaveform result. When referring the one-loop result to the classical averaged momenta $\bar p_a = \frac12 (p_a+p'_a)$,the two waveforms are found to agree at the Newtonian and first post-Newtonian levels,as well as at the first-and-a-half post-Newtonian level, i.e. for the leading-order quadrupolar tail.However, we find that there are significant differences at the second-and-a-half post-Newtonian level,$O\left( \frac{G^2}{c^5} \right)$, i.e.when reaching: (i) the first post-Newtonian correction to the linear quadrupole tail; (ii) Newtonian-level linear tailsof higher multipolarity (odd octupole and even hexadecapole); (iii) radiation-reaction effects on the worldlines;and (iv) various contributions of cubically nonlinear origin (notably linked tothe quadrupole$\times$ quadrupole$\times$ quadrupole coupling in the wavezone).These differences are reflected at the sub-sub-sub-leading level in the soft expansion, $ \sim \om \ln \om $, i.e. $O\left(\frac{1}{t^2} \right)$in the time domain.Finally, we computed the first four terms of the low-frequency expansion of the Multipolar-Post-Minkowskian waveform and checkedthat they agree with the corresponding existing classical soft graviton results.