Comparing one-loop gravitational bremsstrahlung amplitudes to the multipolar-post-Minkowskian waveform

Abstract

We compare recent one-loop-level, scattering-amplitude-based, computations of the classical part of the gravitational bremsstrahlung waveform to the frequency-domain version of the corresponding multipolar-post-Minkowskian waveform result. When referring the one-loop result to the classical averaged momenta ${\overline{p}}_{a}=\frac{1}{2}({p}_{a}+{p}_{a}^{\ensuremath{'}})$, 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, $O(\frac{{G}^{2}}{{c}^{5}})$, i.e., when reaching he following: (i) the first post-Newtonian correction to the linear quadrupole tail; (ii) Newtonian-level linear tails of higher multipolarity (odd octupole and even hexadecapole); (iii) radiation-reaction effects on the worldlines; and (iv) various contributions of cubically nonlinear origin (notably linked to the quadrupole $\ifmmode\times\else\texttimes\fi{}$ quadrupole $\ifmmode\times\else\texttimes\fi{}$ quadrupole coupling in the wave zone). These differences are reflected at the sub-sub-sub-leading level in the soft expansion, $\ensuremath{\sim}\ensuremath{\omega}\mathrm{ln}\ensuremath{\omega}$, i.e., $O(\frac{1}{{t}^{2}})$ in the time domain. Finally, we computed the first four terms of the low-frequency expansion of the multipolar-post-Minkowskian waveform and checked that they agree with the corresponding existing classical soft graviton results.

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