Quadrupolar bremsstrahlung waveform at the third-and-a-half post-Newtonian accuracy

Abstract

We study the quadrupolar part of the gravitational waveform $h_{ij}$ (encoded in the helicity-($-2)$ radiative quadrupole moment $U_2=\frac{1}{2!}{\bar{m}}^i{\bar{m}}^j{U}_{ij}\in \frac{R}{4G}{\bar{m}}^i{\bar{m}}^j{h}_{ij}\equiv W$) 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 {\it frequency-domain} value of $U_2$ up to the 2-loop level, i.e. $O(G^4)$ contributions to $h_{ij}(\ddot{I}‰,\hat{I}¸,\ddot{I}•)$, corresponding to $O(G^3)$ contributions to ${\hat{U}}_2(\ddot{I}‰,\hat{I}¸,\ddot{I}•)$. 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 1-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.

31 pages