Gravitational bremsstrahlung waveform at the fourth post-Minkowskian order and the second post-Newtonian level
Donato Bini, Thibault Damour, Andrea Geralico
Abstract
Using the multipolar post-Minkowskian formalism, we compute the frequency-domain waveform generated by the gravitational scattering of two nonspinning bodies at the fourth post-Minkowskian order ($O({G}^{4})$, or two-loop order), and at the fractional second post-Newtonian accuracy [$O({v}^{4}/{c}^{4})$]. The waveform is decomposed in spin-weighted spherical harmonics and the needed radiative multipoles, ${U}_{\ensuremath{\ell}m}(\ensuremath{\omega}),{V}_{\ensuremath{\ell}m}(\ensuremath{\omega})$, are explicitly expressed in terms of a small number of master integrals. The basis of master integrals contains both (modified) Bessel functions, and solutions of inhomogeneous Bessel equations with Bessel-function sources. We show how to express the latter in terms of Meijer G functions. The low-frequency expansion of our results is checked against existing classical soft theorems. We also complete our previous results on the $O({G}^{2})$ bremsstrahlung waveform by computing the $O({G}^{3})$ spectral densities of radiated energy and momentum, in the rest frame of one body, at the thirtieth order in velocity.