Gravitational bremsstrahlung waveform at the fourth post-Minkowskian order and the second post-Newtonian level

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(G4), or two-loop order), and at the fractional second post-Newtonian accuracy [O(v4/c4)]. The waveform is decomposed in spin-weighted spherical harmonics and the needed radiative multipoles, Um(ω),Vm(ω), 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(G2) bremsstrahlung waveform by computing the O(G3) spectral densities of radiated energy and momentum, in the rest frame of one body, at the thirtieth order in velocity.