Gravitational Bremsstrahlung Waveform at the fourth Post-Minkowskian order and the second Post-Newtonian level

TL;DR

This paper calculates the gravitational waveform for scattering of two nonspinning bodies at high precision, combining post-Minkowskian and post-Newtonian methods, and provides explicit formulas for radiative multipoles and energy emission.

Contribution

It presents the first computation of the gravitational waveform at the fourth post-Minkowskian order and second post-Newtonian level, with explicit expressions for radiative multipoles and energy flux.

Findings

Waveform computed at $O(G^4)$ and $O(v^4/c^4)$ accuracy.

Explicit expressions for radiative multipoles in terms of master integrals.

Extended previous results with $O(G^3)$ spectral densities of radiated energy and momentum.

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, $U_{\ell m}(\omega), V_{\ell m}(\omega)$, 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 againg 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.