  With the aim of computing bound waveforms from scattering amplitudes, we
explore gravitational two-body dynamics using the Schwinger-Dyson equations and
Bethe-Salpeter recursion. We show that the tree-level scattering waveform
admits a natural analytic continuation, in rapidity, to the bound waveform,
which we confirm from an independent calculation, in the Post-Newtonian
expansion, of the time-domain multipoles at large eccentricity. We demonstrate
consistency of this scattering-to-bound map with the Damour-Deruelle
prescription for orbital elements in the quasi-Keplerian parametrization (which
enters into the evaluation of the multipoles) and with the analytic
continuation, in the binding energy, of radiated energy and angular momentum at
3PM.
