Post-Minkowskian expansion of the Prompt Response in a Schwarzschild background

We study the early-time component of the Green's function of a Schwarzschild black hole, traveling on the curved light cone and usually denoted as the prompt response. Working in a post-Minkowskian approximation, we show for the first time that the prompt response is given by the residue of poles at $ω=0$ present in the complex Fourier domain. The contribution of the high-frequency arcs, previously assumed to generate the prompt response, vanishes. The analytical expression of the prompt response in this scheme is a polynomial of order $\ell$ in the observer's retarded time, with $\ell$ the multipole number. We validate the model against numerical predictions, obtaining good agreement for a compact source far from the black hole. We provide a phenomenologically-corrected expression to improve the match as the source is moved closer. We investigate the polynomial structure of the prompt response for sources close to the black hole through a series of numerical fits. Our work is a fundamental step in the broader effort to develop first-principles, analytical models for binary black hole coalescence signals, valid close to the merger and during the early ringdown stage.