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

TL;DR

This paper derives an analytical model for the prompt response of a Schwarzschild black hole's Green's function using a post-Minkowskian expansion, revealing its polynomial structure and validating it against numerical data.

Contribution

It introduces a novel analytical expression for the prompt response based on residue analysis, improving understanding of early-time signals in black hole perturbations.

Findings

Prompt response is given by residues at ω=0 in the Fourier domain.

The response is a polynomial in retarded time of order ℓ.

Model matches numerical predictions for distant sources, with corrections for closer sources.

Abstract

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 $\omega=0$ 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.