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

TL;DR

The paper demonstrates that the Schwarzschild prompt response, previously attributed to high-frequency arcs, actually arises from poles at $\omega=0$ in a post-Minkowskian small-$\omega$ expansion, yielding a time-domain polynomial of degree $\ell$ in the observer's retarded time. By decomposing the Green's function into two frequency-domain components and applying Leaver's Coulomb-wave framework, the authors derive explicit $\ell=2$ results up to $\mathcal{O}[(M/r')^3]$ and show a cancellation between these components, leading to a closed-form prompt expression. Comparisons with numerical RWZHyp solutions reveal good agreement for distant sources, with dephasing for closer sources that can be mitigated by semi-phenomenological time-shifts and light-cone curvature corrections; polynomial fits suggest the prompt retains a polynomial-like structure even near the horizon, though the Coulomb-wave PM approach loses convergence there. The work points to future improvements via Mano-Suzuki-Takasugi representations (MST) or hypergeometric methods and extensions to Kerr spacetimes, aiming to deliver robust, first-principles prompt models for binary black-hole coalescences near merger and during early ringdown.

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 $ω=0$ 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.

Figures (8)

• Figure 1: Real component ('$+$' polarization) (top) and amplitude (bottom) of the Green's function $\ell=2,\,m=1$ multipole as observed at $\mathcal{I}^+$ vs the retarded time of the observer $u$. This quantity has been rescaled with the retarded time at which the first signal travelling on the flat light cone would arrive. Results shown for different location of the initial impulse $r'$, as displayed in the labels. In dashed grey, results of numerical integrations obtained through the RWZHyp code Bernuzzi:2010tyBernuzzi:2011aj, as described in Sec. \ref{['sec:background']}. In bright purple, the prediction obtained in the flat spacetime limit.
• Figure 2: Schematic representation of the non-analytical behavior of the Laplace-transformed Green's function component $\tilde{G}^{(1)}$, in the complex-frequency plane. In a PM expansion, this quantity is superposition of: a contribution which has integrable poles up to order $\ell+1$ in $\omega=0$, starting from the lowest PM order $\mathcal{O}[(\omega M)^0]$; a term with a branch cut along the negative imaginary axis, starting from the first PM order $\mathcal{O}(\omega M)$
• Figure 3: Real component ('$+$' polarization) (top) and amplitude (bottom) of the Green's function $\ell=2,\,m=1$ multipole as observed at $\mathcal{I}^+$ vs the retarded time of the observer $u$. This quantity has been rescaled with the retarded time at which the first signal travelling on the curved light cone would arrive to $\mathcal{I}^+$. Each column shows results for different location of the initial impulse $r'$, as displayed by the titles on the top axes. In dashed black, results of numerical integrations obtained through the RWZHyp code Bernuzzi:2010tyBernuzzi:2011aj, as described in Sec. \ref{['sec:background']}. In color, the prediction obtained through a PM expansion up to $\mathcal{O}[(M/r')^n]$, with $n$ shown in the legend.
• Figure 4: Same as in Fig. \ref{['fig:OG3_largeR']}, but the PM predictions have been shifted in the retarded time $u$ to minimize the mismatch with the numerical waveform, as discussed in the text.
• Figure 5: In black, shift in the retarded time $u$ of the observer at $\mathcal{I}^+$ needed to minimize the mismatch between the numerical waveform and the predictions, i.e. to pass from results in Fig. \ref{['fig:OG3_largeR']} to Fig. \ref{['fig:OG3_largeR_aligned']}, vs $r'$.