A-BHPT-toolkit: Analytic Black Hole Perturbation Theory Package for Gravitational Scattering Amplitudes

TL;DR

The paper tackles the challenge of obtaining analytic black hole perturbation theory (BHPT) scattering amplitudes by introducing a public MST-based package that delivers both analytic post-Minkowskian (PM) expansions in the low-frequency limit and exact numerical results for arbitrary frequencies. The authors implement a robust Mathematica workflow to determine MST coefficients and the phase factor to a desired $ ext{$\\epsilon$}$-order, solving the connection equation for the renormalized angular momentum $ u$ order-by-order and assembling the phase from MST data. As an application, they compute scattering phase shifts and inelasticity for massless spins $s = 0, 1, 2$ scattering off Kerr black holes through the third PM order and compare with exact numerical solutions, reproducing known results and providing new analytic coefficients. This analytic toolkit facilitates on-shell EFT matching of tidal responses and supports gravitational Raman scattering studies, offering improved reproducibility and analytic control for gravitational-scattering calculations in BHPT.

Abstract

Applications of effective field theory (EFT) and scattering amplitudes to gravitational problems have recently produced many unique results that advanced our understanding of the dynamics of compact binaries. Many of these results were made possible by comparing gravitational scattering amplitudes in EFT with exact expressions from general relativity. However, the latter expressions are not easily available as they require intricate solution techniques for the Teukolsky master equation, such as the Mano-Suzuki-Takasugi (MST) method. In this paper, we develop and present the first public package that enables computations of gravitational scattering amplitudes in black hole perturbation theory within the MST framework. Our package supports both analytic computations to a given post-Minkowskian (PM) order in the low-frequency limit and exact numerical computations for an arbitrary frequency of the perturbing field. As an application, we compute scattering phase shifts and inelasticity parameters for massless spin - 0, 1, and 2 fields resulting from scattering off a rotating Kerr black hole through the third PM order and compare these results with the exact numerical solutions.

Figures (8)

• Figure 1: Flowchart of the phase determination procedure. The manual steps are essentially choosing to what order each intermediate expression should be expanded to, while Mathematica carries out the automatic ones.
• Figure 2: The expansion of $Re[\nu]$ up to fourth order in $\epsilon$ and the numerical result of $Re[\nu]$ for $s = 2,\ l = 2,\ m = 0$. The analytic expansion and numerical result match well until around $\epsilon = 0.7$, which defines the radius of convergence of the analytic expression for the renormalized angular momentum.
• Figure 3: The analytic expansion of $Re\left[_2\delta_{20\omega}\right]$ for $q = 0.5$ and $P = -1$ up to third order in $\epsilon$ and the corresponding numerical result. Once again, agreement is seen until around $\epsilon = 0.75$, where the numerical result exhibits an inflection point, the same value of $\epsilon$ shown to be the convergence radius of the analytic expression for $\nu$ in this case.
• Figure 4: The analytic expansion of $Re\left[_0\delta_{00\omega}\right]$ for $q = 0.5$ up to third order in $\epsilon$ and the corresponding numerical result.
• Figure 5: The analytic expansion of $Re\left[_0\delta_{01\omega}\right]$ for $q = 0.5$ up to third order in $\epsilon$ and the corresponding numerical result.