Mixing of spherical and spheroidal modes in perturbed Kerr black holes

TL;DR

This paper calculates the mixing coefficients between spherical and spheroidal harmonics in Kerr black hole perturbations, providing data and fits to improve gravitational-wave modeling.

Contribution

It introduces computed and tabulated mixing coefficients between spherical and spheroidal harmonics for Kerr black holes, aiding gravitational-wave source analysis.

Findings

Computed complex-valued mixing coefficients as functions of Kerr parameter

Provided tables and analytical fits for these coefficients

Facilitated improved black-hole perturbation modeling

Abstract

The angular dependence of the gravitational radiation emitted in compact binary mergers and gravitational collapse is usually separated using spin-weighted spherical harmonics ${}_sY_{\ell m}$ of spin weight $s$, that reduce to the ordinary spherical harmonics $Y_{\ell m}$ when $s=0$. Teukolsky first showed that the perturbations of the Kerr black hole that may be produced as a result of these events are separable in terms of a different set of angular functions: the spin-weighted spheroidal harmonics ${}_sS_{\ell m n}$, where $n$ denotes the "overtone index" of the corresponding Kerr quasinormal mode frequency $\omega_{\ell m n}$. In this paper we compute the complex-valued scalar products of the ${}_sS_{\ell m n}$'s with the ${}_sY_{\ell m}$'s ("spherical-spheroidal mixing coefficients") and with themselves ("spheroidal-spheroidal mixing coefficients") as functions of the dimensionless Kerr parameter $j$. Tables of these coefficients and analytical fits of their dependence on $j$ are available online for use in gravitational-wave source modeling and in other applications of black-hole perturbation theory.