The Spectra of Gravitational Atoms

TL;DR

This paper analytically and numerically computes the spectra of ultralight scalar and vector fields around rotating black holes, crucial for understanding boson clouds and their astrophysical implications.

Contribution

It introduces a combined analytical and numerical approach to determine the spectra for various field types and black hole spins, extending previous methods to larger coupling constants.

Findings

Analytical spectra for small coupling constants using matched asymptotic expansions.

Numerical methods for spectra at larger coupling constants and arbitrary spins.

Conjectured analytic forms for magnetic mode instability rates.

Abstract

We compute the quasi-bound state spectra of ultralight scalar and vector fields around rotating black holes. These spectra are determined by the gravitational fine structure constant $\alpha$, which is the ratio of the size of the black hole to the Compton wavelength of the field. When $\alpha$ is small, the energy eigenvalues and instability rates can be computed analytically. Since the solutions vary rapidly near the black hole horizon, ordinary perturbative approximations fail and we must use matched asymptotic expansions to determine the spectra. Our analytical treatment relies on the separability of the equations of motion, and is therefore only applicable to the scalar field and the electric modes of the vector field. However, for slowly-rotating black holes, the equations for the magnetic modes can be written in a separable form, which we exploit to derive their energy eigenvalues and conjecture an analytic form for their instability rates. To check our conjecture, and to extend all results to large values of $\alpha$, we solve for the spectra numerically. We explain how to accurately and efficiently compute these spectra, without relying on separability. This allows us to obtain reliable results for any $\alpha \gtrsim 0.001$ and black holes of arbitrary spin. Our results provide an essential input to the phenomenology of boson clouds around black holes, especially when these are part of binary systems.