Bounds on the mass of superradiantly unstable scalar fields around Kerr black holes

TL;DR

This paper numerically bounds the mass of superradiantly unstable scalar fields around Kerr black holes, providing analytical fits useful for astrophysical applications and showing bounds increase with angular momentum parameters.

Contribution

It introduces numerical bounds on scalar field mass in Kerr black holes and provides accurate analytical fits as a function of black hole spin, advancing understanding of superradiant instabilities.

Findings

Normalized upper bound on scalar mass increases with angular momentum numbers.

Analytical fit to mass bounds achieves 0.1% accuracy for fundamental modes.

Higher-order modes up to ll=m=20 are also accurately fitted.

Abstract

In this work we compute numerical bounds on the mass $\mu$ of superradiantly unstable scalar fields in a Kerr black hole background using the continued fraction method. We show that the normalized upper bound on the mass $\mu$ increases with the angular momentum number $\ell$ and the azimuthal number $m$, approaching the most stringent analytical bound known to date when $\ell=m \gg 1$. We also provide an analytical fit to the numerically determined mass bound as a function of the dimensionless spin parameter $a/M$ of the black hole with an accuracy of the order $0.1\%$ for the fundamental mode with $\ell=m=1$, and of the order $1\%$ for higher-order modes (up to $\ell=m=20$). We argue that this analytical fit is particularly useful in astrophysical scenarios, since the lowest $\ell=m$ modes are capable of producing the strongest observable imprints of superradiance.