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

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.