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

In this work we compute numerical bounds on the mass μ 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 μ increases with the angular momentum number and the azimuthal number m, approaching the most stringent analytical bound known to date when =m 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 =m=1, and of the order $1%$ for higher-order modes (up to =m=20). We argue that this analytical fit is particularly useful in astrophysical scenarios, since the lowest =m modes are capable of producing the strongest observable imprints of superradiance.