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 ℓ and the azimuthal number m, approaching the most stringent analytical bound known to date when ℓ=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 ℓ=m=1, and of the order 1% for higher-order modes (up to ℓ=m=20). We argue that this analytical fit is particularly useful in astrophysical scenarios, since the lowest ℓ=m modes are capable of producing the strongest observable imprints of superradiance.