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Due to coherent superradiant amplification, massive bosonic fields can
trigger an instability in spinning black holes, tapping their energy and
angular momentum and forming macroscopic Bose-Einstein condensates around them.
This phenomenon produces gaps in the mass-spin distribution of astrophysical
black holes, a continuous gravitational-wave signal emitted by the condensate,
and several environmental effects relevant for gravitational-wave astronomy and
radio images of black holes. While the spectrum of superradiantly unstable mode
is known in great detail for massive scalar (spin-0) and vector (spin-1)
perturbations, so far only approximated results were derived for the case of
massive tensor (spin-2) fields, due to the nonseparability of the field
equations. Here, solving a system of ten elliptic partial differential
equations, we close this program and compute the spectrum of the most unstable
modes of a massive spin-2 field for generic black-hole spin and boson mass,
beyond the hydrogenic approximation and including the unique dipole mode that
dominates the instability in the spin-2 case. We find that the instability
timescale for this mode is orders of magnitude shorter than for any other
superradiant mode, yielding much stronger constraints on massive spin-2 fields.
These results pave the way for phenomenological studies aimed at constraining
beyond Standard Model scenarios, ultralight dark matter candidates, and
extensions to General Relativity using gravitational-wave and electromagnetic
observations, and have implications for the phase diagram of vacuum solutions
of higher-dimensional gravity.
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