  We consider the imprint of superradiant instabilities of nonevaporating
primordial black holes (PBHs) on the spectrum of the cosmic microwave
background (CMB). In the radiation dominated era, PBHs are surrounded by a
roughly homogeneous cosmic plasma which endows photons with an effective mass
through the plasma frequency. In this setting, spinning PBHs are unstable to a
spontaneous spindown through the well-known "black-hole bomb" mechanism. At
linear level, the photon density is trapped by the effective photon mass and
grows exponentially in time due to superradiance. As the plasma density
declines due to cosmic expansion, the associated energy around PBHs is released
and dissipated in the CMB. We evaluate the resulting spectral distortions of
the CMB in the redshift range 10^3 < z < 2x10^6. Using the existing COBE/FIRAS
bounds on CMB spectral distortions, we derive upper limits on the fraction of
dark matter that can be associated with spinning PBHs in the mass range
10^{-8}*Msun < M < 0.2*Msin. For maximally-spinning PBHs, our limits are much
tighter than those derived from microlensing or other methods. Future data from
the proposed PIXIE mission could improve our limits by several orders of
magnitude.
