Constraining primordial black-hole bombs through spectral distortions of the cosmic microwave background

We consider the imprint of super-radiant 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 the linear level, the photon density is trapped by the effective photon mass and grows exponentially in time due to super-radiance. 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) less than or similar to z less than or similar to 2 x 10(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) M-circle dot less than or similar to M less than or similar to 0.2M(circle dot). 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.