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 ${10}^3\lesssim z\lesssim 2\times {10}^6$. 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 ${10}^{-8}{M}_{\odot }\lesssim M\lesssim 0.2M_{\odot }$. 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.

Figure 1

Upper limits on the mass fraction of DM in PBHs with masses in the range ${10}^{-9}{M}_{\odot }<M<{10}^2{M}_{\odot }$. The solid blue curve is the theoretical constraint derived in this paper using COBE/FIRAS data [29]. The dashed red line is the expected limit from the proposed PIXIE experiment [39]. Our limits are plotted for maximally spinning PBHs with $⟨\tilde{a}⟩=1$ and scale roughly as $1/⟨\tilde{a}⟩$ (see text for details). The limits from other methods are adopted from Ref. 1.

Figure 2

Contour plots in the BH Regge plane [27, 30] corresponding to an instability time scale shorter than the age of the Universe at a given redshift, assuming a mean cosmic gas density as in Eq. (6). Thick and thin curves correspond to polar and axial modes, respectively [30]. In both families, the rightmost part of each curve is described by $\tilde{a}\sim 4M{\omega }_p$. Roughly speaking, PBHs in the mass range $7\times {10}^{-9}{M}_{\odot }<M<0.2M_{\odot }$ go through a cosmic era (at some ${10}^3<z<2\times {10}^6$) when the super-radiant instability is effective.