If primordial black holes (PBHs) form directly from inhomogeneities in the early Universe, then the number in the mass range $10^5-{10}^{12}{M}_{\odot }$ is severely constrained by upper limits to the $\mu$ distortion in the cosmic microwave background (CMB). This is because inhomogeneities on these scales will be dissipated by Silk damping in the redshift interval $5\times {10}^4\lesssim z\lesssim 2\times {10}^6$. If the primordial fluctuations on a given mass scale have a Gaussian distribution and PBHs form on the high-$\sigma$ tail, as in the simplest scenarios, then the $\mu$ constraints exclude PBHs in this mass range from playing any interesting cosmological role. Only if the fluctuations are highly non-Gaussian, or form through some mechanism unrelated to the primordial fluctuations, can this conclusion be obviated.

• Received 25 October 2017

DOI:https://doi.org/10.1103/PhysRevD.97.043525