Limits on primordial black holes from $\mu$ distortions in cosmic microwave background

If primordial black holes (PBHs) form directly from inhomogeneities in the early universe, then the number in the mass range $10^6 -10^9M_{\odot}$ 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 $5\times 10^4\lesssim z\lesssim2\times 10^6$. 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.