  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 $5\times
10^4\lesssim z\lesssim2\times 10^6$. 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.
