  We re-examine the constraints on the density perturbation spectrum, including
its spectral index $n$, from the production of primordial black holes. The
standard cosmology, where the Universe is radiation dominated from the end of
inflation up until the recent past, was studied by Carr, Gilbert and Lidsey; we
correct two errors in their derivation and find a significantly stronger
constraint than they did, $n \lesssim 1.25$ rather than their 1.5. We then
consider an alternative cosmology in which a second period of inflation, known
as thermal inflation and designed to solve additional relic over-density
problems, occurs at a lower energy scale than the main inflationary period. In
that case, the constraint weakens to $n \lesssim 1.3$, and thermal inflation
also leads to a `missing mass' range, $10^{18} g \lesssim M \lesssim 10^{26}
g$, in which primordial black holes cannot form. Finally, we discuss the effect
of allowing for the expected non-gaussianity in the density perturbations
predicted by Bullock and Primack, which can weaken the constraints further by
up to 0.05.
