Constraints on the density perturbation spectrum from primordial black holes

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.