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 can not 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 .