Primordial black holes from inflationary models with and without broken scale invariance

We review the formalism of primordial black hole (PBH) production and show that the mass variance at the horizon crossing has been systematically overestimated in previous studies. We derive the correct expression. The difference is maximal at the earliest formation times and still very significant for PBH masses $\sim {10}^{15}\mathrm{g},$ an accurate estimate requiring numerical calculations. In particular, this would lead to weaker constraints on the spectral index n. We then derive constraints on inflationary models from the fact that primordial black holes must not overclose the Universe. This is done both for the scale-free case of the power spectrum studied earlier and for the case where a step in the mass variance is superimposed. In the former case we find various constraints on n, depending on the parameters. In the latter case these limits can be much more strengthened, so that one could find from an observational limit on n, a constraint on the allowed height of the step.