  We estimate the maximum allowed amplitude for the power spectrum of the
primordial curvature perturbations, ${\cal P_R}(k)$, on all scales from the
absence of any detection signals of sub-solar mass black holes. In particular
we analyze the constraints on the PBHs and we focus on the low mass limit where
the Hawking radiation is expected to significantly influence the big bang
observables, considering also different early cosmic histories. We derive the
upper bounds for the variance of density perturbations, $\sigma(M)$, for any
possible reheating temperature as well as for the cosmological scenario of a
scalar condensate domination. We expect our results to have considerable
implications for models designed to generate PBHs, especially in the low mass
range, and provide additional constraints to a large class of inflationary
models.
