  Non-thermal histories for the early universe have received notable attention
as they are a rich source of phenomenology, while also being well motivated by
top-down approaches to beyond the Standard Model physics. The early (pre-BBN)
matter phase in these models leads to enhanced growth of density perturbations
on sub-Hubble scales. Here we consider whether primordial black hole formation
associated with the enhanced growth is in conflict with existing observations.
Such constraints depend on the tilt of the primordial power spectrum, and we
find that non-thermal histories are tightly constrained in the case of a
significantly blue spectrum. Alternatively, if dark matter is taken to be of
non-thermal origin we can restrict the primordial power spectrum on scales
inaccessible to CMB and LSS observations. We establish constraints for a wide
range of scalar masses (reheat temperatures) with the most stringent bounds
resulting from the formation of $10^{15}$ g black holes. These black holes
would be evaporating today and are constrained by FERMI observations. We also
consider whether the breakdown of the coherence of the scalar oscillations on
sub-horizon scales can lead to a Jean's pressure preventing black hole
formation and relaxing our constraints. Our main conclusion is that primordial
black hole constraints, combined with existing constraints on non-thermal
WIMPs, favor a primordial spectrum closer to scale invariance or a red tilted
spectrum.
