Lyman-$\alpha$ forest constraints on Primordial Black Holes as Dark
  Matter
  The renewed interest in the possibility that primordial black holes (PBHs)
may constitute a significant part of the dark matter has motivated revisiting
old observational constraints, as well as developing new ones. We present new
limits on the PBH abundance, from a comprehensive analysis of high-resolution,
high-redshift Lyman-$\alpha$ forest data. Poisson fluctuations in the PBH
number density induce a small-scale power enhancement which departs from the
standard cold dark matter prediction. Using a grid of hydrodynamic simulations
exploring different values of astrophysical parameters, {we obtain a
marginalized upper limit on the PBH mass of $f_{\rm PBH}M_{\rm PBH} \sim
60~M_{\odot}$ at $2\sigma$, when a Gaussian prior on the reionization redshift
is imposed, preventing its posterior distribution to peak on very high values,
which are disfavoured by the most recent estimates obtained both through Cosmic
Microwave Background and Inter-Galactic Medium observations. Such bound weakens
to $f_{\rm PBH}M_{\rm PBH} \sim 170~M_{\odot}$, when a conservative flat prior
is instead assumed. Both limits significantly improves previous constraints
from the same physical observable.} We also extend our predictions to
non-monochromatic PBH mass distributions, ruling out large regions of the
parameter space for some of the most viable PBH extended mass functions.
