Press-Schechter primordial black hole mass functions and their
  observational constraints
  We present a modification of the Press-Schechter (PS) formalism to derive
general mass functions for primordial black holes (PBHs), considering their
formation as being associated to the amplitude of linear energy density
fluctuations. To accommodate a wide range of physical relations between the
linear and non-linear conditions for collapse, we introduce an additional
parameter to the PS mechanism, and that the collapse occurs at either a given
cosmic time, or as fluctuations enter the horizon. We study the case where
fluctuations obey Gaussian statistics and follow a primordial power spectrum of
broken power-law form with a blue spectral index for small scales. We use the
observed abundance of super-massive black holes (SMBH) to constrain the
extended mass functions taking into account dynamical friction. We further
constrain the modified PS by developing a method for converting existing
constraints on the PBH mass fraction, derived assuming monochromatic mass
distributions for PBHs, into constraints applicable for extended PBH mass
functions. We find that when considering well established monochromatic
constraints there are regions in parameter space where all the dark matter can
be made of PBHs. Of special interest is the region for the characteristic mass
of the distribution ~10^2 M_Sun, for a wide range of blue spectral indices in
the scenario where PBHs form as they enter the horizon, where the linear
threshold for collapse is of the order of the typical overdensities, as this is
close to the black hole masses detected by LIGO which are difficult to explain
by stellar collapse.
