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 with 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 supermassive 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 ${\sim}10^2\, \mathrm{M}_\odot$, 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.