Critical collapse and the primordial black hole initial mass function

It has normally been assumed that primordial black holes (PBHs) always form with mass approximately equal to the mass contained within the horizon at that time. Recent work studying the application of critical phenomena in gravitational collapse to PBH formation has shown that in fact, at a fixed time, PBHs with a range of masses are formed. When calculating the PBH initial mass function it is usually assumed that all PBHs form at the same horizon mass. It is not clear, however, that it is consistent to consider the spread in the mass of PBHs formed at a single horizon mass, whilst neglecting the range of horizon masses at which PBHs can form. We use the excursion set formalism to compute the PBH initial mass function, allowing for PBH formation at a range of horizon masses, for two forms of the density perturbation spectrum. First we examine power-law spectra with $n>1$, where PBHs form on small scales. We find that, in the limit where the number of PBHs formed is small enough to satisfy the observational constraints on their initial abundance, the mass function approaches that found by Niemeyer and Jedamzik under the assumption that all PBHs form at a single horizon mass. Second, we consider a flat perturbation spectrum with a spike at a scale corresponding to horizon mass $\sim 0.5 M_{\odot}$, and compare the resulting PBH mass function with that of the MACHOs (MAssive Compact Halo Objects) detected by microlensing observations. The predicted mass spectrum appears significantly wider than the steeply-falling spectrum found observationally.