Primordial black holes as dark matter: converting constraints from monochromatic to extended mass distributions

The model in which Primordial Black Holes (PBHs) constitute a non-negligible fraction of the dark matter has (re)gained popularity after the first detections of binary black hole mergers. Most of the observational constraints to date have been derived assuming a single mass for all the PBHs, although some more recent works tried to generalize constraints to the case of extended mass functions. Here we derive a general methodology to obtain constraints for any PBH Extended Mass Distribution (EMD) and any observables in the desired mass range. Starting from those obtained for a monochromatic distribution, we convert them into constraints for EMDs by using an equivalent, effective mass Meq that depends on the specific observable. We highlight how limits of validity of the PBH modelling affect the EMD parameter space. Finally, we present converted constraints on the total abundance of PBH from microlensing, stellar distribution in ultra-faint dwarf galaxies and CMB accretion for Lognormal and Power Law mass distributions, finding that EMD constraints are generally stronger than monochromatic ones.