The Maximal-Density Mass Function for Primordial Black Hole Dark Matter
  The advent of gravitational wave astronomy has rekindled interest in
primordial black holes (PBH) as a dark matter candidate. As there are many
different observational probes of the PBH density across different masses,
constraints on PBH models are dependent on the functional form of the PBH mass
function. This complicates general statements about the mass functions allowed
by current data, and, in particular, about the maximum total density of PBH.
Numerical studies suggest that some forms of extended mass functions face
tighter constraints than monochromatic mass functions, but they do not preclude
the existence of a functional form for which constraints are relaxed. We use
analytical arguments to show that the mass function which maximizes the
fraction of the matter density in PBH subject to all constraints is a finite
linear combination of monochromatic mass functions. We explicitly compute the
maximum fraction of dark matter in PBH for different combinations of current
constraints, allowing for total freedom of the mass function. Our framework
elucidates the dependence of the maximum PBH density on the form of
observational constraints, and we discuss the implications of current and
future constraints for the viability of the PBH dark matter paradigm.
