The Maximal-Density Mass Function for Primordial Black Hole Dark Matter

TL;DR

This paper derives the maximal possible density of primordial black holes as dark matter by analytically determining the optimal mass function that satisfies all observational constraints, providing a comprehensive framework for assessing PBH viability.

Contribution

It introduces an analytical method to find the maximum PBH density by representing the mass function as a finite combination of monochromatic functions, considering all current observational constraints.

Findings

The maximum PBH density is achieved by a finite linear combination of monochromatic mass functions.

The framework clarifies how observational constraints influence the maximum allowed PBH dark matter fraction.

Current constraints can be incorporated to evaluate the viability of PBH as dark matter candidates.

Abstract

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.