Constraints on primordial black holes with extended mass functions

Constraints on primordial black holes in the range ${10}^{-18}{M}_{\odot }$ to $10^3{M}_{\odot }$ are reevaluated for a general class of extended mass functions. Whereas previous work has assumed that PBHs are produced with one single mass, instead there is expected to be a range of masses even in the case of production from a single mechanism; constraints therefore change from previous literature. Although tightly constrained in the majority of cases, it is shown that, even under conservative assumptions, primordial black holes in the mass range ${10}^{-10}{M}_{\odot }$ to ${10}^{-8}{M}_{\odot }$ could still constitute the entirety of the dark matter. This stresses both the importance for a comprehensive reevaluation of all respective constraints that have previously been evaluated only for a monochromatic mass function and the need to obtain more constraints in the allowed mass range.

Figure 1

Summary of previous literature for the idealized case in which the entire PBH dark matter consists of PBHs of a single mass $M$ (monochromatic mass function): the constraint “curtain” on the dark-matter fraction $f\equiv {\rho }_{\mathrm{PBH}}/{\rho }_{\mathrm{DM}}$ for a variety of effects associated with PBHs of mass $M$ in units of solar mass $M_{\odot }$. Only the strongest constraints are included. We show constraints from extragalactic $\gamma$ rays from evaporation (EG) [21], femtolensing of $\gamma$-ray bursts (F) [22], white-dwarf explosions (WD) [23], neutron-star capture (NS) [24], Kepler microlensing of stars (K) [11], MACHO/EROS/OGLE microlensing of stars (ML) [25] and quasar microlensing (ML) [26], millilensing of quasars (mLQ) [27], pulsar timing (PT) (SKA forecast) [28], and accretion effects on the optical thickness ($\tau$) [29]. More details can be found in [17]. Whereas these constraints are only valid for the case of a monochromatic mass function, instead all realistic cases with extended mass spectra require a rederivation of those constraints and an integration of these over the whole mass range; this will be done in the present work.

Figure 2

Maximum PBH dark-matter fraction $f\equiv {\rho }_{\mathrm{PBH}}/{\rho }_{\mathrm{DM}}$ as a function of ${\sigma }_f$ and $M_f$ for the extended mass function specified in Eq. (1). The region enclosed by the red dashed contour in the middle of the plot indicates the possibility of 100% primordial black-hole dark matter (see the legend). The various contours are at ${\log }_{10}(f)=0(\text{red dashed line}),-0.2,-0.4,-0.6,-0.8,-1,-2,\dots ,-6$.