Primordial black hole constraints for extended mass functions

TL;DR

This work addresses whether extended primordial black hole (PBH) mass functions can weaken the existing constraints on PBHs as dark matter. It develops a general formalism using a normalized mass-distribution function $\psi(M)$ and analyzes three representative mass-function families—lognormal, power-law, and critical-collapse—over the broad mass range $10^{-18}-10^{4} M_{\odot}$. The main finding is that extending the mass function does not substantially relax the bounds; the allowed PBH fraction decreases with increasing width, yielding three mass windows where PBHs could in principle constitute all of the dark matter under robust bounds, but with all bounds included, the total PBH dark matter fraction is limited to order $10\%$. The paper also provides a practical method to combine independent observables into extended-mass-function constraints, with implications for interpreting LIGO-era PBH scenarios and for future PBH-related cosmological and astrophysical constraints.

Abstract

We revisit the cosmological and astrophysical constraints on the fraction of the dark matter in primordial black holes (PBHs) with an extended mass function. We consider a variety of mass functions, all of which are described by three parameters: a characteristic mass and width and a dark matter fraction. Various observations then impose constraints on the dark matter fraction as a function of the first two parameters. We show how these constraints relate to those for a monochromatic mass function, demonstrating that they usually become more stringent in the extended case than the monochromatic one. Considering only the well-established bounds, and neglecting the ones that depend on additional astrophysical assumptions, we find that there are three mass windows, around $4\times 10^{-17}M_\odot,$ $2\times 10^{-14}M_\odot$ and $25-100M_\odot$, where PBHs can constitute all dark matter. However, if one includes all the bounds, PBHs can only constitute of order $10\%$ of the dark matter.

Figures (3)

• Figure 1: Upper left panel: Constraints from different observations on the fraction of PBH DM, $f_{\rm PBH}\equiv \Omega_{\rm PBH}/\Omega_{\rm DM}$, as a function of the PBH mass $M_c$, assuming a monochromatic mass function. The purple region on the left is excluded by evaporations Carr:2009jm, the red region by femtolensing of gamma-ray bursts (FL) Barnacka:2012bm, the brown region by neutron star capture (NS) for different values of the dark matter density in the cores of globular clusters Capela:2013yf, the green region by white dwarf explosions (WD) Graham:2015apa, the blue, violet, yellow and purple regions by the microlensing results from Subaru (HSC) Niikura:2017zjd, Kepler (K) Griest:2013aaa, EROS Tisserand:2006zx and MACHO (M) Allsman:2000kg, respectively. The dark blue, orange, red and green regions on the right are excluded by Planck data Ali-Haimoud:2016mbv, survival of stars in Segue I (Seg I) Koushiappas:2017chw and Eridanus II (Eri II) Brandt:2016aco, and the distribution of wide binaries (WB) Monroy-Rodriguez:2014ula, respectively. The black dashed and solid lines show, respectively, the combined constraint with and without the constraints depicted by the colored dashed lines. Other panels: Same as the upper left panel but for a lognormal PBH mass function with $\sigma=2$ (upper right) and for a power-law PBH mass function with $\gamma=-1$ (lower left) and $\gamma=1$ (lower right).
• Figure 2: Upper panels: Combined observational constraints on $M_c$ and $\sigma$ for a lognormal PBH mass function. The color coding shows the maximum allowed fraction of PBH DM. In the white region $\log_{10}f_{\rm max}<-3$, while the solid, dashed, dot-dashed and dotted contours correspond to $f_{\rm max}=1$, $f_{\rm max}=0.5$, $f_{\rm max}=0.2$ and $f_{\rm max}=0.1$, respectively. In the left panel only the constraints depicted by the solid lines in Fig. \ref{['mono']} are included, whereas the right panel includes all the constraints. Lower panels: Same as the upper left panel but for a power-law mass function with $\gamma<0$ (left) and $\gamma>0$ (right).
• Figure 3: Observational constraints on $M_c$ and $\sigma$ for a lognormal PBH mass function, assuming 100% PBH DM. The left panel presents a zoom into the high mass region relevant for the LIGO events, while the right panel presents a zoom into the low mass region. The color coding is the same as in Fig. \ref{['mono']}.