Primordial black holes as dark matter

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}–10^{17}\mathrm{g}$, $10^{20}–10^{24}\mathrm{g}$ and $1–10^3{M}_{\odot }$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, nonsphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate-mass PBHs of $1–10^3{M}_{\odot }$ is of special interest in view of the recent detection of black-hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

Figure 1

Effect of critical collapse on the fraction $f$ as a function of black-hole mass in units of solar mass for a quasimonochromatic mass function (left panel) and for axionlike curvaton (red line) as well as running-mass inflation (blue line) (right panel). The latter models are specified in Sec. 3. Solid lines show the PBH abundances for the horizon-mass collapse estimate; dashed lines show the same models with critical collapse included.

Figure 2

Fraction $f$ as a function of black-hole mass in units of solar mass for axionlike curvaton (red curve) as well as running-mass inflation (blue curve); the models are specified in Sec. 6. Left panel: Effect of nonsphericity (dotted lines), with parameters $\kappa =9/\sqrt{10\pi }$ and $\gamma =1/2$ [see Eq. (27)]. Right panel: Effect of non-Gaussianity (dot-dashed curves), where we chose $f_{\mathrm{NL}}=+0.005$ (lower curve) and $f_{\mathrm{NL}}=-0.005$ (upper curve). In both cases the parameter choices are made for illustrative purpose (see main text for details). Critical collapse is assumed throughout the plots.

Figure 3

Constraints on $f(M)$ for a variety of evaporation (magenta), dynamical (red), lensing (cyan), large-scale structure (green) and accretion (orange) effects associated with PBHs. The effects are extragalactic $\gamma$-rays from evaporation (EG) [11], femtolensing of $\gamma$-ray bursts (F) [187], white-dwarf explosions (WD) [188], neutron-star capture constraints (NS) [36], Kepler microlensing of stars (K) [189], MACHO/EROS/OGLE microlensing of stars [27] and quasar microlensing (broken line) [190] (ML), survival of a star cluster in Eridanus II (E) [191], wide binary disruption (WB) [37], dynamical friction on halo objects (DF) [33], millilensing of quasars (mLQ) [32], generation of large-scale structure through Poisson fluctuations (LSS) [14] and accretion effects (WMAP and FIRAS) [15]. Only the strongest constraint is usually included in each mass range, but the accretion limits are shown with broken lines since they are highly model dependent. Where a constraint depends on some extra parameter which is not well known, we use a typical value. Most constraints cut off at high $M$ due to the incredulity limit. See the original references for more accurate forms of these constraints.

Figure 4

Four windows in which PBHs could conceivably provide the dark-matter density. Upper left panel: (A) Intermediate-mass black holes. The constraints in this mass range are EROS and MACHO microlensing bounds [27] (in blue), dynamical constraints (in red) from the lifetime of the central star cluster in the Eridanus II dwarf galaxy [191], as well as dynamical constraints (in green) from the existence of wide-binary star systems [37]. Upper right panel: (B) Sublunar black holes. In this case the constraints (in blue) are again the femtolensing of GRBs from [187], while the limits from neutron-star capture (in green) are taken from [36]. The red-shaded region denotes microlensing constraints from the Kepler survey [189], while the red-shaded region on the left shows constraints from white-dwarf explosions [188]. Lower left panel: (C) Subatomic black holes. The constraints here (red-shaded region) stem from nondetections of extragalactic $\gamma$ rays that would be observable from the evaporation of PBHs of these masses [11, 35] and (in blue) femtolensing of $\gamma$-ray bursts (GRBs) taken from Fermi data [187]. Lower right panel: (D) Planck-mass relics from PBH evaporations. This shows the mass range of the initial PBHs if they derive from inflation [62] but there are no observational constraints on such relics. Details on all these regimes and the meaning of the constraints can be found in the subsections on the respective scenarios.

Figure 5

Constraints on the dark-matter fraction of primordial black holes in the intermediate-mass range $M_{\odot }<M<10^3{M}_{\odot }$. Excluded regions are shaded. EROS constraints are taken from Ref. [27] and are depicted in blue. Wide-binary (WB) constraints [225, 253] correspond to the green region in the plot. The latest constraints from the survival of the star cluster near the core of Eridanus II [191] are shown in the red-shaded areas. For all red curves we assume a cluster age of 3 Gyr. The various constraints are due to different choices of values for the velocity dispersion $\sigma$ and $\rho$, the dark-matter density in the center of the galaxy. Specifically, we chose $(\sigma ,\rho )=(5\mathrm{km}{\mathrm{s}}^{-1},0.1M_{\odot }{\mathrm{pc}}^{-3})$ (red solid curve), $(\sigma ,\rho )=(10\mathrm{km}{\mathrm{s}}^{-1},0.1M_{\odot }{\mathrm{pc}}^{-3})$ (red dashed curve), $(\sigma ,\rho )=(5\mathrm{km}{\mathrm{s}}^{-1},0.01M_{\odot }{\mathrm{pc}}^{-3})$ (red dot-dashed curve), and $(\sigma ,\rho )=(10\mathrm{km}{\mathrm{s}}^{-1},0.01M_{\odot }{\mathrm{pc}}^{-3})$ (red dotted curve).

Figure 6

The differential dark-matter fraction $\mathrm{d}f/\mathrm{d}M$ in the intermediate-mass range $M_{\odot }<M<10^3{M}_{\odot }$ for the axionlike curvaton model (red, solid curve) as well as for running-mass inflation (blue, dashed curve). The parameter choices are $P_{\zeta }(k_{\mathrm{f}})=3.08\times {10}^{-3}$, $M_{\min }=6\times {10}^{-7}{M}_{\mathrm{f}}$, $\lambda =1.2$ for the axion-curvaton model (see Sec. 3b), and $a=0.011$, $b=0.0245$, and $c=-0.00304345$ for the running-mass model (see Sec. 3a). These choices are made in order to yield a dark-matter fraction of 1, as well as to be most compatible with the constraints in the intermediate-mass window (cf. Fig. 5). Critical collapse, with ${\delta }_c=0.45$, $k=3.3$, and $\gamma =0.36$, has been applied to obtain both of the curves. Featured are also mass bins or ranges I–XVI used to demonstrate the comparison with constraints as described in the text. The same mass bins can also be seen in Fig. 5 showing the constraints.