Generalized constraints on the curvature perturbation from primordial black holes

Primordial black holes (PBHs) can form in the early Universe via the collapse of large density perturbations. There are tight constraints on the abundance of PBHs formed due to their gravitational effects and the consequences of their evaporation. These abundance constraints can be used to constrain the primordial power spectrum, and hence models of inflation, on scales far smaller than those probed by cosmological observations. We compile, and where relevant update, the constraints on the abundance of PBHs before calculating the constraints on the curvature perturbation, taking into account the growth of density perturbations prior to horizon entry. We consider two simple parametrizations of the curvature perturbation spectrum on the scale of interest: constant and power-law. The constraints from PBHs on the amplitude of the power spectrum are typically in the range ${10}^{-2}–{10}^{-1}$ with some scale dependence.

Figure 1

The limits on the initial mass fraction of PBHs as a function of PBH mass (in grams). The solid lines represent the tightest limits for each mass range and the dotted lines are the weaker limits where there is an overlap between constraints. As discussed in Sec. 2 we have not considered the diffuse gamma-ray background constraint which applies for $2\times {10}^{13}\mathrm{g}<M_{\mathrm{PBH}}<5\times {10}^{14}\mathrm{g}$ as it depends significantly on the PBH mass function.

Figure 2

Generalized constraints on the amplitude of the power spectrum of primordial curvature perturbations as a function of comoving wave number (in units of ${\mathrm{Mpc}}^{-1}$). We have assumed that the power spectrum is scale-invariant over the (relatively small) range of scales which contributes to a given constraint. Deviations from scale-invariance consistent with slow-roll inflation lead to small changes in the constraints (see text for further details).