Generalised constraints on the curvature perturbation from primordial black holes

TL;DR

This work addresses how primordial black hole (PBH) abundance limits translate into constraints on the primordial curvature perturbation spectrum across an extremely wide range of scales. By relating the PBH initial mass fraction $\beta(M_{\rm PBH})$ to the mass variance $\sigma_{\rm hor}^2(R)$ and computing $\sigma_{\rm hor}^2(R)$ from ${\cal P}_{\cal R}(k)$ using gauge-transformed density perturbations, the authors derive generalized, time-evolution-aware constraints on ${\cal P}_{\cal R}(k)$ for Gaussian perturbations with a delta-function mass function. They find that the allowed amplitude of the curvature power spectrum typically lies in ${\cal P}_{\cal R}(k)\lesssim 10^{-2}-10^{-1}$, with mild dependence on the assumed spectral index near the scale of interest, spanning $k$ from about $10^{-2}$ to $10^{23}\, {\rm Mpc}^{-1}$. These generalized limits complement cosmological-scale measurements and provide a robust means to test inflationary scenarios that produce enhanced small-scale power.

Abstract

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 parameterizations 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.

Figures (2)

• 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. \ref{['abund']} we have not considered the diffuse gamma-ray background constraint which applies for $2 \times 10^{13} \, {\rm g} < M_{\rm PBH}< 5 \times 10^{14} \, {\rm g}$ as it depends significantly on the PBH mass function.
• Figure 2: Generalised constraints on the amplitude of the power spectrum of primordial curvature perturbations as a function of comoving wavenumber (in units of ${\rm Mpc}^{-1}$). We have assumed that the power spectrum is scale-invariant over the (relatively small) range of scale which contribute 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).