Can Primordial Black Holes be a Significant Part of Dark Matter ?

Abstract

The computation of PBH (primordial black hole) production from primordial perturbations has recently been improved by considering a more accurate relation between the primordial power spectrum and the PBH mass variance. We present here exact expressions which are valid for primordial spectra of arbitrary shape and which allow accurate numerical calculations. We then consider the possibility to have a significant part of dark matter in the form of PBHs produced by a primordial spectrum of inflationary origin possessing a characteristic scale. We show that in this model the relevant PBH mass is constrained to lie in the range $5\times 10^{15} {\rm g}\lesssim M \lesssim 10^{21} {\rm g}$. This is much less than the mass range coming from the QCD phase transition, allowing the two mechanisms to be easily distinguished.

Figures (3)

• Figure 1: The quantities $\sigma^2_H(t_k)$ (dotted line) and $k^3\Phi^2(k,t_k)\propto \delta^2_H(t_k)$ (solid line) are displayed for the same parameter value $p=0.5$. The overall normalization is arbitrary (and different) for each curve. It is obvious that the two quantities $\sigma^2_H(t_k)$ and $\delta^2_H(t_k)$ have very different shapes in the vicinity of the characteristic scale $k_s$ where $\sigma^2_H(t_k)$ exhibits a spiky structure. Their respective maxima have different locations. Note the same asymptotic ratio between large and small values of $k$ (with $t_k \ll t_{eq}$).
• Figure 2: The quantity $\beta(M)$ is shown for our model containing a jump in the inflaton potential derivative for the parameters $p=7.913\times 10^{-4},~M_s=10^{18}{\rm g}$. As can be seen, $\beta(M)$ acquires a well localized bump in the vicinity of $M_s\equiv M(t_{k_s})$. Note that we have $M_s\approx 1.6~M_{peak}$ and $\beta_-\equiv \beta(M\ll M_{peak}) \approx 10^{-12}\times \beta(M_{peak})$, a value sufficient to avoid the severe constraint from the contribution to the $\gamma$-ray background of evaporated PBHs with $M\leq M_*$.
• Figure 3: The allowed region in parameter space $(p, M_s)$ is shown. The solid line indicates the points for which $\Omega_{PBH,0}(M_{peak})=0.3$. Below the solid line, the gravitational constraint at $M_{peak}$ is violated and this region is therefore excluded. Below the dotted line, the $\gamma$-ray background constraint is violated. It is seen that for $10^{21}{\rm g}\lesssim M_s$, the allowed parameter values $p$ yield $\Omega_{PBH,0}(M_{peak})<0.3$ which becomes rapidly negligible with growing $M_s$. For the values of $p$ shown here, $M_s\approx 1.6 ~M_{peak}$.