Constraints on the primordial curvature perturbation from primordial black holes

TL;DR

This work addresses constraining the amplitude of the primordial curvature perturbation on small scales by examining the present-day abundance of primordial black holes (PBHs). It extends the sub-horizon PBH formation framework to calculate constraints on the curvature perturbation power ${\cal A}_{\cal R}$ from PBHs that form without horizon exit during inflation, as a function of the reheat temperature $T_{\rm RH}$, and compares them to conventional super-horizon PBH bounds. The authors derive the evolution of the Bardeen potential in the radiation era, relate PBH abundance to the curvature perturbation through the mass variance $\sigma_{\Psi}$, and compute present-day PBH densities via Press-Schechter with a threshold $|\Psi_{\rm c}|=1/2$, obtaining tight constraints on ${\cal A}_{\cal R}$ for $T_{\rm RH} < 10^{8}$ GeV (sub-horizon) that are about a factor of 3 stronger than the super-horizon limits. The results show how the bound weakens with increasing $T_{\rm RH}$ as the horizon mass at the end of inflation decreases and fewer PBHs survive to today, highlighting PBHs as a powerful, complementary probe of inflationary perturbations across a broad range of scales.

Abstract

We calculate the constraints on the primordial curvature perturbation at the end of inflation from the present day abundance of Primordial Black Holes (PBHs), as a function of the reheat temperature T_{\rm RH}. We first extend recent work on the formation of PBHs on scales which remain within the horizon during inflation and calculate the resulting constraints on the curvature perturbation. We then evaluate the constraint from PBHs that form, more conventionally, from super-horizon perturbations. The constraints apply for T_{\rm RH} < 10^{8} GeV and the inclusion of sub-horizon PBHs leads to a limit which is roughly three times tighter than the bound from super-horizon PBHs.

Figures (2)

• Figure 1: The differential mass fraction of sub-horizon PBHs, eq. (\ref{['omega_pbh1']}), as a function of $x_{\rm e}$ for ${\cal A}^{1/2}=0.02$ and $T_{\rm RH}= 1$ GeV. The distribution is sharply peaked at small $x_{\rm e}$, demonstrating that the majority of PBHs form at essentially the same time.
• Figure 2: The constraint on the amplitude of the curvature perturbation power spectrum at the end of inflation from the present day abundance of PBHs as a function of reheat temperature $T_{\rm RH}$. The solid and dotted lines are for sub-horizon PBHs. The solid line assumes that the linear theory calculation is still valid on scales on which the comoving curvature perturbation becomes larger than one before the end of inflation, while the dotted line ignores PBH formation on these scales (and hence provides a conservative evaluation of the constraint). The long and short dashed lines are for super-horizon PBHs using a density contrast threshold $\delta_{\rm c}=1/3$ and $0.45$ respectively.