Generating Primordial Black Holes Via Hilltop-Type Inflation Models

TL;DR

This work investigates whether single-field Hilltop-type inflation and running-mass models can generate primordial black holes (PBHs) from enhanced small-scale perturbations without conflicting with current cosmological bounds. By expressing the PBH condition in terms of the end-of-inflation spectrum and slow-roll parameters, the authors show that the end-of-inflation potential must flatten (characterized by a small $\mathcal{B}=\epsilon_e/\epsilon_*$) to allow PBH formation, and they test this within Hilltop and running-mass frameworks under WMAP5 constraints. The results delineate parameter regions (in terms of $N$, $p$, $q$, $\eta_p$, $\eta_q$, $\alpha$, $\mu_0^2$, etc.) where PBHs can form, and they highlight how the viability depends strongly on the total number of $e$-folds, with larger $N$ typically expanding the allowed space. These findings aid inflationary model discrimination by linking the small-scale PBH production to specific potential shapes and cosmological parameter bounds, and they anticipate tighter constraints from Planck-era data.

Abstract

It has been shown that black holes would have formed in the early Universe if, on any given scale, the spectral amplitude of the Cosmic Microwave Background (CMB) exceeds 10^(-4). This value is within the bounds allowed by astrophysical phenomena for the small scale spectrum of the CMB, corresponding to scales which exit the horizon at the end of slow-roll inflation. Previous work by Kohri et. al. (2007) showed that for black holes to form from a single field model of inflation, the slope of the potential at the end of inflation must be flatter than it was at horizon exit. In this work we show that a phenomenological Hilltop model of inflation, satisfying the Kohri et. al. criteria, could lead to the production of black holes, if the power of the inflaton self-interaction is less than or equal to 3, with a reasonable number or e-folds. We extend our analysis to the running mass model, and confirm that this model results in the production of black holes, and by using the latest WMAP year 5 bounds on the running of the spectral index, and the black hole constraint we update the results of Leach et. al. (2000) excluding more of parameter space.

Figures (7)

• Figure 1: For the figures on the left, increasing the variant decreases $N$, and the opposite is true for the variants defining the figures on the right.
• Figure 2: Plot of the running mass model. The solid line represents a larger $\alpha$ than the dashed-dotted line.
• Figure 3: A plot of the maximum and minimum values of $\log(\log(N))$ versus $p$ and $q$, from a range values of $\eta_p$ and $\eta_q$.
• Figure 4: A plot of the minimum values of $\log(N)$ versus $\eta_p$ and $\eta_q$, from a range values of $p$ and $q$.
• Figure 5: Plot of $\log(\mathcal{B})$ versus $\log(n')$ for the Hilltop model with $N=60$ (figure on left), $N=100$ (figure on right) and $n_s=0.95$. The hatched region is excluded, representing $\log(n')>-2$ and $\log(\mathcal{B})<-8$. The region $\log(\mathcal{B})>-6$ does not lead to the formation of PBHs, and is represented by the tan colour in the figure. PBHs can form in the region $-8\leq\log(\mathcal{B})\leq-6$ without violating astrophysical or cosmological bounds, and is represented by the light orange region. The yellow dots correspond to $\{p,q\}=\{3,4\}$, the green dots to $\{p,q\}=\{2,3\}$ and the blue dots to $\{p,q\}=\{2,2.5\}$.