Primordial black holes from single field models of inflation

TL;DR

This paper addresses whether dark matter can be comprised of primordial black holes (PBH) within a single-field inflation framework. It develops a simple toy model with a near-inflection point in the inflaton potential that induces a temporary ultra-slow-roll phase, producing a broad peak in the curvature power spectrum at small scales while keeping the large-scale spectrum consistent with CMB observations. The authors compute the exact power spectrum, showing that the peak arises from the nontrivial inflaton dynamics near the quasi-inflection point and yields a PBH mass distribution broad enough to potentially account for all dark matter, with an example giving $\Omega_{\rm PBH}^{\rm eq} \approx 0.42$ and a mass relation $M_{\rm PBH}=M_\odot\,e^{2(N-36.28)}$. They also discuss the associated Hawking evaporation for light PBHs, the resulting early structure formation, and gravitational-wave signatures in the stochastic background and binary mergers detectable by AdvLIGO and LISA. Overall, the work provides an economical, testable single-field mechanism linking late-stage inflationary dynamics to PBH dark matter and distinctive gravitational-wave signals accessible to upcoming observatories.

Abstract

Primordial black holes (PBH) have been shown to arise from high peaks in the matter power spectra of multi-field models of inflation. Here we show, with a simple toy model, that it is also possible to generate a peak in the curvature power spectrum of single-field inflation. We assume that the effective dynamics of the inflaton field presents a near-inflection point which slows down the field right before the end of inflation and gives rise to a prominent spike in the fluctuation power spectrum at scales much smaller than those probed by Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) observations. This peak will give rise, upon reentry during the radiation era, to PBH via gravitational collapse. The mass and abundance of these PBH is such that they could constitute the totality of the Dark Matter today. We satisfy all CMB and LSS constraints and predict a very broad range of PBH masses. Some of these PBH are light enough that they will evaporate before structure formation, leaving behind a large curvature fluctuation on small scales. This broad mass distribution of PBH as Dark Matter will be tested in the future by AdvLIGO and LISA interferometers.

Figures (6)

• Figure 1: Top figure: Single field potential $V(\phi)$ (arbitrary units) with an inflection point (vertical line) at $\phi=1.191\,v$, and an asymptotically flat plateau. The model parameters are $a=1,\, b=b_c(1)-\beta,\, \beta=1\times10^{-4}$ and $\kappa^2v^2 = 0.108$. Bottom figure: The exact evolution in phase space of the inflaton field (blue) for the same potential, compared with the slow-roll approximation (dashed red).
• Figure 2: Left panel: The integrand of the number of e-folds as a function of $\phi/v$. The dashed-red curve corresponds to the SRA with $\beta=1\times10^{-4}$. The exact calculation (blue, green and brown curves) is also shown for three different values of the resonance parameter $\beta=(1,\,2,\,4)\times10^{-4}$. Right panel: The number of e-folds of inflation $N$ as a function of $\phi/v$, in the SRA (dashed-red) and the exact calculation (blue). In both figures, the model parameters are $a=1,\, b=b_c(1)-\beta$ and $\kappa^2v^2 = 0.108$.
• Figure 3: The exact parameter $\epsilon'(N)/\epsilon(N)$ (in green, dashed section corresponds to negative values), together with the $\epsilon(N)$ parameter in the exact calculation (in blue) and SRA (in red), along the evolution during inflation. Fig. (a) corresponds to $a=0.2$, $b=b_c(0.2)-\beta$, $\beta=10^{-4}$ and $\kappa^2v^2=0.00333$, while Fig. (b) is for our representative parameter choice, $a=1$, $b=b_c(1)-\beta$, $\beta=10^{-4}$ and $\kappa^2v^2=0.108$. Both choices give $\Delta N=35$ and $n_s=0.954$ on CMB scales.
• Figure 4: The exact matter power spectrum (blue line), for model parameters: $a=1$, $b=b_c(1)-\beta$, $\kappa^2 v^2 = 0.108$ and $\beta = 1\times10^{-4}$. We have also plotted the SRA (red-dashed line), and the range of values allowed by Planck (2015), by compact minihalos (green line) and by PBH (black dashed line), at 95% c.l. (figure adapted from ref. Bringmann:2011ut).
• Figure 5: The curvature power spectra obtained from the exact inflaton evolution, for different values of the model parameters ($a,\,\kappa^2v^2$) for which there is a significant peak at small scales, while leaving CMB scales essentially unaffected, with $\Delta N=35$ and $n_s=0.954$. In all cases $\beta = 10^{-4}$.