Single Field Double Inflation and Primordial Black Holes

TL;DR

The paper investigates how a single scalar field within scalar-tensor theories can realize two distinct inflationary epochs, separated by a regime where slow-roll breaks down, to reconcile Planck-scaleCMB observations with the production of large curvature perturbations at small scales that form primordial black holes (PBHs). It develops an exact dynamical treatment of the inflaton, shows that slow-roll can fail during the transition between phases, and proposes a UV-complete potential with non-minimal coupling and running self-coupling that yields a first phase of inflation (≈30–40 $e$-folds) followed by a second, hilltop-like phase (≈20–30 $e$-folds) that generates a peak in the curvature power spectrum. The second phase can produce PBHs whose abundance can saturate the dark matter density, with a mass function that is approximately lognormal in slow-roll regimes but may acquire a low-mass tail due to slow-roll violations; the model remains compatible with current $n_s$ and $r$ constraints while providing a testable PBH signature at small scales. Together, these results connect UV-complete inflationary dynamics to PBH phenomenology and offer a concrete framework to explore PBH dark matter within single-field inflation.

Abstract

Within the framework of scalar-tensor theories, we study the conditions that allow single field inflation dynamics on small cosmological scales to significantly differ from that of the large scales probed by the observations of cosmic microwave background. The resulting single field double inflation scenario is characterised by two consequent inflation eras, usually separated by a period where the slow-roll approximation fails. At large field values the dynamics of the inflaton is dominated by the interplay between its non-minimal coupling to gravity and the radiative corrections to the inflaton self-coupling. For small field values the potential is, instead, dominated by a polynomial that results in a hilltop inflation. Without relying on the slow-roll approximation, which is invalidated by the appearance of the intermediate stage, we propose a concrete model that matches the current measurements of inflationary observables and employs the freedom granted by the framework on small cosmological scales to give rise to a sizeable population of primordial black holes generated by large curvature fluctuations. We find that these features generally require a potential with a local minimum. We show that the associated primordial black hole mass function is only approximately lognormal.

Figures (2)

• Figure 1: The tensor-to-scalar ratio $r$ vs. the spectral tilt $n_{s}$ from $\beta_{\lambda}/\lambda = 0.01$ (bottommost line) to $\beta_{\lambda}/\lambda = 10$ (topmost line). We show the $1\sigma$ observational bounds Ade:2015lrj for a running $n_{s}$ (black boundaries) and for a not running $n_{s}$ (gray boundaries). We choose $v_2 \approx v_3 = 0.106$, $\xi = 122.7$, $m_{\Phi} = 0.12 M_{P}$.
• Figure 2: Left: The Hubble slow-roll parameter $\epsilon_{H}$ obtained with the exact approach (solid line) and using the slow-roll approximation (dashed line) as functions of the number of $e$-folds $N$. Right: The power spectrum $\mathcal{P}_{\zeta}$ as a function of the wave number $k.$ The potential parameters are given by the first (black lines) and the second (gray lines) benchmark point in Table \ref{['tab:bench']}. The purple band highlights the curvature fluctuations yielding a PBH population that, depending on the value of the criticality parameter $\zeta_c$, can match the measured DM abundance.