Revisiting small-scale fluctuations in $α$-attractor models of inflation

Abstract: Cosmological $\alpha$-attractors stand out as particularly compelling models to describe inflation in the very early universe, naturally meeting tight observational bounds from cosmic microwave background (CMB) experiments. We investigate $\alpha$-attractor potentials in the presence of an inflection point, leading to enhanced curvature perturbations on small scales. We study both single- and multi-field models, driven by scalar fields living on a hyperbolic field space. In the single-field case, ultra-slow-roll dynamics at the inflection point is responsible for the growth of the power spectrum, while in the multi-field set-up we study the effect of geometrical destabilisation and non geodesic motion in field space. The two mechanisms can in principle be distinguished through the spectral shape of the resulting scalar power spectrum on small scales. These enhanced scalar perturbations can lead to primordial black hole (PBH) production and second-order gravitational wave (GW) generation. Due to the existence of universal predictions in $\alpha$-attractors, consistency with current CMB constraints on the large-scale spectral tilt implies that PBHs can only be produced with masses smaller than $10^8\,\text{g}$ and are accompanied by ultra-high frequency GWs, with a peak expected to be at frequencies of order $10\,\text{kHz}$ or above.