Hypernatural inflation

We construct a model of natural inflation in the context of alpha-attractor supergravity, in which both the dilaton field and the axion field are light during inflation, and the inflaton may be a combination of the two. The T-model version of this theory is defined on the Poincare disk with radius vertical bar Z vertical bar = 1. It describes a Mexican hat potential with the flat axion direction corresponding to a circle of radius vertical bar Z vertical bar < 1. The axion decay constant f(a) in this theory can be exponentially large because of the hyperbolic geometry of the Poincare disk. Depending on initial conditions, this model may describe alpha-attractor inflation driven by the radial component of the inflaton field, natural inflation driven by the axion field, or a sequence of these two regimes. We also construct the E-model version of this theory, which have similar properties. In addition, we describe generalized alpha-attractor models where the potential can be singular at the boundary of the moduli space, and show that they can provide a simple solution for the problem of initial conditions for the models with plateau potentials.