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We investigate inflation models in Jordan frame supergravity, in which an inflaton non-minimally couples to the scalar curvature. By imposing the condition that an inflaton would have the canonical kinetic term in the Jordan frame, we construct inflation models with asymptotically flat potential through pole inflation technique and discuss their relation to the models based on Einstein frame supergravity. We also show that the model proposed by Ferrara et al. has special position and the relation between the Kahler potential and the frame function is uniquely determined by requiring that scalars take the canonical kinetic terms in the Jordan frame and that a frame function consists only of a holomorphic term (and its anti-holomorphic counterpart) for symmetry breaking terms. Our case corresponds to relaxing the latter condition.
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Inflationary perturbations are approximately Gaussian and deviations from Gaussianity are usually calculated using in-in perturbation theory. This method, however, fails for unlikely events on the tail of the probability distribution: in this regime non-Gaussianities are important and perturbation theory breaks down for $|zeta| gtrsim |f_{rm scriptscriptstyle NL}|^{-1}$. In this paper we show that this regime is amenable to a semiclassical treatment, $hbar to 0$. In this limit the wavefunction of the Universe can be calculated in saddle-point, corresponding to a resummation of all the tree-level Witten diagrams. The saddle can be found by solving numerically the classical (Euclidean) non-linear equations of motion, with prescribed boundary conditions. We apply these ideas to a model with an inflaton self-interaction $propto lambda dotzeta^4$. Numerical and analytical methods show that the tail of the probability distribution of $zeta$ goes as $exp(-lambda^{-1/4}zeta^{3/2})$, with a clear non-perturbative dependence on the coupling. Our results are relevant for the calculation of the abundance of primordial black holes.
We constructed 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 |Z| = 1. It describes a Mexican hat potential with the flat axion direction corresponding to a circle of radius |Z| < 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 has 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.
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