The Planck value of the spectral index can be interpreted as $n_s = 1 - 2/N$ in terms of the number of e-foldings $N$. An appealing explanation for this phenomenological observation is provided by $alpha$-attractors: the inflationary predictions of t
hese supergravity models are fully determined by the curvature of the Kahler manifold. We provide a novel formulation of $alpha$-attractors which only involves a single chiral superfield. Our construction involves a natural deformation of no-scale models, and employs these to construct a De Sitter plateau with an exponential fall-off. Finally, we show how analogous structures with a flat Kahler geometry arise as a singular limit of such $alpha$-scale models.
We discuss the possibility to construct supergravity models with a single superfield describing inflation as well as the tiny cosmological constant $V sim 10^{{-120}}$. One could expect that the simplest way to do it is to study models with a supersy
mmetric Minkowski vacuum and then slightly uplift them. However, due to the recently proven no-go theorem, such a tiny uplifting cannot be achieved by a small modification of the parameters of the theory. We illustrate this general result by investigation of models with a single chiral superfield recently proposed by Ketov and Terada. We show that the addition of a small constant or a linear term to the superpotential of a model with a stable supersymmetric Minkowski vacuum converts it to an AdS vacuum, which results in a rapid cosmological collapse. One can avoid this problem and uplift a supersymmetric Minkowski vacuum to a dS vacuum with $V_{0}sim 10^{-120}$ without violating the no-go theorem by making these extra terms large enough. However, we show that this leads to a strong supersymmetry breaking in the uplifted vacua.
We provide strong evidence for universality of the inflationary field range: given an accurate measurement of $(n_s,r)$, one can infer $Delta phi$ in a model-independent way in the sub-Planckian regime for a range of universality classes of inflation
ary models. Both the tensor-to-scalar ratio as well as the spectral tilt are essential for the field range. Given the Planck constraints on $n_s$, the Lyth bound is strengthened by two orders of magnitude: whereas the original bound gives a sub-Planckian field range for $r lesssim 2 cdot 10^{-3}$, we find that $n=0.96$ brings this down to $r lesssim 2 cdot 10^{-5}$.
We study to what extent the spectral index $n_s$ and the tensor-to-scalar ratio $r$ determine the field excursion $Deltaphi$ during inflation. We analyse the possible degeneracy of $Delta phi$ by comparing three broad classes of inflationary models,
with different dependence on the number of e-foldings $N$, to benchmark models of chaotic inflation with monomial potentials. The classes discussed cover a large set of inflationary single field models. We find that the field range is not uniquely determined for any value of $(n_s, r)$; one can have the same predictions as chaotic inflation and a very different $Delta phi$. Intriguingly, we find that the field range cannot exceed an upper bound that appears in different classes of models. Finally, $Delta phi$ can even become sub-Planckian, but this requires to go beyond the single-field slow-roll paradigm.
The bosonic sector of various supergravity theories reduces to a homogeneous space G/H in three dimensions. The corresponding algebras g are simple for (half-)maximal supergravity, but can be semi-simple for other theories. We extend the existing lit
erature on the Kac-Moody extensions of simple Lie algebras to the semi-simple case. Furthermore, we argue that for N=2 supergravity the simple algebras have to be augmented with an su(2) factor.
