We provide a unified description of cosmological $alpha$-attractors and late-time acceleration, in excellent agreement with the latest Planck data. Our construction involves two superfields playing distinctive roles: one is the dynamical field and it
s evolution determines inflation and dark energy, the other is nilpotent and responsible for a landscape of vacua and supersymmetry breaking. We prove that the attractor nature of the theory is enhanced when combining the two sectors: cosmological attractors are very stable with respect to any possible value of the cosmological constant and, interestingly, to any generic coupling of the inflationary sector with the field responsible for uplifting. Finally, as related result, we show how specific couplings generate an arbitrary inflaton potential in a supergravity framework with varying Kahler curvature.
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.
