Cosmological attractors from $\alpha$-scale supergravity

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 these supergravity models are fully determined by the curvature of the Kähler 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 falloff. Finally, we show how analogous structures with a flat Kähler geometry arise as a singular limit of such $\alpha$-scale models.

Figure 1

The monomial powers $n_{\pm }$ for the $\alpha$-scale models as a function of $\alpha$.

Figure 2

Scalar potential and imaginary mass of the model defined by Eq. (17) in terms of $\phi$ for $\alpha =\{1,4,9\}$. The blue line represents the instability occurring at $\alpha =1$.

Figure 3

The $(n_s,r)$ predictions for $N=50$ and $N=60$ of the model Eq. (17) superimposed on the Planck constraints. The predictions interpolate between (1) and (2) for small and order- $1\alpha$ and those of linear inflation for large $\alpha$.