Superconformal Inflationary $α$-Attractors

TL;DR

This paper generalizes the universal inflationary attractor paradigm by introducing the curvature-controlled parameter α, which tunes the relation between the geometric inflaton field and its canonical counterpart via a hyperbolic tanh mapping. In the α→∞ limit, predictions reproduce chaotic inflation with $n_s=1-2/N$ and $r=8/N$ (for quadratic potentials), while α→0 yields a universal attractor with $n_s=1-2/N$ and $r=12α/N^2$, aligning with Planck-favored regions. The authors construct a broad class of α-attractors within supergravity, derive their kinetic terms from an $SU(1,1)/U(1)$-type Kähler potential, and show how the curvature $R_K=-2/(3α)$ governs the interpolation between models. The work provides a unified framework connecting diverse inflationary scenarios and outlines observationally distinct regimes across the $(n_s,r)$ plane that future data could probe.

Abstract

Recently a broad class of superconformal inflationary models was found leading to a universal observational prediction $n_s=1-2/N$ and $r=12/N^2$. Here we generalize this class of models by introducing a parameter $α$ inversely proportional to the curvature of the inflaton Kahler manifold. In the small curvature (large $α$) limit, the observational predictions of this class of models coincide with the predictions of generic chaotic inflation models. However, for sufficiently large curvature (small $α$), the predictions converge to the universal attractor regime with $n_s=1-2/N$ and $r=12α/N^2$, which corresponds to the part of the $n_s-r$ plane favored by the Planck data.

Figures (2)

• Figure 1: The cosmological observables $n_s$ and $r$ for the theory with a potential $V_{0} \Bigl(1- e^{-\sqrt {2\over 3\alpha} \varphi}\Bigr)^2$ for $N=60$. As shown by the thick blue line, $n_s$ and $r$ for this model depend on $\alpha$ and continuously interpolate between the prediction of the simplest chaotic inflationary model with $V \sim \varphi^{2}$ for $\alpha \rightarrow \infty$, the prediction of the Starobinsky model for $\alpha = 1$ (the lowest red circle), and the prediction $n_{s} =1-\frac{2}{N}$, $r = 0$ for $\alpha \to 0$. The red dots on the thick blue line correspond to $\alpha = 10^{3},\, 10^{2},\, 10,\, 1$, from the top down.
• Figure 2: The cosmological observables $(n_s,r)$ for different scalar potentials $\tanh^{2n} ({\varphi \over \sqrt{6 \alpha}})$ with $2n = (2/3, 1, 2, 3, 4)$ for $N=60$. These continuously interpolate between the predictions of the simplest inflationary models with the monomial potentials $\varphi^{2n}$ for $\alpha \rightarrow \infty$, and the attractor point $n_{s} =1-2/N$, $r = 0$ for $\alpha \to 0$, shown by the red star. The different trajectories form a fan-like structure for $\alpha \gg n^2$. The set of dark red dots at the upper parts of the interpolating straight lines corresponds to $\alpha = 100$. The set of dark blue dots corresponds to $\alpha = 10$. The lines gradually merge for $\alpha = O(1)$.