Large Field Inflation and Double $α$-Attractors

TL;DR

This work develops cosmological $\alpha$-attractors within supergravity, showing that inflationary predictions interpolate between two universal points as the curvature parameter $\alpha$ varies. In the small-$\alpha$ limit, the models yield plateau-like outcomes with $n_s = 1 - \frac{2}{N}$ and $r \sim \frac{12\alpha}{N^2}$, while in the large-$\alpha$ limit they reproduce chaotic-inflation predictions with $n_s = 1 - \frac{2}{N}$ and $r \sim \frac{8}{N}$; a second attractor arises where the potential near its minimum is quadratic. The authors embed these ideas in both disk and half-plane superconformal constructions and interpret $\alpha$ as the curvature of the Kähler moduli space, $R_k = -\frac{2}{3\alpha}$. The results connect geometric properties of the underlying supergravity theory to robust, observation-relevant predictions, offering a framework to interpret current Planck and BICEP2 constraints through the lens of moduli-space geometry.

Abstract

We consider a broad class of inflationary models that arise naturally in supergravity. They are defined in terms of a parameter $α$ that determines the curvature and cutoff of these models. As a function of this parameter, we exhibit that the inflationary predictions generically interpolate between two attractor points. At small cutoff $α$, the resulting inflationary model is of plateau-type with $n_s = 1 - 2 / N$ and $r = 12 α/ N^2$. For $α= 1$, these predictions coincide with predictions of the Starobinsky model and Higgs inflation. In contrast, for large cutoff $α$, the theory asymptotes to quadratic inflation, with $n_s = 1 - 2 / N$, $r = 8 / N$. Both universal predictions can be attributed to a stretching of the moduli space. For intermediate values of $α$, the predictions interpolate between these two critical points, thus covering the sweet spots of both Planck and BICEP2.

Figures (5)

• Figure 1: The cosmological observables $n_s$ and $\log r$ for a representative class of large and small field models with different potentials $V(\phi/\Lambda)$ converge to the predictions of the simplest model with the quadratic potential in the large $\Lambda$ limit.
• 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 bright blue 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 $\alpha = O(1)$. The upper blue contours correspond to BICEP2 results, the lower contours correspond to Planck 2013.
• Figure 3: The double $\alpha$-attractor in the $(n_s, r)$ plane for different chaotic models with $f(x)= x+c x^2$ with $c=(0, 0.4, 0.8, 1.6, 3.2, 10)$ (in order from right to the left) for 60 e-folds and $M=1$. The dots correspond to $\log \alpha = (1,2)$. The dots which would correspond to $\alpha = 1$ practically merge with each other and are covered by the bright blue star corresponding to $\alpha = 0$. The lower attractor at small $\alpha$ is at small $r$, the upper attractor at large $\alpha$ is at the point where all models lead to a $\phi^2$ model.
• Figure 4: a) A periodic potential in the theory (\ref{['newJordan']}) for $\alpha = 1$; b) The same potential in terms of the canonically normalized field $\varphi$ for $\alpha = 1$. The emergence of an infinitely long plateau of the potential, with a sufficiently sharp fall-off, is responsible for the attractor behavior of this class of models for $\alpha \lesssim 1$; c) The same potential in terms of the canonically normalized field $\varphi$ for $\alpha = 10^{4}$. The uniform horizontal stretching of the inflationary potential is responsible for the attractor behavior in this class of theories in the limit $\alpha\to \infty$. Because of the stretching of the potential, it became very flat. Therefore inflation may occur near each of the many minima of the potential.
• Figure 5: 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 $\alpha \rightarrow \infty$ (red star), the prediction of the Starobinsky model for $\alpha = 1$ (the lowest red dot), and the prediction $n_{s} =1-2/N$, $r = 0$ for $\alpha \to 0$ (blue star). The red dots on the thick blue line correspond to $\log \alpha = \{3, \ldots, 0\}$, from the top down.