Escher in the Sky

TL;DR

The paper argues that inflationary predictions are governed not by the detailed inflaton potential but by the geometry of the moduli space, specifically a hyperbolic disk of radius √(3α). Through disk and half-plane formulations in supergravity, it shows how α-attractors yield universal predictions n_s ≈ 1 − 2/N and r ≈ 12α/N^2, with small α driving a plateau-like potential and robust observational consequences. It connects this cosmological framework to the SU(1,1) symmetry of superconformal theories, Escher-inspired geometric intuitions, and the FRW open-geometry analogy, providing concrete constructions via Kähler potentials and superpotentials that reproduce the attractor behavior across a broad class of models. The work highlights a geometry-driven path to consistent early-un universe physics, suggesting that future measurements of primordial gravitational waves could probe the underlying moduli-space geometry and its associated symmetries.

Abstract

The cosmological models called $α$-attractors provide an excellent fit to the latest observational data. Their predictions $n_{s} = 1-2/N$ and $r = 12α/N^{2}$ are very robust with respect to the modifications of the inflaton potential. An intriguing interpretation of $α$-attractors is based on a geometric moduli space with a boundary: a Poincare disk model of a hyperbolic geometry with the radius $\sqrt{3α}$, beautifully represented by the Escher's picture Circle Limit IV. In such models, the amplitude of the gravitational waves is proportional to the square of the radius of the Poincare disk.

Figures (12)

• Figure 1: Blue, brown and green lines show the potentials of the T-models with $V \sim \tanh^2{\varphi\over\sqrt {6 \alpha}}$ for $\alpha = 1, 2, 3$ correspondingly. The red line in the center shows the potential of the GL model Goncharov:1983mw.
• Figure 2: Predictions of the simplest $\alpha$-attractor T-model with the potential $V\sim \tanh^2{\varphi\over\sqrt {6 \alpha}}$ for N = 60 cut through the most interesting part of the Planck 2015 plot for $n_{s}$ and $r$Planck:2015xua.
• Figure 3: E-model potential $\alpha\mu^{2}(1-e^{-{\sqrt {2\over 3 \alpha}}\varphi})^{2}$ in units of $\alpha\mu^{2} = 1$ for $\alpha = 1, 2, 3, 4$. Smaller $\alpha$ correspond to more narrow minima of the potentials. The blue line shows the potential of the Starobinsky model, which belongs to the class of E-models with $\alpha = 1$.
• Figure 4: Predictions of E-models with $V\sim (1-e^{-{\sqrt {2\over 3 \alpha}}\varphi})^{2}$.
• Figure 5: A computer generated version of Escher's picture Circle Limit IV (Heaven and Hell) by V. Bulatov, http://bulatov.org/math/1201/. It presents a Poincaré disk model of a hyperbolic geometry. The radius square of the disk in the context of our cosmological models is $R^2=3\alpha$. The curvature of this manifold ${\cal R}_{ \mathbb{H}^2}=-{2\over 3 \alpha}$. To see angels and devils moving in the Poincaré disk click here: http://youtu.be/milmZUVSjro