Escher in the Sky

TL;DR

This paper discusses $ ext{alpha}$-attractor cosmological models, highlighting their robust predictions, geometric interpretation via hyperbolic geometry, and the relation between gravitational wave amplitude and the Poincare disk radius.

Contribution

It provides a geometric interpretation of $ ext{alpha}$-attractors using hyperbolic geometry and relates gravitational wave amplitude to the disk radius.

Findings

Predictions $n_s = 1 - 2/N$ and $r = 12 ext{alpha}/N^2$ are robust.

The geometric model uses a Poincare disk with radius $ ext{sqrt}(3 ext{alpha})$.

Gravitational wave amplitude is proportional to the square of the disk radius.

Abstract

The cosmological models called $\alpha$-attractors provide an excellent fit to the latest observational data. Their predictions $n_{s} = 1-2/N$ and $r = 12\alpha/N^{2}$ are very robust with respect to the modifications of the inflaton potential. An intriguing interpretation of $\alpha$-attractors is based on a geometric moduli space with a boundary: a Poincare disk model of a hyperbolic geometry with the radius $\sqrt{3\alpha}$, 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.