Landscape of Modular Cosmology

TL;DR

This paper explores the global structure of $SL(2,\

Contribution

It reveals that the sharp ridges in the potential are geometrically equivalent to inflationary plateaus, providing new insights into the structure of modular cosmological models.

Findings

The potential's ridges are geometrically equivalent to inflationary plateaus.

The fundamental domain and its images cover the upper half-plane.

Hyperbolic geometry explains the apparent sharpness of ridges.

Abstract

We investigate the global structure of the recently discovered family of $SL(2,\mathbb{Z})$-invariant potentials describing inflationary $\alpha$-attractors. These potentials have an inflationary plateau consisting of the fundamental domain and its images fully covering the upper part of the Poincar\'e half-plane. Meanwhile, the lower part of the half-plane is covered by an infinitely large number of ridges, which, at first glance, are too sharp to support inflation. However, we show that this apparent sharpness is just an illusion created by hyperbolic geometry, and each of these ridges is physically equivalent to the inflationary plateau in the upper part of the Poincar\'e half-plane.