# Inflationary $\alpha$-attractor cosmology: A global dynamical systems perspective

## TL;DR

This paper employs a dynamical systems approach to analyze flat FLRW $ ext{α}$-attractor inflationary models, providing a comprehensive description of solution spaces, asymptotics, and inflationary attractors with high analytic accuracy.

## Contribution

It introduces a regularized dynamical systems framework for $ ext{α}$-attractor models, revealing the structure of solution spaces and the nature of inflationary attractors.

## Key findings

- Solution spaces are fully characterized including asymptotics.
- Inflationary viability linked to a center manifold near de Sitter state.
- Center manifold expansion describes inflation with arbitrary analytic accuracy.

## Abstract

We study flat FLRW $\alpha$-attractor $\mathrm{E}$- and $\mathrm{T}$-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of $e$-folds, is associated with a solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this `inflationary attractor solution.' A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.