Dynamical systems in perturbative scalar field cosmology

TL;DR

This paper develops a new regular dynamical system framework for analyzing linear scalar perturbations in flat cosmological models with scalar fields, providing a comprehensive global view and extending to more general potentials.

Contribution

It introduces a regular, compact dynamical system approach for scalar perturbations, enabling global analysis and extension to complex scalar field potentials.

Findings

Constructed a global solution space with known solutions on invariant sets.

Provided new insights into curvature perturbations using dynamical systems.

Extended the framework to more general scalar field potentials.

Abstract

We derive a new \emph{regular} dynamical system on a 3-dimensional \emph{compact} state space describing linear scalar perturbations of spatially flat Robertson-Walker geometries for relativistic models with a minimally coupled scalar field with an exponential potential. This enables us to construct the global solution space, illustrated with figures, where known solutions are shown to reside on special invariant sets. We also use our dynamical systems approach to obtain new results about the comoving and uniform density curvature perturbations. Finally we show how to extend our approach to more general scalar field potentials. This leads to state spaces where the state space of the models with an exponential potential appears as invariant boundary sets, thereby illustrating their role as building blocks in a hierarchy of increasingly complex cosmological models.