Quintessence from a state space perspective

TL;DR

This paper introduces a new dynamical systems approach to analyze quintessence models with bounded potentials, providing clearer insights into their evolution and observational properties.

Contribution

It develops regular, bounded dynamical systems variables for quintessence models, enabling better visualization and understanding of scalar field evolution.

Findings

New variables lead to a regular dynamical system on a bounded state space.

Trajectories illustrate different quintessence evolution types.

Graphs of $w_\varphi(N)$ characterize evolution behaviors.

Abstract

We use dynamical systems methods to study quintessence models in a spatially flat and isotropic spacetime with matter and a scalar field with potentials for which $\lambda(\varphi)=-V_{,\varphi}/V$ is bounded, thereby going beyond the exponential potential for which $\lambda(\varphi)$ is constant. The scalar field equation of state parameter $w_\varphi$ plays a central role when comparing quintessence models with observations, but with the dynamical systems used to date $w_\varphi$ is an indeterminate, discontinuous, function on the state space in the asymptotically matter dominated regime. Our first main result is the introduction of new variables that lead to a \emph{regular} dynamical system on a \emph{bounded} three-dimensional state space on which $w_\varphi$ is a \emph{regular} function. The solution trajectories in the state space then provide a visualization of different types of quintessence evolution, and how initial conditions affect the transition between the matter and scalar field dominated epochs; this is complemented by graphs $w_\varphi(N)$, where $N$ is the $e$-fold time, which enables characterizing different types of quintessence evolution.