Quintessence from a state space perspective

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.