Tracking Quintessence

Tracking quintessence, in a spatially flat and isotropic space-time with aminimally coupled canonical scalar field and an asymptotically inversepower-law potential $V(\varphi)\propto\varphi^{-p}$, $p>0$, as$\varphi\rightarrow0$, is investigated. This is done by introducing a newthree-dimensional \emph{regular} dynamical system, which enables a rigorousexplanation of the tracking feature: 1) The dynamical system has a trackerfixed point $\mathrm{T}$ with a two-dimensional stable manifold that pushes anopen set of nearby solutions toward a single tracker solution originating from$\mathrm{T}$. 2) All solutions, including the tracker solution and thesolutions that track/shadow it, end at a common future attractor fixed pointthat depends on the potential. Thus, the open set of solutions that shadow thetracker solution share its properties during the tracking quintessence epoch.We also discuss similarities and differences of underlying mechanisms fortracking, thawing and scaling freezing quintessence, and, moreover, weillustrate with state space pictures that all of these types of quintessenceexist simultaneously for certain potentials.