Tracking Quintessence

Abstract

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

Figures (5)

• Figure 1: The ski-slope state space and the $\bar{\varphi}=0$ boundary. The dotted lines correspond to a cut off in $v$ in order to obtain a finite figure, since $v\rightarrow \infty$ when $\bar{\varphi}\rightarrow 0$, which also results in a cut off for small $\bar{\varphi}$ on the scalar field dominant boundary $v\bar{\varphi} = 1/\sqrt{3}$.
• Figure 2: The orbit structure on the scalar field dominant boundary $v\bar{\varphi}=1/\sqrt{3}$ projected onto $(\bar{\varphi},u)$ with $0<\bar{\varphi}\leq1$ on the horizontal axis and $u\in [-\sqrt2, \sqrt2]$ on the vertical axis for the cases $-\sqrt{6} < \lambda_+ <0$ and $\lambda_+ \leq -\sqrt{6}$, respectively. These two cases are illustrated by the models given in eq. \ref{['lambda.hyp.gen']} with $p=1/2$, $\nu=2$ and the two values $\alpha = -1$ and $\alpha=-10$, which result in $\lambda_+=-1$ and $\lambda_+ = - 10$, respectively, since $\lambda_+ = p\nu\alpha$ for these models. The dotted line corresponds to the cut off in $\bar{\varphi}$ on the scalar field dominant boundary in Figure \ref{['Fig1']}.
• Figure 3: Tracker orbits in the ski-slope state-space and the respective graph of $w_\varphi(N)$ for the potential \ref{['lambda.hyp.gen']}, illustrating that $w_\varphi$ can decrease, be a constant, and increase for the tracker orbit.
• Figure 4: The tracker orbit in the ski-slope state-space and their graphs $w_\varphi(N)$ for the potential \ref{['lambda.hyp.gen']} with a minimum, together with the $\Lambda$CDM orbit, a thawing quintessence reference orbit originating from $\mathrm{FL}_0^{\varphi_*}$, and also a scaling freezing quintessence reference orbit when $\lambda_+ \ll -1$ in figure (b), with associated graphs of $w_\varphi$ in (c) and (d), where $w_\varphi$ for the thawing orbit begins with $w_\varphi \approx -1$, while the tracker orbit begins with $w_\varphi \approx -2/(2+p)$, which for $p=1/2$ yields $w_\varphi \approx -0.8$, and where the dashed scaling freezing orbit begins with $w_\varphi \approx 0$.
• Figure 5: The tracker orbit (full line), overshooting (dashed line) and undershooting (dotted line) orbits in the ski-slope state-space and their graphs $w_\varphi(N)$ for the monotonically decreasing positive potential \ref{['lambda.hyp.gen']} with $\nu=1$, $p=3$ and $\alpha=0$.