Quintessence: Quadratic potentials

TL;DR

The work examines global quintessence dynamics in a flat FRW universe with dust for two quadratic potentials $V = \Lambda \pm \tfrac{1}{2} m^2\varphi^2$. It develops regular unconstrained dynamical systems on compact state spaces for both cases, deriving monotone quantities and mapping the full solution space to identify attractors, fixed points, and invariant boundaries. A new dynamical-systems formulation is provided for the hilltop case $V=\Lambda - \tfrac{1}{2} m^2\varphi^2$, revealing a torus-like compact state space with kinaton, FL, and de Sitter fixed points and a ΛCDM–like subset connected by heteroclinic orbits; a one-parameter quintessence attractor surface emerges, governing thawing behavior. The analysis shows that interior trajectories are primarily heteroclinic, while observationally viable thawing quintessence models reside on a two-dimensional attractor surface that connects matter domination to late-time acceleration, providing a global geometric understanding of quintessence dynamics and its observational implications.

Abstract

Arguably one can use a canonical scalar field $\varphi$, minimally coupled to gravity, with quadratic potentials $V = Λ\pm \frac12 m^2\varphi^2$ to explore some general features of slow-roll and hilltop thawing quintessence, respectively. For each of these two potentials, and pressure-free matter, we introduce a regular unconstrained dynamical system on a compact state space, where the formulation for the hilltop case is new. Together with a derivation of monotonic functions in the two global state space settings, this enables us to obtain global results and to introduce figures that illustrate the global solution spaces of these models, in which we situate the observationally viable quintessence solutions.

Figures (7)

• Figure 1: The potentials $V = \Lambda \pm \frac{1}{2} m^2\varphi^2$.
• Figure 2: Inflationary attractor (separatrix) orbits $\mathrm{dS}^\pm \rightarrow \mathrm{dS}^0$ on the $v= 1/\sqrt{3}$ ($\Omega_\varphi = 1$) boundary for $V(\varphi) = \Lambda(1 + \frac{1}{2} \bar{m}^2\,\varphi^2)$, where $\bar{m} = \frac{m}{\sqrt{2\Lambda}}$.
• Figure 3: State space $(\bar{\varphi},u,v)$ depiction of thawing quintessence orbits for the potential $V(\varphi) = \Lambda(1 + \bar{m}^2\,\varphi^2)$, where $\bar{m} = \frac{m}{\sqrt{2\Lambda}}$, and orbits with frozen $\varphi =\varphi_*$ on the $v=0$ boundary, responsible for the thawing quintessence mechanism in this state space representation.
• Figure 4: Orbit structure on the $\bar{s}=0$ (figure to the left) and $\bar{\Omega}_\mathrm{m}=0$ (figure to the right) boundaries, where in the latter case the orbits have been projected onto the plane $\bar{s}=0$.
• Figure 5: The $V(\varphi)=0$ surface 'originating' from $\theta=0$; in addition, there is a corresponding $V(\varphi)=0$ surface, not depicted in order to avoid clutter, at $\theta=\pi$, since $V=0 \Rightarrow \bar{s} = |\bar{\varphi}| = |\sin\theta|$.