Quintessence

TL;DR

Late-time cosmic acceleration may originate from dynamical dark energy in the form of quintessence rather than a cosmological constant. The authors reformulate the field equations as a regular, bounded 3D dynamical system on the state space spanned by $\bar{\varphi}$, $u$, and $v$, with $\lambda(\bar{\varphi})=-\frac{d\ln V}{d\varphi}$, enabling a global analysis of thawing, scaling freezing, and tracking quintessence. They identify the attractor structure through boundary fixed points (e.g., $\mathrm{FL}_0^{\varphi_*}$, $\mathrm{T}$, $\mathrm{P}$) and their connections via unstable manifolds, and they derive simple analytic Padé approximations for $w_{\mathrm{DE}}(N)$ and $\Omega_{\mathrm{DE}}(N)$ that apply across representative potentials such as $V\propto e^{-\lambda\varphi}$ and inverse-power laws. This framework clarifies initial-condition sensitivity, unifies diverse quintessence behaviors under a single dynamical-systems viewpoint, and provides practical tools for predicting cosmological observables across a broad class of potentials.

Abstract

Recent observations suggest that the accelerated expansion of the Universe at late times is caused by a temporally changing dark energy component, rather than the constant one in the standard $Λ$CDM scenario. In this context quintessence, i.e. a canonical scalar field minimally coupled to gravity, plays a prominent role. There are, however, three main types of quintessence models: thawing quintessence, scaling freezing quintessence, and tracking quintessence. Dynamical systems reformulations of the field equations for a broad set of scalar field potentials, including some new ones, allow us to use dynamical systems methods to derive global and asymptotic features, visualised in bounded state space pictures clearly illustrating the relationships and properties of the different types of quintessence, clarifying initial data issues, and yielding simple and accurate approximations.

Figures (4)

• Figure 1: Solutions $\mathrm{FL}_\pm^{\varphi_*} \rightarrow \mathrm{FL}_0^{\varphi_*}$ on the matter-dominated boundary $v=0$ ($\Omega_\mathrm{m}=1$), which provide the thawing quintessence attractor mechanism, and solutions on the unstable $\mathrm{FL}_0^{\varphi_*}$ manifold surface (${\bf U}\mathrm{FL}_0^{\varphi_*}$) for $V \propto \exp(-\lambda\varphi)$. Figure (a) also depicts solutions $\mathrm{K}_\pm^\mp \rightarrow \mathrm{FL}_\pm^{\varphi_*}$ on the boundaries $u=\pm \sqrt{2}$ ($\Omega_V=0$). Figure (b) illustrates that there are classes of potentials that continuously deform the $\Lambda$CDM ${\bf U}\mathrm{FL}_0^{\varphi_*}$ with $u=0$ in Figure (a) to a thawing quintessence ${\bf U}\mathrm{FL}_0^{\varphi_*}$. The boundary $\mathrm{P}^-\rightarrow\mathrm{P}^+$ of ${\bf U}\mathrm{FL}_0^{\varphi_*}$ in Figure (b) is replaced with the scaling solution $\mathrm{S}^-\rightarrow\mathrm{S}^+$ in Figure (c) when $\lambda >\sqrt{3}$, where the scaling matter-dominant condition $\lambda \gg 1$ implies that $\Omega_\varphi$ never becomes observationally significant for either type of quintessence.
• Figure 2: (a) Depicts the surface of thawing quintessence attractor solutions with the scaling freezing attractor solution $\mathrm{S}^- \rightarrow \mathrm{P}^+$ as its interior boundary for the potential $V = M_-^4e^{-\lambda_-\varphi}+ M_+^4e^{-\lambda_+\varphi}$. (b) Depicts the inflationary $\alpha$-attractor solution for $V \propto \frac{e^{-\nu(1 + \tanh\frac{\varphi}{\sqrt{6\alpha}})} - e^{-2\nu}}{1 - e^{-2\nu}}$ on the scalar field boundary $v=1/\sqrt{3}$ ($\Omega_\varphi = 1$) and the surface of thawing quintessence attractor solutions, where only those with $\bar{\varphi}_*\approx 1$ are cosmologically viable.
• Figure 3: Illustration of the three cases of tracking quintessence attractor solutions with (a) $w_\varphi|_\mathrm{T} = -0.7\,\rightarrow\, w_\varphi|_\mathrm{P} = -0.9$, (b) $w_\varphi = w_\varphi|_\mathrm{T} = w_\varphi|_\mathrm{P} = -0.8$, (c) $w_\varphi|_\mathrm{T} = -0.9\,\rightarrow\, w_\varphi|_\mathrm{P} = -0.7$, for $V = V_*\sinh^{-p}(\nu\varphi)$ (the ring denotes when $\Omega_\varphi = 0.68$ for these solutions), and how they form the interior boundary of the thawing quintessence attractor solutions.
• Figure 4: The figures depict the tracking attractor quintessence solution $\mathrm{T}\rightarrow\mathrm{dS}$ (the ring denotes when $\Omega_\varphi = 0.68$), showing that it is the interior boundary of the surface of thawing quintessence solutions. The $\mathrm{FL}_0^{\varphi_*}\rightarrow\mathrm{T}$ solution on the boundary $\bar{\varphi}=0$ illustrates that the tracker fixed point $\mathrm{T}$ is a stable focus on this boundary; (a) shows the key quintessence aspects for $V \propto \varphi^{-p}$ whereas (b) depicts them for $V \propto \exp(\frac{p}{r}\varphi^{-r}) - 1$. Note that the tracking quintessence solution $\mathrm{T}\rightarrow\mathrm{dS}$ in (b) is the unstable centre manifold of $\mathrm{T}$.