Figure 1

The phase plane for the best-fit $\lambda$ found below, approximately 0.1. The point (0, 0) corresponds to the radiation-dominated epoch. The point (0, 0) is unstable and the point (0.0082, 0.9999) is stable and thus an attractor. The line segment $(x,0)$ is a saddle line. The initial conditions best matching observations lead to the red/dotted trajectory, while the other trajectories have different initial conditions. The outer thin solid line corresponds to $x^2+y^2=1-{\Omega }_{b0}-{\Omega }_{r0}$, giving the correct present dark sector energy density, while the angle $B$ gives the required equation of state $w_{\varphi }=-{sin}^{2}B\simeq -0.78$. So the present-day Universe must lie in the vicinity of the point $A$ which is the intersection between the two thin solid lines, which the dotted line indeed passes through.Reuse & Permissions

Figure 2

The evolution of density fractions for radiation (magenta/dot-dashed line), baryons (blue/dashed line), kinetic term $x^2$ (black/solid line), and potential term $y^2$ (red/dotted line), for the best-fit model found in subsection 4d.Reuse & Permissions

Figure 3

The $1\mathrm{\text{−}}d$ and $2\mathrm{\text{−}}d$ probability distribution for the equation of state today, $w_0$, and the potential parameter, ${log}_{10}(\lambda )$.Reuse & Permissions

Figure 4

The evolution of the equation of state for the scalar, shown for 8 models drawn from the Markov chain. It behaves as the cold dark matter at higher redshifts and dark energy for the lower redshifts. For comparison the evolution of the total $w$ in the $\Lambda \mathrm{CDM}$ model is shown as the thicker black line, which runs more or less centrally through the set.Reuse & Permissions

Figure 5

The adiabatic sound speed squared $c_a^2$ (solid line) and the equation of state (circled line) for the scalar, showing $c_a^2\simeq w_{\varphi }$. At redshifts greater than 6, they are both vanishing.Reuse & Permissions