Figure 1

The DBI solutions using quartic/quadratic potential/tension functions of Eq. (9) are plotted in the $w\mathrm{\text{−}}{w}^{\prime }$ plane. The initial thawing behavior, the values today (denoted by x’s along the curves) with property $w^{\prime }\approx 1+w$, and future attractors to a constant $w$ determined by the value of ${\mu }^2$ are all evident.Reuse & Permissions

Figure 2

The high Hubble friction during matter domination freezes the field to $w=-1$ for many e-folds in expansion, despite an initial field velocity measured in terms of the Lorentz factor ${\gamma }_i$. This pseudoattractor ensures that models with different ${\gamma }_i$ then follow the same trajectory, as shown by the convergence of tracks from the left side (early times) to the middle (later times). (Tracks start in the plot at ${\Omega }_{\varphi }={10}^{-10}$, with ${\varphi }_i=-4$ and the ${\gamma }_i$ as labeled.) The light, green curves diverging from the middle to the right side (today) show this is not a true attractor since the thawing rate does depend on the initial field value ${\varphi }_i$ (here fixing ${\gamma }_i=1$). However, the late time, true attractor from DBI dynamics will bring all these trajectories together; indeed at present all models with $\kappa {\varphi }_i<1$ have the same dynamics.Reuse & Permissions

Figure 3

Model parameters can be constrained by comparison to distance data, here taken to be within 2% of $\Lambda \mathrm{CDM}$. Above the solid curves for each value of ${\mu }^2$ the deviation is less than 2% (as ${\varphi }_i$ gets large, $w$ has deviated less from the value $w=-1$ imposed by the matter dominated freezing). Below the solid curves the distance deviation is larger, but often by a small amount: only within the dotted, black contour is the deviation more than 2.1% for ${\mu }^2=50$, and similarly the dashed, red contour bounds the deviation to 2.4% for ${\mu }^2=40$.Reuse & Permissions

Figure 4

Solutions for the potential function are exhibited that deliver constant values of the equation of state $w$. Solid portions of the curves correspond to the region over which the field has rolled by the present. Short dotted arcs near $\varphi =0$ show the $V\sim {\varphi }^2$ asymptotic behavior. The potentials do not contain an explicit cosmological constant [i.e. $V(\varphi )$ has a true zero minimum], but the equations of state can approach $w=-1$ due to the DBI dynamics.Reuse & Permissions

Figure 5

The presence of the multibrane potential $U$ alters the conditions for attractor solutions and opens up new routes to approach $w=-1$. When ${\lambda }^2/U\to 0$, then the cosmological constant is the asymptotic solution. When this combination goes to a constant value, then $w\to \mathrm{\text{const}}$ given by Eq. (26), and when the combination goes to $\infty$ then the standard DBI solution is unchanged. For a power law potential $U\sim {\varphi }^p$, these correspond to $p<-2$, $=-2$, and $>-2$.Reuse & Permissions

Figure 6

As in Fig. 5 but for a form of the standard potential $V$ that would not give an accelerating attractor in standard DBI theory. Here, however, the multibrane potential can give constant $w$ (for $p=-2$) or a cosmological constant (for $p<-2$).Reuse & Permissions