Dark energy from $\alpha$-attractors

A class of inflation theories called $\alpha$-attractors has been investigated recently with interesting properties interpolating between quadratic potentials, the Starobinsky model, and an attractor limit. Here we examine their use for late-time cosmic acceleration. We generalize the class and demonstrate how it can interpolate between thawing and freezing dark energy, and reduce the fine-tuning of initial conditions, allowing $w\approx -1$ for a prolonged period or as a de Sitter attractor.

Figure 1

The potential is plotted as a function of the field $\varphi$ in units of $\sqrt{\alpha }$, for various values $n$. At small field values $\varphi \ll \sqrt{\alpha }$ the $\alpha$-model resembles the quadratic potential $V\propto {\varphi }^2$ (short, dotted green curve), while at large values it flattens to an uplifted plateau.

Figure 2

The present value of the equation of state ratio $w_0$ is plotted as a function of the initial field value ${\varphi }_i$ in units of $\sqrt{\alpha }$, for various values of $n$ (solid curves). For contrast, this relation is also shown for a quadratic potential (dotted green curve, where the x-axis is simply ${\varphi }_i$). For a given value ${\varphi }_i$, the $\alpha$-model can achieve $w_0$ closer to $-1$ (as preferred by observations). Conversely for a given bound on the deviation of $w_0$ from $-1$, the $\alpha$-model can be less fine-tuned in the initial field value.

Figure 3

The first derivative of the potential is plotted as a function of the field $\varphi$ in units of $\sqrt{\alpha }$, for various values $n$ (increasing from top to bottom curves). Note that all derivatives approach zero at large field values (since the potential plateau is asymptotically flat).

Figure 4

The dynamics in the $w-w^{\prime }$ phase space transitions between thawing and freezing dark energy, with an attractor behavior both toward a cosmological constant and toward common dynamics as $\alpha \to 0$. Models with $p=2/3$, 1, and 2 (red, blue, magenta, or from outer to inner) are shown, for $\alpha =1$ (solid curves) and $\alpha =0.1$ (dotted curves). The present is denoted by x’s and the curves are stopped at $a=220$ for clarity. Note the $p=2$, $\alpha =0.1$ case has not quite turned around since it is very close to the potential maximum.

Figure 5

The dynamics in the $w-w^{\prime }$ phase space increasingly resembles a quadratic potential (dashed, long magenta curve) as $\alpha \to \infty$. Models with $n=0$, 1, and 2 (black, blue, red, or from longest to shortest curves for each line type) are shown, for $\alpha =1$ (solid curves) and $\alpha =10$ (dotted curves). For $\alpha =10$, the curves, and the endpoints at $a=1$, move noticeably closer to the quadratic result.

Figure 6

The second derivative of the potential, giving the effective mass squared, is plotted as a function of the field $\varphi$ in units of $\sqrt{\alpha }$, for various values $n$ (increasing from top to bottom curves at small $\varphi$). Note that all derivatives approach zero at large field values (since the potential plateau is asymptotically flat).