 Solution

Write $r={\varphi }_0^2/M_{\mathrm{Pl}}^2$. As $t\to 0$,so $\varphi \to -\infty$ andBecause $2/3<r<2$, the power $r/2$ lies in $(1/3,1)$; thus $\ddot{a}<0$ and the early expansion decelerates. As $t\to \infty$, $\varphi \to 0$ and $\sinh (2\lambda t)\sim e^{2\lambda t}/2$, givingso the late universe has accelerating expansion.

The transition satisfies $\ddot{a}/a=H^2+\dot{H}=0$. Substitution givesor $r{\cosh }^2(2\lambda t)=2$. ThereforeThis is the transition described by the exact hyperbolic-sine scalar-field cosmology.