Lemma: On-shell power-law background

The data (eq:background-data)–(eq:potential) solve (eq:einstein)–(eq:scalar). The gradient of $\bar{\varphi }$ is timelike and future-increasing.

Proof. The Hubble function and scalar derivative are

The nonzero background curvature expressions give

On the displayed scalar,

Thus the energy density is $3M^2{p}^2/t^2$, and $-2M^2\dot{H}={\dot{\bar{\varphi }}}^2$. These verify both independent Einstein equations.

The scalar equation is equivalent to $\ddot{\bar{\varphi }}+3H\dot{\bar{\varphi }}+V^{\prime }(\bar{\varphi })=0$. Its left side is

because $A^2=2p{M}^2$. Finally, ${\bar{g}}^{-1}(\mathrm{d}\bar{\varphi },\mathrm{d}\bar{\varphi })=-A^2/t^2<0$ and $\dot{\bar{\varphi }}>0$.