Nonincreasing Hubble rate for a canonical inflaton ($\dot{H}=-4\pi G{\dot{\varphi }}^2\le 0$)

For flat expansion with a canonical scalar and no additional fluid, differentiating the Friedmann equation gives $\dot{H}=-4\pi G{\dot{\varphi }}^2\le 0$. Expansion can still accelerate if $-\dot{H}/H^2<1$. Equality at an isolated turning point does not imply eternal de Sitter expansion.