 Solution

With spatial flatness, the scalar as the only source, and $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$, the Friedmann equation isThe scalar equation gives $\dot{\rho }=\dot{\varphi }(\ddot{\varphi }+V_{,\varphi })=-3H{\dot{\varphi }}^2$, which is the energy-conservation equation $\dot{\rho }+3H(\rho +P)=0$. Differentiating the Friedmann equation on the expanding branch yields $6M_{\mathrm{Pl}}^{2}H\dot{H}=-3H{\dot{\varphi }}^2$, so $\dot{H}=-{\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2)$. Combining this with $\ddot{a}/a=H^2+\dot{H}$ gives the Friedmann acceleration equation:Thus cosmic inflation occurs exactly when $V(\varphi )>{\dot{\varphi }}^2$ in this canonical scalar model. Equivalently $P<-\rho /3$, or the first Hubble slow-roll parameter ${ϵ}_H=-\dot{H}/H^2$ is smaller than one.