Potential and Hubble slow-roll parameter relation ($\eta \simeq 4{ϵ}_V-2{\eta }_V$)

For a canonical inflaton under the slow-roll approximation, $\dot{\varphi }/H\simeq -M_{\mathrm{Pl}}^2{V}^{\prime }/V$ and $ϵ\simeq {ϵ}_V=(M_{\mathrm{Pl}}^2/2)(V^{\prime }/V{)}^2$. Differentiating $\log {ϵ}_V$ along this trajectory givesHere $\eta =d\log ϵ/d\log a$, not the alternate convention $-\ddot{\varphi }/(H\dot{\varphi })$. The logarithmic definition requires $ϵ>0$; exact de Sitter expansion is treated as a limit.