Hubble slow-roll parameter

A Hubble slow-roll parameter measures the fractional change of the Hubble rate or of a preceding slow-roll parameter per e-fold. Small values make the inflationary background evolve adiabatically.

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.

The first Hubble slow-roll parameter isFor a canonical inflaton, $ϵ={\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2{H}^2)\ge 0$.

Using the e-fold variable $N=\log a$, define