Ultra-slow-roll inflation

Ultra-slow-roll inflation occurs on an exceptionally flat potential. The homogeneous field velocity decays as $\dot{\varphi }\propto a^{-3}$, so $ϵ\propto a^{-6}$ and the second Hubble slow-roll parameter approaches $-6$.

In ultra-slow-roll inflation, $z=a\sqrt{2ϵ}\propto a^{-2}\propto {\eta }^2$. The comoving curvature mode equation isand its Bunch-Davies mode is proportional to $(1+ik\eta )e^{-ik\eta }/{\eta }^3$. Unlike ordinary slow roll, the superhorizon curvature perturbation grows as $a^3$.

For a perturbation $X$, a normalized commutatorcompares its quantum commutator with its root-mean-square phase-space fluctuations. A limit $\mathcal{C}\to 0$ signals effective classicality.