Growing curvature perturbation in ultra-slow-roll inflation
= Growing curvature perturbation in ultra-slow-roll inflation
In ultra-slow-roll inflation, $z=a\sqrt{2\epsilon}\propto a^{-2}\propto\eta^2$. The comoving curvature mode equation is
$$
\mathcal R_k''+\frac4\eta\mathcal R_k'+k^2\mathcal R_k=0,
$$
and 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 $a^3$.