Superhorizon curvature perturbation in ultra-slow-roll inflation

We study the growth of superhorizon modes in the curvature perturbation during an ultra-slow-roll or a large-$\eta$ phase in single-field inflation. In a simple toy model, we derive the two-point correlation function of the curvature perturbation and show that the requirement for causality restricts the growth rate and hence puts a lower limit on the value of $\eta$. The toy model is then realized by considering an inflation potential with an inflection point. Our study is useful to assessing the growth of the curvature perturbation that seeds the formation of primordial black holes.