The evolution of the primordial curvature perturbation in the ultraslow-roll inflation

Abstract A plateau on the background inflaton potential can lead inflation into the ultraslow-roll (USR) stage, during which the primordial curvature perturbation $${{{\mathscr {R}}}}$$ R can still remarkably evolve outside the horizon. In this paper, the evolution of the Fourier mode $${{{\mathscr {R}}}}_k$$ R k and the resulting power spectrum $${{\mathscr {P}}}_{{{\mathscr {R}}}}(k)$$ P R ( k ) are systematically investigated. It is found that, when a scale crosses the horizon well before or after the USR stage, $${{{\mathscr {R}}}}_k$$ R k only revolves around the origin in the complex plane. In contrast, if a scale crosses the horizon around the start of the USR stage, the argument of $${{{\mathscr {R}}}}_k$$ R k will be nearly frozen and $${{{\mathscr {R}}}}_k$$ R k will have a subsequent linear evolution accordingly, in which it may stop rather near or extremely far away from the origin, naturally explaining the entire growing stage of $${{\mathscr {P}}}_{{{\mathscr {R}}}}(k)$$ P R ( k ) on small scales from the minimum to the peak. Moreover, there is a sharp dip in $${{\mathscr {P}}}_{{{\mathscr {R}}}}(k)$$ P R ( k ) , but the lowest point cannot reach 0 exactly. Our work provides a whole picture for $${{{\mathscr {R}}}}_k$$ R k and $${{\mathscr {P}}}_{{{\mathscr {R}}}}(k)$$ P R ( k ) in the USR inflation and will help our understanding of the relevant primordial black hole and gravitational wave physics.