Eur. Phys. J. C (2023) 83: 783
https://doi.org/10.1140/epjc/s10052-023-11959-0

Regular Article - Theoretical Physics

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

Department of Physics, College of Sciences, Northeastern University, 110819, Shenyang, China

^d linan@mail.neu.edu.cn

Received: 26 June 2023
Accepted: 25 August 2023
Published online: 5 September 2023

Abstract

A plateau on the background inflaton potential can lead inflation into the ultraslow-roll (USR) stage, during which the primordial curvature perturbation $\mathcal{R}$ can still remarkably evolve outside the horizon. In this paper, the evolution of the Fourier mode ${\mathcal{R}}_k$ and the resulting power spectrum ${\mathcal{P}}_{\mathcal{R}}(k)$ are systematically investigated. It is found that, when a scale crosses the horizon well before or after the USR stage, ${\mathcal{R}}_k$ 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 ${\mathcal{R}}_k$ will be nearly frozen and ${\mathcal{R}}_k$ 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 ${\mathcal{P}}_{\mathcal{R}}(k)$ on small scales from the minimum to the peak. Moreover, there is a sharp dip in ${\mathcal{P}}_{\mathcal{R}}(k)$, but the lowest point cannot reach 0 exactly. Our work provides a whole picture for ${\mathcal{R}}_k$ and ${\mathcal{P}}_{\mathcal{R}}(k)$ in the USR inflation and will help our understanding of the relevant primordial black hole and gravitational wave physics.