An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation: an extended study

Ji-Xiang Zhao; Nan Li

doi:10.1140/epjc/s10052-024-12578-z

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2024) 84: 222
https://doi.org/10.1140/epjc/s10052-024-12578-z

Regular Article - Theoretical Physics

An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation: an extended study

Ji-Xiang Zhao and Nan Li^b

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

^b linan@mail.neu.edu.cn

Received: 24 November 2023
Accepted: 18 February 2024
Published online: 2 March 2024

Abstract

Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation $R_{k}$ ${{\mathscr {R}}}_k$ and its power spectrum $P_{R}$ ${{\mathscr {P}}}_{{\mathscr {R}}}$ can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of $R_{k}$ ${{\mathscr {R}}}_k$ in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of $R_{k}$ ${{\mathscr {R}}}_k$ and its time derivative $R_{k, N}$ ${{\mathscr {R}}}_{k,N}$ on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, $R_{k}$ ${{\mathscr {R}}}_k$ on five typical scales are investigated in order. Our analytical approximation predicts that $R_{k}$ ${{\mathscr {R}}}_k$ first revolves around the origin in the complex plane, but if it crosses the horizon around the start of the USR stage, there appears a subsequent linear evolution towards or away from the origin. This behavior naturally explains the shape of $P_{R}$ ${{\mathscr {P}}}_{{\mathscr {R}}}$ from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of $P_{R}$ ${{\mathscr {P}}}_{{\mathscr {R}}}$ is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics.

© The Author(s) 2024

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