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

• Published in: The European physical journal. C, Particles and fields Particles and fields, 2024-03, Vol.84 (3), p.222-14, Article 222
• Main Authors: Zhao, Ji-Xiang, Li, Nan
• Format: Article
• Language: English
• Subjects: Approximation; Astronomy; Astrophysics and Cosmology; Asymptotic methods; Black holes; Curvature; Elementary Particles; Evolution; Gravitational waves; Hadrons; Heavy Ions; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Perturbation; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Regular Article - Theoretical Physics; String Theory; Wave physics
• ISSN: 1434-6052; 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-024-12578-z

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 and its power spectrum P R can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of R k in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of R k and its time derivative R k , N on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, R k on five typical scales are investigated in order. Our analytical approximation predicts that 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 from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of P R is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics.