An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation

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

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