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

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.