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

Abstract

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