Dip and non-linearity in the curvature perturbation from inflation with a transient non-slow-roll stage

Abstract

We consider models of inflation that contain a transient non-slow-roll stage and investigate the conditions under which a dip appears in the power spectrum of the curvature perturbation. Using the $δN$ formalism, we derive a general relation between the comoving curvature perturbation ${\cal{R}}$ and the scalar field perturbation $δφ$ and its velocity perturbation $δπ$. Compared with the result obtained in linear perturbation theory, it turns out that properly taking account of the $δπ$ contribution is essential to reproduce the dip in the power spectrum. Namely, the curvature perturbation is proportional to a specific linear combination of $δφ$ and $δπ$ at the linear order. We also investigate the non-linearity at the dip scale and find that models with a bump or an upward step exhibit much larger non-linearity than ultra-slow-roll and Starobinsky's linear potential models. Finally, we demonstrate the importance of non-linearity by computing the probability density functions (PDFs) for the above-mentioned models and show that highly asymmetric PDFs are realised for models with a bump or a step.