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

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 ℛ and the scalar field perturbation δπ 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 models mentioned above and show that highly asymmetric PDFs are realised for models with a bump or a step.