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 $\delta N$ formalism, we derive a general relation between the comoving curvature perturbation $\mathcal{R}$ and the scalar field perturbation $\delta \phi$ and its velocity perturbation $\delta \pi$. Compared with the result obtained in linear perturbation theory, it turns out that properly taking account of the $\delta \pi$ contribution is essential to reproduce the dip in the power spectrum. Namely, the curvature perturbation is proportional to a specific linear combination of $\delta \phi$ and $\delta \pi$ 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.

35 pages, 15 figures

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