A complete analysis of inflation with piecewise quadratic potential

TL;DR

This paper thoroughly analyzes the curvature perturbation in single-field inflation with a piecewise quadratic potential, revealing how potential derivatives influence the power spectrum and non-Gaussian features, with implications for primordial black hole formation.

Contribution

It introduces a comprehensive method combining perturbative and non-perturbative techniques to study inflation with discontinuous potential derivatives and identifies a new parameter affecting spectrum features.

Findings

Spectrum can grow as fast as $k^5( ext{log} k)^2$ with fine-tuning.

The second derivatives of the potential significantly influence the spectrum and statistics.

Non-Gaussian tails impact primordial black hole formation.

Abstract

We conduct a thorough study of the comoving curvature perturbation $\mathcal{R}$ in single-field inflation with two stages, represented by a piecewise quadratic potential, where both the first and second derivatives are allowed to be discontinuous at the transition point. We calculate the evolution of $\mathcal{R}$ by combining the perturbative and non-perturbative methods consistently, and obtain the power spectrum and the non-Gaussian features in the probability distribution function. We find that both the spectrum and the statistics of $\mathcal{R}$ depend significantly on the second derivatives of the potential at both the first and second stages. Furthermore, we find a new parameter constructed from the potential parameters, which we call $\alpha$, plays a decisive role in determining various features in the spectrum such as the amplitude, the slope, and the existence of a dip. In particular, we recover the typical $k^4$ growth of the spectrum in most cases, but the maximum growth rate of $k^5(\log k)^2$ can be obtained by fine-tuning the parameters. Then, using the $\delta N$ formalism valid on superhorizon scales, we give fully nonlinear formulas for $\cal{R}$ in terms of the scalar field perturbation $\delta\phi$ and its time derivative. In passing, we point out the importance of the nonlinear evolution of $\delta\phi$ on superhorizon scales. Finally, using the Press-Schechter formalism for simplicity, we discuss the effect of the non-Gaussian tails of the probability distribution function on the primordial black hole formation.