On the duality in constant-roll inflation

TL;DR

This paper explores a duality in observable parameters and potentials in constant-roll inflation, revealing specific conditions under which dualities in the inflationary models and perturbations occur, especially in small field scenarios.

Contribution

It identifies a duality between large and small $oldsymbol{oldsymbol{ extit{ exteta}}}_H$ in constant-roll inflation and characterizes conditions where this duality applies to the potential and primordial perturbations.

Findings

Duality exists between $n_s$, $r$, and the potential for large and small $ exteta}_H$ when $oldsymbol{oldsymbol{ extepsilon}}_H$ is negligible.

Quadratic potential form remains invariant under $ exteta}_H o 3 - exteta}_H$ in small field approximation.

Logarithmic duality between large and small $ exteta}_H$ for curvature perturbations in quadratic potential models.

Abstract

There is a duality in the observables $n_s$, $r$ and the inflaton potential between large and small $\eta_H$ for the constant-roll inflation if the slow-roll parameter $\epsilon_H$ is negligible. In general, the duality between $\eta_H$ and $\bar{\eta}_H$ does not hold for the background evolution of the inflation. For some particular solutions for the constant-roll inflation with $\eta_H$ being a constant, we find that in the small field approximation, the potential takes the quadratic form and it remains the same when the parameter $\eta_H$ changes to $\bar{\eta}_H=3-\eta_H$. If the scalar field is small and the contribution of $\epsilon_H$ is negligible, we find that there exists the logarithmic duality and the duality between large and small $\eta_H$ for the primordial curvature perturbation in inflationary models with the quadratic potential.