# On the constant-roll inflation with large and small $\eta_H$

## TL;DR

This paper investigates the duality between large and small $ta_H$ in constant-roll inflation, concluding that the duality does not hold and providing observational constraints on $ta_H$ for different inflation models.

## Contribution

It clarifies the non-existence of duality between large and small $ta_H$ in constant-roll inflation and derives observational bounds on $ta_H$ for models with increasing or decreasing $psilon_H$.

## Key findings

- Duality between large and small $ta_H$ does not exist.
- Constraints on $ta_H$ for models with increasing $psilon_H$.
- Constraints on $ta_H$ for models with decreasing $psilon_H$.

## Abstract

We study the seemingly duality between large and small $\eta_H$ for the constant-roll inflation with the second slow-roll parameter $\eta_H$ being a constant. In the previous studies, only the constant-roll inflationary models with small $\eta_H$ are found to be consistent with the observations. The seemingly duality suggests that the constant-roll inflationary models with large $\eta_H$ may be also consistent with the observations. We find that the duality between the constant-roll inflation with large and small $\eta_H$ does not exist because both the background and scalar perturbation evolutions are very different. By fitting the constant-roll inflationary models to the observations, we get $-0.016\le\eta_H\le-0.0078$ at the 95\% C.L if we take $N=60$ for the models with increasing $\epsilon_H$ in which inflation ends when $\epsilon_H=1$, and $3.0135\le \eta_H\le 3.021$ at the 68\% C.L., and $3.0115\le \eta_H\le 3.024$ at the 95\% C.L. for the models with decreasing $\epsilon_H$.