On the constant-roll inflation with large and small ${\eta }_H$

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 ${\epsilon }_H$.

13 pages, 4 figures