The observational constraint on constant-roll inflation

We discuss the constant-roll inflation with constant ϵ2 and constant η¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \eta $$\end{document}. By using the method of Bessel function approximation, the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor to scalar ratio are derived up to the first order of ϵ1. The model with constant ϵ2 is ruled out by the observations at the 3σ confidence level, and the model with constant η¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \eta $$\end{document} is consistent with the observations at the 1σ confidence level. The potential for the model with constant η¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \eta $$\end{document} is also obtained from the Hamilton-Jacobi equation. Although the observations constrain the constant-roll inflation to be the slow-roll inflation, the ns-r results from the constant-roll inflation are not the same as those from the slow-roll inflation even when η¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \eta $$\end{document} ~ 0.01.