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Register a new userOn the constant-roll inflation with large and small $eta_H$
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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 $-0.016leeta_Hle-0.0078$ 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 $3.0135le eta_Hle 3.021$ at the 68% C.L., and $3.0115le eta_Hle 3.024$ at the 95% C.L. for the models with decreasing $epsilon_H$.
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The primordial power spectra of scalar and tensor perturbations during slow-roll inflation are usually calculated with the method of Bessel function approximation. For constant-roll or ultra slow-roll inflation, the method of Bessel function approximation may be invalid. We compare the numerical results with the analytical results derived from the Bessel function approximation, and we find that they differ significantly on super-horizon scales if the constant slow-roll parameter $eta_H$ is not small. More accurate method is needed for calculating the primordial power spectrum for constant-roll inflation.
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