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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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We discuss the constant-roll inflation with constant $epsilon_2$ and constant $bareta$. 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 $epsilon_1$. The model with constant $epsilon_2$ is ruled out by the observations at the $3sigma$ confidence level, and the model with constant $bareta$ is consistent with the observations at the $1sigma$ confidence level. The potential for the model with constant $bareta$ is also obtained from the Hamilton-Jacobi equation. Although the observations constrain the constant-roll inflation to be slow-roll inflation, the $n_s-r$ results from the constant-roll inflation are not the same as those from the slow-roll inflation even when $baretasim 0.01$.
For the constant-roll tachyon inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts and the tensor to scalar ratio up to the first order by using the method of Bessel function approximation. The derived $n_s-r$ results for the constant-roll inflation are also compared with the observations, we find that only one constant-roll inflation is consistent with the observations and observations constrain the constant-roll inflation to be slow-roll inflation. The tachyon potential is also reconstructed for the constant-roll inflation which is consistent with the observations.
We study the inflationary period driven by a fermionic field which is non-minimally coupled to gravity in the context of the constant-roll approach. We consider the model for a specific form of coupling and perform the corresponding inflationary analysis. By comparing the result with the Planck observations coming from CMB anisotropies, we find the observational constraints on the parameters space of the model and also the predictions the model. We find that the values of $r$ and $n_{s}$ for $-1.5<betaleq-0.9$ are in good agreement with the observations when $|xi|=0.1$ and $N=60$.
In this paper, we study a non-canonical extension of a supergravity-motivated model acting as a vivid counterexample to the cosmic no-hair conjecture due to its unusual coupling between scalar and electromagnetic fields. In particular, a canonical scalar field is replaced by the string-inspired Dirac-Born-Infeld one in this extension. As a result, exact anisotropic inflationary solutions for this Dirac-Born-Infeld model are figured out under a constant-roll condition. Furthermore, numerical calculations are performed to verify that these anisotropic constant-roll solutions are indeed attractive during their inflationary phase.
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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