Running scalar spectral index in warm natural inflation

The validity of inflation models is mainly evaluated according to the consistency of the predicted scalar spectral index n s , the tensor scalar ratio r, and the running scalar spectral index α s with cosmic microwave background observations. In warm inflation (WI) scenarios, one can find exact analytical solutions for α s in principle, but long expressions may be obtained. Previous studies for WI scenarios have only shown approximate analytical solutions or numerical results for α s . In this study, we present a general analytical expression of α s without approximation in WI. By providing an analytical expression, even if it is mathematically redundant, we believe that α s will be studied across a broader range of WI models in the future. The obtained analytical expression of α s is used in the study of warm natural inflation (WNI). Although ns and r have been previously investigated, α s is omitted in previous studies on WNI. Our study of α s completes previous phenomenological studies on WNI. In particular, the lower limit of the symmetry-breaking scale in WNI becomes more concrete in this study.