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