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