Running scalar spectral index in warm natural inflation

TL;DR

This paper derives an exact analytical expression for the running scalar spectral index in warm inflation models, specifically applying it to warm natural inflation, thereby enhancing the understanding of inflationary predictions and constraints.

Contribution

It provides the first exact analytical formula for  in warm inflation, enabling broader analysis of models and refining the lower limit of the symmetry-breaking scale in warm natural inflation.

Findings

Derived a general analytical expression for  in warm inflation.

Applied the formula to warm natural inflation, refining model constraints.

Enhanced the understanding of inflationary parameter space.

Abstract

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.