Running scalar spectral index in warm natural inflation

TL;DR

The paper addresses whether warm inflation can consistently accommodate a running scalar spectral index by deriving general analytical expressions for $\alpha_{\mathrm{s}}$ in WI with linear ($p=1$) and cubic ($p=3$) dissipation and applying them to warm natural inflation (WNI). It provides exact formulas for $n_{\mathrm{s}}$ and $\alpha_{\mathrm{s}}$ in both dissipative cases, validates them against numerical WI results, and uses them to constrain the symmetry-breaking scale in WNI. The numerical analysis shows that WNI with either dissipation type can fit Planck data for $n_{\mathrm{s}}$ and $r$, while $\alpha_{\mathrm{s}}$ imposes tighter bounds, yielding a lower bound on the decay constant $f$ around $\tilde{f} \gtrsim 4.0-5.0$, depending on $N$. Overall, the work strengthens the phenomenology of WI and demonstrates that $\alpha_{\mathrm{s}}$ provides a meaningful discriminator among WI models, with implications for the underlying particle physics of the inflaton.

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

Figures (4)

• Figure 1: Comparison with a previous study.
• Figure 2: $n_{\mathrm{s}}$-$r$ (left) and $n_{\mathrm{s}}$--$\alpha_{\mathrm{s}}$ (right) in CNI ($Q=0$ case).
• Figure 3: $n_{\mathrm{s}}$, $r$ and $\alpha_{\mathrm{s}}$ in the WNI with the linear dissipative coefficient.
• Figure 4: The same as Fig. \ref{['FIG:WNI_linear']} but with the cubic dissipative coefficient.