Reexamination of the warm inflation curvature perturbations spectrum

The two approaches to compute perturbations in warm inflation are examined. It is shown that both approaches lead to different expressions for the amplitude of the primordial spectrum, with a difference between them of $\Upsilon/(4H)$ at leading order, where $\Upsilon$ is the dissipation coefficient. In terms of observables, this discrepancy can lead to the spectral index differing by up to order $\mathcal{O}(10^{-3})$, which is within precision demands for current CMB data. Thus, it is important to resolve this ambiguity to have reliable predictions from warm inflation. For this we prove the extent of this discrepancies by deriving a formula for the spectral index and the tensor-to-scalar ratio in each approach. In doing so, we find disparities to be more noticeable in a regime where dissipation is comparable with the expansion rate, which is a very important regime from a phenomenological point of view. To determine the extent of the discrepancy, several cases are examined, including quadratic, quartic and hybrid potentials with quadratic and $T-$dependent dissipative coefficients. The origin of the discrepancy is found to be due to the approximation performed in one of the methods, which underestimates the variation of the momentum perturbation with expansion. Once this is corrected, both approaches are then in agreement.