It was pointed out previously [1] that a
sufficiently negative running of the spectral index of curvature
perturbations from (ordinary i.e. cold) inflation is able to prevent eternal inflation from
ever occurring. Here, we reevaluate those original results, but in the
context of warm inflation, in which a substantial radiation component (produced by the inflaton) exists throughout the inflationary period.
We demonstrate that the same general
requirements
found in the context
of ordinary (cold) inflation also hold true in warm inflation; indeed an even tinier amount of negative running
is sufficient to prevent eternal inflation. This is particularly pertinent, as models featuring negative running are more generic in warm inflation scenarios.
Finally, the condition for the existence of eternal inflation in cold inflation — that the curvature perturbation amplitude exceed unity on superhorizon scales — becomes more restrictive in the case of warm inflation. The curvature perturbations must be even larger, i.e. even farther out on the potential, away from the part of the potential where observables, e.g. in the Cosmic Microwave Background, are produced.