Negative running prevents eternal inflation

Understand

Current data from the Planck satellite and the BICEP2 telescope favor, at around the $2 \sigma$ level, negative running of the spectral index of curvature perturbations from inflation.

• We show that for negative running $\alpha < 0$, the curvature perturbation amplitude has a maximum on scales larger than our current horizon size.
• A condition for the absence of eternal inflation is that the curvature perturbation amplitude always remain below unity on superhorizon scales.
• For current bounds on $n_{\rm S}$ from Planck, this corresponds to an upper bound of the running $\alpha < - 4 \times 10^{-5}$, so that even tiny running of the scalar spectral index is sufficient to prevent eternal inflation from occurring, as long as the running remains negative on scales outside the horizon.