Negative running prevents eternal inflation
  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. In single-field inflation models,
negative running is associated with a finite duration of inflation: we show
that eternal inflation may not occur even in cases where inflation lasts as
long as $10^4$ e-folds.
