  Slow-roll inflation can become eternal if the quantum variance of the
inflaton field around its slowly rolling classical trajectory is converted into
a distribution of classical spacetimes inflating at different rates, and if the
variance is large enough compared to the rate of classical rolling that the
probability of an increased rate of expansion is sufficiently high. Both of
these criteria depend sensitively on whether and how perturbation modes of the
inflaton interact and decohere. Decoherence is inevitable as a result of
gravitationally-sourced interactions whose strength are proportional to the
slow-roll parameters. However, the weakness of these interactions means that
decoherence is typically delayed until several Hubble times after modes grow
beyond the Hubble scale. We present perturbative evidence that decoherence of
long-wavelength inflaton modes indeed leads to an ensemble of classical
spacetimes with differing cosmological evolutions. We introduce the notion of
per-branch observables---expectation values with respect to the different
decohered branches of the wave function---and show that the evolution of modes
on individual branches varies from branch to branch. Thus single-field
slow-roll inflation fulfills the quantum-mechanical criteria required for the
validity of the standard picture of eternal inflation. For a given potential,
the delayed decoherence can lead to slight quantitative adjustments to the
regime in which the inflaton undergoes eternal inflation.
