  A formalism to describe the false-vacuum decay in non-perturbative regimes
was proposed recently. Here, we extend it to the presence of Einstein gravity
and calculate the corresponding effective potential and decay rate for a
$\lambda \phi^4$ scalar field theory. A comparison with the usual perturbative
decay rate shows that the higher the coupling $\lambda$, the greater the decay
probability. From the running of the self-interaction coupling, we conclude
that the theory becomes weakly coupled in the infrared limit, which proves that
Einstein gravity made the weak coupling approximation even more reliable as the
universe cooled down. We comment on future applications of these results to
cosmological phase transitions, gravitational-wave astronomy, and condensed
matter systems.
