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Recently, the calculation of tunneling actions, that control the exponentialsuppression of the decay of metastable vacua, has been reformulated as anelementary variational problem in field space. This paper extends thisformalism to include the effect of gravity. Considering tunneling potentials$V_t(\phi)$ that go from the false vacuum $\phi_+$ to some $\phi_0$ on thestable basin of the scalar potential $V(\phi)$, the tunneling action is theminimum of the functional $S_E[V_t]=6\pi^2m_P^4\int_{\phi_+}^{\phi_0}(D+V_t')^2/(V_t^2D)d\phi $, where $D\equiv[(V_t')^2+6(V-V_t)V_t/m_P^2]^{1/2}$, $V_t'=dV_t/d\phi$ and $m_P$ is the reducedPlanck mass. This one-line simple result applies equally to AdS, Minkowski ordS vacua decays and reproduces the Hawking-Moss action in the appropriatecases. This formalism provides new handles for the theoretical understanding ofdifferent features of vacuum decay in the presence of gravity.

A Fresh Look at the Calculation of Tunneling Actions including Gravitational Effects