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

Recently, the calculation of tunneling actions, that control the exponential suppression of the decay of metastable vacua, has been reformulated as an elementary variational problem in field space. This paper extends this formalism to include the effect of gravity. Considering tunneling potentials $V_t(\phi)$ that go from the false vacuum $\phi_+$ to some $\phi_0$ on the stable basin of the scalar potential $V(\phi)$, the tunneling action is the minimum 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 reduced Planck mass. This one-line simple result applies equally to AdS, Minkowski or dS vacua decays and reproduces the Hawking-Moss action in the appropriate cases. This formalism provides new handles for the theoretical understanding of different features of vacuum decay in the presence of gravity.