Fresh look at the calculation of tunneling actions including gravitational effects

Abstract

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 Vt(φ) that go from the false vacuum φ+ to some φ0 on the stable basin of the scalar potential V(φ), the tunneling action is the minimum of the functional SE[Vt]=6π2mP4∫φ+φ0(D+Vt′)2/(Vt2D)dφ, where D[(Vt′)2+6(V-Vt)Vt/mP2]1/2, Vt′=dVt/dφ, and mP is the reduced Planck mass. This one-line simple result applies equally to anti-de Sitter, Minkowski, or de Sitter 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.