False vacuum decay with gravity: Negative mode problem

Abstract

Motivated by work of Lavrelashvili, Rubakov and Tinyakov (LRT), who claim that the WKB approximation would break down even at energy scale far below the Planck one, the decay of false vacuum under the presence of gravity is carefully investigated. First we approach the issue from a different point of view, namely, by solving the Wheeler-DeWitt equation under the boundary condition appropriate for the false vacuum decay. The resulting wave function shows no sign of the break-down of the WKB approximation. Then, based on the canonical Hamiltonian form of the Euclidean path integral, we develop a method to analyze the number of negative modes around the Euclidean bounce solution describing the tunneling process and evaluate it in some limiting cases. Again, we find a strong evidence that LRT's claim is superficial.