SLAC-PUB-2463 January 1980 (T) # GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY $ ^{*} $ Sidney Coleman $ ^{\dagger} $ Stanford Linear Accelerator Center Stanford University, Stanford, California 94305 and Frank De Luccia Institute for Advanced Study Princeton, New Jersey 88548 ## ABSTRACT It is possible for a classical field theory to have two stable homogeneous ground states, only one of which is an absolute energy minimum. In the quantum version of the theory, the ground state of higher energy is a false vacuum, rendered unstable by barrier penetration. There exists a well-established semi-classical theory of the decay of such false vacua. In this paper, we extend this theory to include the effects of gravitation. Contrary to naive expectation, these are not always negligible, and may sometimes be of critical importance, especially in the late stages of the decay process. Submitted to Physical Review ### 1. Introduction Consider the theory of a single scalar field defined by the action, $$ \mathrm{~\boldmath~s~}=\int\mathrm{d}^{4}\mathrm{\boldmath~x~}\left[\frac{1}{2}(\partial_{\mu}\phi)^{2}-\mathrm{\boldmath~u~}(\phi)\right]\qquad, $$ where U is as shown in Fig. 1. That is to say, U has two local minima, $\phi_{\pm}$, only one of which, $\phi_{-}$, is an absolute minimum. The classical field theory defined by Eq. (1.1) possesses two stable homogeneous equilib- brium states, $\phi = \phi_{+}$ and $\phi = \phi_{-}$. In the quantum version of the theory, though, only the second of these corresponds to a truly stable state, a true vacuum. The first decays through barrier penetration; it is a false vacuum. This is a prototypical case; false vacua occur in many field theories. In particular, they occur in some unified electroweak and grand unified theories, and it is this that gives the theory of vacuum decay possible physical importance. For simplicity, though, we will restrict ourselves here to the theory defined by Eq. (1.1); the extension of our methods to more elaborate field theories is straightforward. The decay of the false vacuum is very much like the nucleation processes associated with first-order phase transitions in statistical mechanics. $ ^{1} $ The decay is initiated by the materialization of a bubble of true vacuum within the false vacuum. This is a quantum tunneling event, and has a certain probability of occurrence per unit time per unit volume, $ \SLAC-PUB-2463 January 1980 (T) # GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY $ ^{*} $ Sidney Coleman $ ^{\dagger} $ Stanford Linear Accelerator Center Stanford University, Stanford, California 94305 and Frank De Luccia Institute for Advanced Study Princeton, New Jersey 88548 ## ABSTRACT It is possible for a classical field theory to have two stable homogeneous ground states, only one of which is an absolute energy minimum. In the quantum version of the theory, the ground state of higher energy is a false vacuum, rendered unstable by barrier penetration. There exists a well-established semi-classical theory of the decay of such false vacua. In this paper, we extend this theory to include the effects of gravitation. Contrary to naive expectation, these are not always negligible, and may sometimes be of critical importance, especially in the late stages of the decay process. Submitted to Physical Review ### 1. Introduction Consider the theory of a single scalar field defined by the action, $$ \mathrm{~\boldmath~s~}=\int\mathrm{d}^{4}\mathrm{\boldmath~x~}\left[\frac{1}{2}(\partial_{\mu}\phi)^{2}-\mathrm{\boldmath~u~}(\phi)\right]\qquad, $$ where U is as shown in Fig. 1. That is to say, U has two local minima, $\phi_{\pm}$, only one of which, $\phi_{-}$, is an absolute minimum. The classical field theory defined by Eq. (1.1) possesses two stable homogeneous equilib- brium states, $\phi = \phi_{+}$ and $\phi = \phi_{-}$. In the quantum version of the theory, though, only the second of these corresponds to a truly stable state, a true vacuum. The first decays through barrier penetration; it is a false vacuum. This is a prototypical case; false vacua occur in many field theories. In particular, they occur in some unified electroweak and grand unified theories, and it is this that gives the theory of vacuum decay possible physical importance. For simplicity, though, we will restrict ourselves here to the theory defined by Eq. (1.1); the extension of our methods to more elaborate field theories is straightforward. The decay of the false vacuum is very much like the nucleation processes associated with first-order phase transitions in statistical mechanics. $ ^{1} $ The decay is initiated by the materialization of a bubble of true vacuum within the false vacuum. This is a quantum tunneling event, and has a certain probability of occurrence per unit time per unit volume, $ \