Abstract
We study the negative modes of gravitational instantons representing vacuum decay in asymptotically flat space-time. We consider two different vacuum decay scenarios: the Coleman-de Luccia $\mathrm{O}(4)$-symmetric bubble, and $\mathrm{O}(3) \times \mathbb{R}$ instantons with a static black hole. In spite of the similarities between the models, we find qualitatively different behaviours. In the $\mathrm{O}(4)$-symmetric case, the number of negative modes is known to be either one or infinite, depending on the sign of the kinetic term in the quadratic action. In contrast, solving the mode equation numerically for the static black hole instanton, we find only one negative mode with the kinetic term always positive outside the event horizon. The absence of additional negative modes supports the interpretation of these solutions as giving the tunnelling rate for false vacuum decay seeded by microscopic black holes.
Highlights
False vacuum decay through the nucleation of true vacuum bubbles has many important applications ranging from early universe phase transitions to stability of the Higgs vacuum
The absence of additional negative modes supports the interpretation of these solutions as giving the tunneling rate for false vacuum decay seeded by microscopic black holes
We look at two different Higgs vacuum decay scenarios, vacuum decay in empty space and vacuum decay seeded by black holes
Summary
False vacuum decay through the nucleation of true vacuum bubbles has many important applications ranging from early universe phase transitions to stability of the Higgs vacuum. The black hole acts as a nucleation seed and greatly enhances the vacuum decay rate This effect was investigated initially for vacuum decay in de Sitter space [26], and later for more general scenarios including asymptotically flat space [27,28,29]. We shall give the results of a numerical analysis of the negative modes for vacuum decay with an asymptotically flat black hole nucleation seed In this case we find only one negative mode, and the kinetic term in the action of the perturbations is always positive. We conclude from this that vacuum decay seeded by black holes most likely always has a consistent formulation in terms of bounce solutions
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