1 Introduction

Vacuum decay can be seen as a phase transition in spacetime and a long time ago the relevance of gravity for the process was studied [1]. The standard treatment of this process makes use of a scalar field, known as the inflaton, that drives the transition between the false and true vacuum states. This situation can be described by instanton calculations as, for instance, the Coleman-de Luccia and the Hawking-Moss instantons.

Here we present a different approach, generalizing past works of one of the authors [2,3]. In particular we are going to use (anti-)de Sitter solutions in $N$ spacetime dimensions. In this background we put a spherically symmetric $(N-1)$-brane that splits spacetime into two domains. The system can be described by Israel junction conditions [4], which provide the equations of motion for the timelike brane. The associated solutions are of two kinds: the first one consists of a degenerate brane of zero radius, while the second one consists of a bounce brane collapsing from infinity towards a finite nonzero turning point, and then re-expanding. To model vacuum decay we consider the tunnelling from the zero radius solution to the bounce solution. The corresponding physical picture is the following: a very small brane⁵ inside a de Sitter geometry with cosmological constant $\Lambda _{+}$, due to quantum effects, has a non-vanishing probability to tunnel into a brane, containing a de Sitter spacetime with a different cosmological constant $\Lambda _{-}$. This represents the formation of a bubble of a different vacuum phase that then expands to infinity, realizing a transition of the whole spacetime geometry. We can obtain an expression for the probability of such a process using an effective action for this system.