4 Conclusions

In this contribution we have summarized how it is possible to compute the tunnelling amplitude for a spherical $p$-brane: the process, in view of our short discussion in the introduction, can be used to model the transition between two different vacuum phases, one being the de Sitter spacetime with cosmological constant $\Lambda _{+}$ and the other being the inflating inside of the $p$-brane, which is characterized by the cosmological constant $\Lambda _{-}$. This computation was already performed in four dimensions in the seminal work by Coleman and de Luccia [1] and that result was reproduced in [3]. Here we have carried out its generalization to arbitrary dimensions (referring to [7] for a more detailed analysis).