Nucleation of de Sitter from the anti de Sitter spacetime in scalar field models

We show that, in the framework of Einstein-scalar gravity, the de Sitter (dS) spacetime can be nucleated out of the anti de Sitter (AdS) one. This is done by using a scalar lump solution, which has an \(\text{AdS}_4\) spacetime in the core, allows for \(\text{dS}_4\) vacua and was found to be plagued by tachyonic instabilities. Using the Euclidean action formalism, we compute and compare the probability amplitudes and the free energies of the lump and the \(\text{dS}_4\) vacua. Our results show that the former is generally less favored than the latter, with the most preferred state being a \(\text{dS}_4\) vacuum. The lump, thus, describes a metastable state which eventually decays into the true \(\text{dS}_4\) vacuum. This nucleation mechanism of dS spacetime may provide insights into the short-distance behavior of gravity, in particular for the characterization of string-theory vacua, cosmological inflation and the black-hole singularity problem.