Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections

The physics of tunneling from one spacetime into another is often understood in terms of instantons. For some instantons, it was recently shown in the literature that there are two complementary "interpretations" for their analytic continuations. Dubbed "something-to-something" and "nothing-to-something" interpretations, respectively, the former involves situation in which the initial and final hypersurfaces are connected by a Euclidean manifold, whereas the initial and final hypersurfaces in the latter case are not connected in such a way. We consider a de Sitter space with real projective space $\mathbb{R}\text{P}^3$ spatial sections, as was originally understood by de Sitter himself. This original version of de Sitter space has several advantages over the usual de Sitter space with $\text{S}^3$ spatial sections. In particular, the interpretation of the de Sitter entropy as entanglement entropy is much more natural. We discuss the subtleties involved in the tunneling of such a de Sitter space.