On the bouncing completion of eternal inflation

The duration of our Universe, all of its content, and whether it had a beginning or has always existed has long been the subject of intense investigation. As free particles trajectories are given by geodesics, the inquisition of godesic completeness and extensions become crucial for the investigation of possible eternal Universes. Nevertheless, General Relativity’s invariance under diffeomorphisms imposes an additional difficulty to realize whether the incompleteness has physical significance or if it is merely an inappropriate coordinate choice. In this context, the singularity theorems provide sufficient conditions for geodesic incompleteness without recurring to coordinate charts. However, kinematic alternatives for classifying incomplete space-times that are expanding have been proposed, leading to the Borde-Guth-Vilenkin (BGV) theorem, where no restriction on the matter fields are necessary, such as energy conditions. Notwithstanding, whether the space time admits a metric extension that is compatible with General Relativity, i.e, a 𝒞2 extension, needs to be addressed. In this dissertation, using the pivotal example of the flat patch of the de Sitter space, we manage to find a new global chart for this space - which without considerations of extensibility would be diagnosed as geodesically incomplete. Furthermore, we developed a general protocol for a 𝐶2 extension of a flat Friedmann-Lemaître-Robertson-Walker metric, and the necessary conditions for its application by exhausting all the possible cases in the asymptotic limit, finding necessary and sufficient conditions for extensibility. The incomplete spaces that violate the assumptions have either a scalar curvature singularity or a parallelly propagated singularity, in which cases no 𝒞2 extension is allowed. Moreover, we discuss results for possible cyclic scenarios proposed in the literature. The results obtained in this work were published in Phys. Rev. D 111, 123531 (2025).