Cosmological singularity theorems for f(R) gravity theories - Alani, Ivo Andrés (2016)
In the present work some generalizations of the Hawking singularity theorems in the context of f(R) theories are presented. The main assumptions are: the matter fields stress energy tensor satisfies the condition (Tij-(gij/2)T)kikj ≥ 0 for any generic unit time like field ki; the scalaron takes bounded positive values during its evolution and the resulting space time is globally hyperbolic. Then, if there exist a Cauchy hyper-surface Σ for which the expansion parameter θ of the geodesic congruence emanating orthogonally from Σ satisfies some specific bounds, then the resulting space time is geodesically incomplete. Some mathematical results of reference [92] are very important for proving this. The generalized theorems presented here apply directly for some specific models such as the Hu-Sawicki or Starobinsky ones [27,38]. For other scenarios, some extra assumptions should be implemented in order to have a geodesically incomplete space time. The hypothesis considered in this text are sufficient, but not necessary. In other words, their negation does not imply that a singularity is absent.