Cosmological singularity theorems for f(R) gravity theories

Abstract

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.