Generic instabilities of nonsingular cosmologies in Horndeski theory: A no-go theorem

The null energy condition can be violated stably in generalized Galileon theories, which gives rise to the possibilities of healthy nonsingular cosmologies. However, it has been reported that in many cases cosmological solutions are plagued with instabilities or have some pathologies somewhere in the whole history of the universe. Recently, this was shown to be generically true in a certain subclass of the Horndeski theory. In this short paper, we extend this no-go argument to the full Horndeski theory and show that nonsingular models (with flat spatial sections) in general suffer from either gradient instabilities or some kind of pathology in the tensor sector. This implies that one must go beyond the Horndeski theory to implement healthy nonsingular cosmologies.