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The null energy condition can be violated stably in generalized Galileontheories, which gives rise to the possibilities of healthy non-singularcosmologies. However, it has been reported that in many cases cosmologicalsolutions are plagued with instabilities or have some pathologies somewhere inthe whole history of the universe. Recently, this was shown to be genericallytrue in a certain subclass of the Horndeski theory. In this short paper, weextend this no-go argument to the full Horndeski theory, and show thatnon-singular models (with flat spatial sections) in general suffer either fromgradient instabilities or some kind of pathology in the tensor sector. Thisimplies that one must go beyond the Horndeski theory to implement healthynon-singular cosmologies.

Generic instabilities of non-singular cosmologies in Horndeski theory: a no-go theorem