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Stable non-singular cosmologies in beyond Horndeski theory and disformal transformations

In this note we collect, systemise and generalise the existing results for relations between general Horndeski theories and beyond Horndeski theories via disformal transformations of metric. We derive additional disformal transformation rules relating Lagrangian functions of beyond Horndeski theory and corresponding Horndeski theory and demonstrate that some of them become singular at some moments(s) once one constructs a non-singular cosmological solution in beyond Horndeski theory that is free from ghost, gradient instabilities and strong gravity regime during the entire evolution of the system. The key issue here is that such solutions are banned in Horndeski theory due to existing no-go theorem. The proof of singular behaviour of disformal relations in this case resolves the apparent contradiction between the fact that Horndeski and beyond Horndeski theories appear related by field redefinition but describe different physics in the context of non-singular cosmologies.