Conformal and kinetic couplings as two Jordan frames of the same theory: Conformal and kinetic couplings

Certain scalar-tensor (ST) theories with non-minimal coupling of the scalar field to curvature may admit an Einstein frame representation, where gravity is described by the Einstein-Hilbert action plus the scalar sector. Between them, some theories exactly coincide in their respective Einstein frames. If transformations between Jordan and Einstein frames are invertible, these theories can be associated with two Jordan frames of the unique theory. Such successive dualities can connect theories with non-derivative coupling, like R phi 2, with derivatively coupled theories, like Horndeski and DHOST. In absence of matter, these are equivalent, though looking very different. We show the existence of a successive duality between the R phi 2 theory and a recently found Palatini kinetically coupled theory, which both look in their Einstein frames as Einstein theory minimally coupled to scalar. Transforming singular exact solutions of the latter to Jordan frames, we compare desingularization properties of the above two theories which both violate the null energy condition. It is found that kinetically coupled theory has stronger desingularization features, exhibiting possibility of Genesis-type behavior of the homogeneous and isotropic cosmological solutions.