A geometrical approach to degenerate scalar-tensor theories

Degenerate scalar-tensor theories are recently proposed covariant theories of gravity coupled with a scalar field. Despite being characterised by higher order equations of motion, they do not propagate more than three degrees of freedom, thanks to the existence of constraints. We discuss a geometrical approach to degenerate scalar-tensor systems, and analyse its consequences. We show that some of these theories emerge as a certain limit of DBI Galileons. In absence of dynamical gravity, these systems correspond to scalar theories enjoying a symmetry which is different from Galileon invariance. The scalar theories have however problems concerning the propagation of fluctuations around a time dependent background. These issues can be tamed by breaking the symmetry by hand, or by minimally coupling the scalar with dynamical gravity in a way that leads to degenerate scalar-tensor systems. We show that distinct theories can be connected by a relation which generalizes Galileon duality, in certain cases also when gravity is dynamical. We discuss some implications of our results in concrete examples. Our findings can be helpful for assessing stability properties and understanding the non-perturbative structure of systems based on degenerate scalar-tensor systems.