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Non-minimal derivative couplings of the composite metric

In the context of massive gravity, bi-gravity and multi-gravity non-minimal matter couplings via a specific composite effective metric were investigated recently. Even if these couplings generically reintroduce the Boulware-Deser ghost, this composite metric is unique in the sense that the ghost reemerges only beyond the decoupling limit and the matter quantum loop corrections do not detune the potential interactions. We consider non-minimal {\it derivative} couplings of the composite metric to matter fields for a specific subclass of Horndeski scalar-tensor interactions. We first explore these couplings in the mini-superspace and investigate in which scenario the ghost remains absent. We further study these non-minimal derivative couplings in the decoupling-limit of the theory and show that the equation of motion for the helicity-0 mode remains second order in derivatives. Finally, we discuss preliminary implications for cosmology.