We propose a new class of gravity theories which are characterized by a nontrivial coupling between the gravitational metric and matter mediated by an auxiliary rank-2 tensor. The actions generating the field equations are constructed so that these theories are equivalent to general relativity in a vacuum, and only differ from general relativity theory within a matter distribution. We analyze in detail one of the simplest realizations of these generalized coupling theories. We show that in this case the propagation speed of gravitational radiation in matter is different from its value in vacuum and that this can be used to weakly constrain the (single) additional parameter of the theory. An analysis of the evolution of homogeneous and isotropic spacetimes in the same framework shows that there exist cosmic histories with both an inflationary phase and a dark era characterized by a different expansion rate.
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