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We reflect on the possibility of having a matter action that is invariantonly under transverse diffeomorphisms. This possibility is particularlyinteresting for the dark sector, where no restrictions arise based on the weakequivalence principle. In order to implement this idea we consider a scalarfield which couples to gravity minimally but via arbitrary functions of themetric determinant. We show that the energy-momentum tensor of the scalar fieldtakes the perfect fluid form when its velocity vector is time-like. We analyzethe conservation of this tensor in detail, obtaining a seminal novel result forthe energy density of this field in the kinetic dominated regime. Indeed, inthis regime the fluid is always adiabatic and we obtain an explicit expressionfor the speed of sound. Furthermore, to get insight in the gravitationalproperties of these theories, we consider the fulfillment of the energyconditions, concluding that nontrivial physically reasonable matter violatesthe strong energy condition in the potential domination regime. On the otherhand, we present some shift-symmetric models of particular interest. These are:constant equation of state models (which may provide us with a successfuldescription of dark matter or dark radiation) and models presenting differentgravitational domains (characterized by the focusing or possible defocusing oftime-like geodesics), as it happens in unified dark matter-energy models.

TDiff in the Dark: Gravity with a scalar field invariant under transverse diffeomorphisms