  We reflect on the possibility of having a matter action that is invariant
only under transverse diffeomorphisms. This possibility is particularly
interesting for the dark sector, where no restrictions arise based on the weak
equivalence principle. In order to implement this idea we consider a scalar
field which couples to gravity minimally but via arbitrary functions of the
metric determinant. We show that the energy-momentum tensor of the scalar field
takes the perfect fluid form when its velocity vector is time-like. We analyze
the conservation of this tensor in detail, obtaining a seminal novel result for
the energy density of this field in the kinetic dominated regime. Indeed, in
this regime the fluid is always adiabatic and we obtain an explicit expression
for the speed of sound. Furthermore, to get insight in the gravitational
properties of these theories, we consider the fulfillment of the energy
conditions, concluding that nontrivial physically reasonable matter violates
the strong energy condition in the potential domination regime. On the other
hand, we present some shift-symmetric models of particular interest. These are:
constant equation of state models (which may provide us with a successful
description of dark matter or dark radiation) and models presenting different
gravitational domains (characterized by the focusing or possible defocusing of
time-like geodesics), as it happens in unified dark matter-energy models.
