Symmetry restoration for TDiff scalar fields

We explore the idea of restoring the full diffeomorphism (Diff) invariance in theories with only transverse diffeomorphisms (TDiff) by the introduction of additional fields. In particular, we consider in detail the case of a TDiff invariant scalar field and how Diff symmetry can be restored preserving locality by introducing an additional vector field. We reobtain the corresponding dynamics and energy-momentum tensor from the covariantized action and analyze the potential and kinetic domination regimes. For the former, the theory describes a cosmological constant-type behaviour, while for the latter we show that the theory can describe an adiabatic perfect fluid whose equation of state and speed of sound is obtained in a straightforward way. Furthermore, the reformulation with the full symmetry allows us to analyze the gravitational properties of the theory beyond those particular regimes. In particular, we find the general expression for the effective speed of sound of the non-adiabatic perfect fluid, which provides us with physically reasonable conditions that should be satisfied by the coupling functions. Finally, we investigate the particular models leading to an adiabatic fluid.