Bardeen variables and hidden gauge symmetries in linearized massive gravity

We give a detailed discussion of the use of the (3+1) decomposition and of Bardeen's variables in massive gravity linearized over a Minkowski as well as over a de Sitter background. In Minkowski space the Bardeen "potential" \Phi, that in the massless case is a non-radiative degree of freedom, becomes radiative and describes the helicity-0 component of the massive graviton. Its dynamics is governed by a simple Klein-Gordon action, supplemented by a term (\Box \Phi)^2 if we do not make the Fierz-Pauli tuning of the mass term. In de Sitter the identification of the variable that describes the radiative degree of freedom in the scalar sector is more subtle, and even involves expressions non-local in time. The use of this new variable provides a simple and transparent derivation of the Higuchi bound and of the disappearance of the scalar degree of freedom at a special value of $m_g^2/H^2$. The use of this formalism also allows us to uncover the existence of a hidden gauge symmetry of the massive theory, that becomes manifest only once the non-dynamical components of the metric are integrated out, and that is present both in Minkowski and in de Sitter.