Appearances are deceptive: Can graviton have a mass?

Abstract: We study the dynamics of linear gravitational perturbations on cosmological backgrounds of massive fermionic fields. We observe that, when gravitational and matter action are expanded to quadratic order in gravitational perturbations on cosmological backgrounds, the graviton appears to have an off-shell mass. We derive a consistent set of two equations for the evolution of linear classical and quantum gravitational perturbations on general cosmological backgrounds, and demonstrate that the graviton mass disappears at the level of equations of motion (on-shell). In the case we consider the expansion of the Universe is driven by the one-loop backreaction of fermions, and the dynamical gravitons evolve on the same background. These equations govern the evolution of linear gravitational perturbations on general cosmological matter backgrounds. A concrete one-loop calculation is performed for the simple case of massive Dirac fermions when the temperature of the cosmological fluid changes adiabatically when compared with the expansion rate of the Universe.