  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.
