As the bosons of the weak force are massive, it is legitimate to investigate whether the mediator of gravity could be massive. Gravity is a long range force, so it can not be too massive.
Mass term for a spin 0 field
\[m = \frac{\partial^2 V}{\partial \phi^2}\]If adding a mass:
\[\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2 + g\phi {\rm Tr}(T_{\mu\nu}) - \frac{m^2}{2}\phi^2\]We get a Yukawa potential:
\[V(r) = \frac{g^2}{4\pi} \frac{M_1M_2}{r}e^{-mr}\]Massive Fiertz-Pauli
vDVZ discontinuity
$m\to0$ does not lead to GR!
Veinstein mechanism and screening
dRGT massive gravity
Ghost free an massive theory of gravity