Massive graviton on arbitrary background: derivation, syzygies,\n applications

We give the detailed derivation of the fully covariant form of the quadratic\naction and the derived linear equations of motion for a massive graviton in an\narbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]).\nOur starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free\nmassive gravities and using a simple model of this family, we are able to\nexpress this action and these equations of motion in terms of a single metric\nin which the graviton propagates, hence removing in particular the need for a\n"reference metric" which is present in the non perturbative formulation. We\nshow further how 5 covariant constraints can be obtained including one which\nleads to the tracelessness of the graviton on flat space-time and removes the\nBoulware-Deser ghost. This last constraint involves powers and combinations of\nthe curvature of the background metric. The 5 constraints are obtained for a\nbackground metric which is unconstrained, i.e. which does not have to obey the\nbackground field equations. We then apply these results to the case of Einstein\nspace-times, where we show that the 5 constraints become trivial, and\nFriedmann-Lema\\^{\\i}tre-Robertson-Walker space-times, for which we correct in\nparticular some results that appeared elsewhere. To reach our results, we\nderive several non trivial identities, syzygies, involving the graviton fields,\nits derivatives and the background metric curvature. These identities have\ntheir own interest. We also discover that there exist backgrounds for which the\ndRGT equations cannot be unambiguously linearized.\n