On couplings to matter in massive (bi-)gravity

We investigate the coupling to matter in ghost-free massive (bi-) gravity. When species in the matter sector couple covariantly to only one metric, we show that at one-loop these couplings do not spoil the special structure of the graviton potential. When the same species couples directly to both metrics we show that a ghost is present at the classical level and that loops destroy the special structure of the potential at an unacceptably low scale. We then propose a new 'composite' effective metric built out of both metrics. When matter fields couple covariantly to this effective metric, the would be Boulware-Deser ( BD) ghost is absent in different representative limits. At one-loop such couplings do not detune the special structure of the potential. We conjecture that matter can couple covariantly to that effective metric in all generality without introducing any BD ghost below a cut-off scale parametrically larger than the strong coupling scale. We also discuss alternative couplings to matter where the kinetic and potential terms of the matter field couple to different metrics. In both cases we discuss preliminary implications for cosmology.