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 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 Boulware--Deser 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.
