  Massive gravity theories have been developed as viable IR modifications of
gravity motivated by dark energy and the problem of the cosmological constant.
On the other hand, modified gravity and modified dark matter theories were
developed with the aim of solving the problems of standard cold dark matter at
galactic scales. Here we propose to adapt the framework of ghost-free massive
bigravity theories to reformulate the problem of dark matter at galactic
scales. We investigate a promising alternative to dark matter called dipolar
dark matter (DDM) in which two different species of dark matter are separately
coupled to the two metrics of bigravity and are linked together by an internal
vector field. We show that this model successfully reproduces the phenomenology
of dark matter at galactic scales (MOND) as a result of a mechanism of
gravitational polarisation. The model is safe in the gravitational sector, but
because of the particular couplings of the matter fields and vector field to
the metrics, a ghost in the decoupling limit is present in the dark matter
sector. However, it might be possible to push the mass of the ghost beyond the
strong coupling scale by an appropriate choice of the parameters of the model.
Crucial questions to address in future work are the exact mass of the ghost,
and the cosmological implications of the model.
