  In this paper, we fully investigate cosmological scenarios in which dark
matter is non-minimally coupled to an extra scalar degree of freedom. The
interaction is realized by means of conformal and disformal terms in the
transformed gravitational metric. Considering linear perturbation theory, we
show that the growth rate of dark matter differs from the uncoupled case and
that the well-known Kaiser formula undergoes modification. As a result,
redshift space distortion measurements cease to be a direct probe of the linear
growth rate of total matter, since the distortion factor has an extra,
coupling-dependent term. We study the effect of the coupling in three
cosmological models, two conformally and one disformally coupled, and forecast
the constraints on the coupling, and other cosmological parameters, from future
galaxy surveys.
