The distribution of galaxies provides an ideal laboratory to test for
deviations from General Relativity. In particular, redshift-space distortions
are commonly used to constrain modifications to the Poisson equation, which
governs the strength of dark matter clustering. Here, we show that these
constraints rely on the validity of the weak equivalence principle, which has
never been tested for the dark matter component. Relaxing this restrictive
assumption leads to modifications in the growth of structure that are fully
degenerate with modifications induced by the Poisson equation. This in turns
strongly degrades the constraining power of redshift-space distortions. Such
degeneracies can however be broken and tight constraints on modified gravity
can be recovered by measuring gravitational redshift from the galaxy
distribution, an effect that will be detectable by the coming generation of
large-scale structure surveys.
