The distribution of galaxies provides an ideal laboratory to test the fundamental properties of gravity and dark matter on cosmological scales. In particular, constraints on deviations from general relativity are commonly obtained by measuring the growth of cosmic structure. However, such constraints rely on the validity of the weak equivalence principle, which has never been tested for dark matter. In my seminar, I will show that dropping this restrictive assumption leads to severe degeneracies and makes it challenging to distinguish fundamental gravity modifications from interactions in the dark sector. Luckily, I will demonstrate that it is possible to break such degeneracies and recover tight constraints thanks to measurements of the distortion of time expected from upcoming galaxy surveys.