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I will introduce the formalism to use cosmology to constrain one of the fundamental properties normally attributed to dark matter: its invariance under Lorentz transformations.
If Lorentz invariance is broken, new light degrees of freedom (which are not universally coupled) take part in the gravitational interaction. As a consequence, the equivalence principle is generically violated in the sectors not satisfying Lorentz invariance. The dynamics of gravitational clustering is thus modified. I'll focus on two effects that can be constraint (detected) with current observations: scale dependent enhancement in the growth of large scale structure and scale dependent bias between dark matter and baryon density perturbations. I will show how our analysis can be used to put bounds on the invariance of dark matter under Lorentz transformations at the percent level.
