  We consider a scenario where local Lorentz invariance is violated by the
existence of a preferred time direction at every space-time point. This
scenario can arise in the context of quantum gravity and its description at low
energies contains a unit time-like vector field which parameterizes the
preferred direction. The particle physics tests of Lorentz invariance preclude
a direct coupling of this vector to the fields of the Standard Model, but do
not bear implications for dark matter. We discuss how the presence of this
vector and its possible coupling to dark matter affect the evolution of the
Universe. At the level of homogeneous cosmology the only effect of Lorentz
invariance violation is a rescaling of the expansion rate. The physics is
richer at the level of perturbations. We identify three effects crucial for
observations: the rescaling of the matter contribution to the Poisson equation,
the appearance of an extra contribution to the anisotropic stress and the
scale-dependent enhancement of dark matter clustering. These effects result in
distinctive features in the power spectra of the CMB and density fluctuations.
Making use of the data from Planck and WiggleZ we obtain the most stringent
cosmological constraints to date on departures from Lorentz symmetry. Our
analysis provides the first direct bounds on deviations from Lorentz invariance
in the dark matter sector.
