Horndeski in motion

We study a class of homogeneous but anisotropic cosmologies within the family of shift-symmetric Horndeski theories, where the scalar field features an inhomogeneous profile but it preserves a translational symmetry that is realised as a combination of spatial translations and internal shifts. The spatial gradient of the scalar field introduces a preferred direction, so the resulting cosmologies are of the axisymmetric Bianchi I type. The momentum density of these configurations exhibits a universal evolution and an additional component with non-vanishing momentum density is required to have non-trivial effects. We show the relation of these scenarios with cosmologies of non-comoving components and, in particular, we explain how they provide a specific realisation of moving dark energy models. Among the class of shift-symmetric Horndeski theories, we analyse in more detail the case of Kinetic Gravity Braiding with emphasis on its application to moving dark energy models and its effects on large scale dark flows as well as the CMB dipole and quadrupole.