Horndeski dark matter and beyond
  Starting from the Gleyzes-Langlois-Piazza-Vernizzi action, we derive the most
general effective theory that is invariant under internal shifts and a
$\mathbb{Z}_2$ mirror symmetry in the scalar sector. Contrary to what one may
think, this model presents a dark matter tracker previous to the dark energy
domination. We show that, in an empty universe and to linear order in
perturbations, the scalar mode clusters in exactly the same way as standard
nonrelativistic cold dark matter. This also holds for the subsector of the
theory where the speed of propagation of gravitational waves equals that of
light, in agreement with the recent multimessenger observation. However, the
inclusion of standard model particles introduces nontrivial couplings of the
gravitational scalar mode to baryons, modifying their clustering properties. We
argue that no arrangement of the parameters of the model can reduce the extra
scalar to precisely behave as cold dark matter.
