Violations of energy conservation in Horava-Lifshitz gravity: a new ingredient in the dark matter puzzle

We investigate the interplay between Horava-Lifshitz (HL) gravity and more general theories where the local Hamiltonian constraint is lost, for example due to the time variability of the Lagrangian (e.g. via its parameters) where time is defined on a foliation according to a prescription mimicking Lambda and 4-volume time in unimodualr gravity. In one direction we subject the multitude of parameters in HL to this variability game, mimicking RG flow in a cosmological setting. In the opposite direction, we examine the evolution on the left-over Hamiltonian should the HL algebra of constraints be still applicable, rather than the algebra of General Relativity being restored. Within the projectable theory, the non-vanishing Hamiltonian can be reinterpreted as a pressureless fluid, resulting in essentially the same phenomenologies at macroscopic scales as in the standard cold dark matter paradigm. At high energies and short distances, however, unlike in theories with similar variability based on GR, violations of stress-energy tensor conservation persist, and these are computed here for the full class of projectable HL models. The phenomenological implications are examined: remarkably the driven solution resulting from these energy conservation violations is shown to be the attractor of the system during a free-fall collapse as far as the backreaction is negligible. When the backreaction is taken into account, the driven solution is expected to play an important role towards our understanding of microscopic caustic avoidance, which is one of the most significant issues in many alternatives to particle dark matter scenarios.