  We describe a class of modified gravity theories that deform general
relativity in a way that breaks time reversal invariance and, very mildly,
locality. The algebra of constraints, local physical degrees of freedom, and
their linearized equations of motion, are unchanged, yet observable effects may
be present on cosmological scales, which have implications for the early
history of the universe.
  This is achieved in the Hamiltonian framework, in a way that requires the
constant mean curvature gauge conditions and is, hence, inspired by shape
dynamics.
