Separate Universes beyond General Relativity
  We establish purely geometric or metric-based criteria for the validity of
the separate universe ansatz, under which the evolution of small-scale
observables in a long-wavelength perturbation is indistinguishable from a
separate Friedmann-Robertson-Walker cosmology in their angle average. In order
to be able to identify the local volume expansion and curvature in a
long-wavelength perturbation with those of the separate universe, we show that
the lapse perturbation must be much smaller in amplitude than the curvature
potential on a time slicing that comoves with the Einstein tensor. Interpreting
the Einstein tensor as an effective stress energy tensor, the condition is that
the effective stress energy comoves with freely falling synchronous observers
who establish the local expansion, so that the local curvature is conserved. By
matching the expansion history of these synchronous observers in cosmological
simulations, one can establish and test consistency relations even in the
nonlinear regime of modified gravity theories.
