On the equivalence between the effective cosmology and excursion set
  treatments of environment
  In studies of the environmental dependence of structure formation, the large
scale environment is often thought of as providing an effective background
cosmology: e.g. the formation of structure in voids is expected to be just like
that in a less dense universe with appropriately modified Hubble and
cosmological constants. However, in the excursion set description of structure
formation which is commonly used to model this effect, no explicit mention is
made of the effective cosmology. Rather, this approach uses the spherical
evolution model to compute an effective linear theory growth factor, which is
then used to predict the growth and evolution of nonlinear structures. We show
that these approaches are, in fact, equivalent: a consequence of Birkhoff's
theorem. We speculate that this equivalence will not survive in models where
the gravitational force law is modified from an inverse square, potentially
making the environmental dependence of clustering a good test of such models.
