On the equivalence between the effective cosmology and excursion set treatments of environment

TL;DR

This paper demonstrates the equivalence between the effective cosmology approach and the excursion set method in modeling environmental effects on structure formation, based on Birkhoff's theorem, with implications for modified gravity theories.

Contribution

It establishes the theoretical equivalence of two prevalent methods for modeling environmental dependence in structure formation, highlighting the role of Birkhoff's theorem.

Findings

The two approaches are mathematically equivalent.

This equivalence relies on Birkhoff's theorem.

Potential deviations could test modified gravity models.

Abstract

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.