In attempts to explain dark energy, a number of models have been proposed in which the formation of a large-scale structure depends on the local environment. These models are highly non-linear and difficult to analyse analytically. N-body simulations have therefore been used to study their non-linear evolution. Here we extend excursion set theory to incorporate environmental effects on structure formation. We apply the method to a chameleon model and calculate observables such as the non-linear mass function at various redshifts. The method can be generalized to study other observables and other models of environmentally dependent interactions. The analytic methods described here should be proved to be useful in delineating which models deserve more detailed study with N-body simulations.
Li, B., & Efstathiou, G. (2012). An extended excursion set approach to structure formation in chameleon models. Monthly Notices of the Royal Astronomical Society, 421(2), 1431-1442. https://doi.org/10.1111/j.1365-2966.2011.20404.x