We have revisited the extended excursion set theory in modified gravity models, taking the chameleon model as an example. Instead of specifying their Lagrangian size, here we define the environments by the Eulerian size, chosen to be of the same order of the Compton length of the scalar field by physical arguments. We find that the Eulerian and Lagrangian environments have very different environmental density contrast probability distributions, the former being more likely to have high matter density, which in turn suppress the effect of the fifth force in matter clustering and halo formation. The use of Eulerian environments also evades the unphysical restriction of having an upper mass limit in the case of Lagrangian environments. Two methods of computing the unconditional mass functions, numerical integration and Monte Carlo simulation, are discussed and found to give consistent predictions.
Li, B., & Lam, T. (2012). Excursion set theory for modified gravity: Eulerian versus Lagrangian environments. Monthly Notices of the Royal Astronomical Society, 425(1), 730-739. https://doi.org/10.1111/j.1365-2966.2012.21592.x