We revisit the excursion set approach to calculate void abundances in chameleon-type modified gravity theories, which was previously studied by Clampitt, Cai and Li (2013). We focus on properly accounting for the void-in-cloud effect, i.e., the growth of those voids sitting in over-dense regions may be restricted by the evolution of their surroundings. This effect may change the distribution function of voids hence affect predictions on the differences between modified gravity and GR. We show that the thin-shell approximation usually used to calculate the fifth force is qualitatively good but quantitatively inaccurate. Therefore, it is necessary to numerically solve the fifth force in both over-dense and under-dense regions. We then generalise the Eulerian void assignment method of Paranjape, Lam and Sheth (2012) to our modified gravity model. We implement this method in our Monte Carlo simulations and compare its results with the original Lagrangian methods. We find that the abundances of small voids are significantly reduced in both modified gravity and GR due to the restriction of environments. However, the change in void abundances for the range of void radii of interest for both models is similar. Therefore, the difference between models remains similar to the results from the Lagrangian method, especially if correlated steps of the random walks are used. As Clampitt, Cai and Li (2013), we find that the void abundance is much more sensitive to modified gravity than halo abundances. Our method can then be a faster alternative to N-body simulations for studying the qualitative behaviour of a broad class of theories. We also discuss the limitations and other practical issues associated with its applications.
We use the spherical evolution approximation to investigate nonlinear evolution from the non-Gaussian initial conditions characteristic of the local f_nl model. We provide an analytic formula for the nonlinearly evolved probability distribution function of the dark matter which shows that the under-dense tail of the nonlinear PDF in the f_nl model should differ significantly from that for Gaussian initial conditions. Measurements of the under-dense tail in numerical simulations may be affected by discreteness effects, and we use a Poisson counting model to describe this effect. Once this has been accounted for, our model is in good quantitative agreement with the simulations.
