Voids in modified gravity reloaded: Eulerian void assignment

We revisit the excursion set approach to calculate void abundances in chameleon-type modified gravity theories, which was previously studied by Clampitt, Cai & Li. We focus on properly accounting for the void-in-cloud effect, i.e. the growth of those voids sitting in overdense 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 (MG) and general relativity (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 overdense and underdense regions. We then generalize the Eulerian-void-assignment method of Paranjape, Lam & Sheth 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 MG 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 et al., we find that the void abundance is much more sensitive to MG 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.