We investigate void properties in $f(R)$ models using N-body simulations, focusing on their differences from General Relativity (GR) and their detectability. In the Hu-Sawicki $f(R)$ modified gravity (MG) models, the halo number density profiles of voids are not distinguishable from GR. In contrast, the same $f(R)$ voids are more empty of dark matter, and their profiles are steeper. This can in principle be observed by weak gravitational lensing of voids, for which the combination of a spectroscopic redshift and a lensing photometric redshift survey over the same sky is required. Neglecting the lensing shape noise, the $f(R)$ model parameter amplitudes $|f_{R0}|=10^{-5}$ and $10^{-4}$ may be distinguished from GR using the lensing tangential shear signal around voids by 4 and 8$sigma$ for a volume of 1~(Gpc/$h$)$^3$. The line-of-sight projection of large-scale structure is the main systematics that limits the significance of this signal for the near future wide angle and deep lensing surveys. For this reason, it is challenging to distinguish $|f_{R0}|=10^{-6}$ from GR. We expect that this can be overcome with larger volume. The halo void abundance being smaller and the steepening of dark matter void profiles in $f(R)$ models are unique features that can be combined to break the degeneracy between $|f_{R0}|$ and $sigma_8$.
We explore voids in dark matter and halo fields from simulations of $Lambda$CDM and Hu-Sawicki $f(R)$ models. In $f(R)$ gravity, dark matter void abundances are greater than that of general relativity (GR). However, when using haloes to identify voids, the differences of void abundances become much smaller, but can still be told apart, in principle, at the 2, 6 and 14 $sigma$ level for the $f(R)$ model parameter amplitudes of $|f_{R0}|=10^{-6}$, $10^{-5}$ and $10^{-4}$. In contrast, the abundance of large voids found using haloes in $f(R)$ gravity is lower than in GR. The more efficient halo formation in underdense regions makes $f(R)$ voids less empty of haloes. This counter intuitive result suggests that voids are not necessarily emptier in $f(R)$ if one looks at galaxies in voids. Indeed, the halo number density profiles of voids are not distinguishable from GR. However, the same $f(R)$ voids are more empty of dark matter. This can in principle be observed by weak gravitational lensing of voids, for which the combination of a spec-$z$ and a photo-$z$ survey over the same sky is necessary. For a volume of 1~(Gpc/$h$)$^3$, neglecting the lensing shape noise, $|f_{R0}|=10^{-5}$ and $10^{-4}$ may be distinguished from GR using the lensing tangential shear signal around voids by 4 and 8$sigma$. The line-of-sight projection of large-scale structure is the main systematics that limits the significance of this signal, limiting the constraining power for $|f_{R0}|=10^{-6}$. The halo void abundance being smaller and the steepening of dark matter void profiles in $f(R)$ models are unique features that can be combined to break the degeneracy between $|f_{R0}|$ and $sigma_8$. The outflow of mass from void centers and velocity dispersions are greater in $f(R)$. Model differences in velocity profiles imply potential powerful constraints of the model in phase space and in redshift space.
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.
