Voids in modified gravity: excursion set predictions

We investigate the behaviour of the fifth force in voids in chameleon models using the spherical collapse method. Contrary to Newtonian gravity, we find the fifth force is repulsive in voids. The strength of the fifth force depends on the density inside and outside the void region as well as its radius. It can be many times larger than the Newtonian force and their ratio is in principle unbound. This is very different from the case in haloes, where the fifth force is no more than 1/3 of gravity. The evolution of voids is governed by the Newtonian gravity, the effective dark energy force and the fifth force. While the first two forces are common in both Lambda cold dark matter (Lambda CDM) and chameleon universes, the fifth force is unique to the latter. Driven by the outward-pointing fifth force, individual voids in chameleon models expand faster and grow larger than in a Lambda CDM universe. The expansion velocity of the void shell can be 20-30 per cent larger for voids of a few Mpc h(-1) in radius, while their sizes can be larger by similar to 10 per cent. This difference is smaller for larger voids of the same density. We compare void statistics using excursion set theory; for voids of the same size, their number density is found to be larger in chameleon models. The fractional difference increases with void size due to the steepening of the void distribution function. The chance of having voids of radius similar to 25 Mpc h(-1) can be 2.5 times larger. This difference is about 10 times larger than that in the halo mass function. We find strong environmental dependence of void properties and population in chameleon models. The differences in size and expansion velocity with general relativity are both larger for small voids in high-density regions. In general, the difference between chameleon models and Lambda CDM in void properties (size, expansion velocity and distribution function) is larger than the corresponding quantities for haloes. This suggests that voids might be better candidates than haloes for testing gravity.