The abundance of voids and the excursion set formalism

We present measurements of the number density of voids in the dark matter distribution from a series of N-body simulations of a \Lambda CDM cosmology. We define voids as spherical regions of \rho_v = 0.2\rho_m around density minima in order to relate our results to the predicted abundances using the excursion set formalism. Using a linear underdensity of \delta_v = -2.7, from a spherical evolution model, we find that a volume conserving model, which does not conserve number density in the mapping from the linear to nonlinear regime, matches the measured abundance to within 16% for a range of void radii 1< r(Mpc/h)<15. This model fixes the volume fraction of the universe which is in voids and assumes that voids of a similar size merge as they expand by a factor of 1.7 to achieve a nonlinear density of \rho_v = 0.2\rho_m today. We find that the model of Sheth & van de Weygaert (2004) for the number density of voids greatly overpredicts the abundances over the same range of scales. We find that the volume conserving model works well at matching the number density of voids measured from the simulations at higher redshifts, z=0.5 and 1, as well as correctly predicting the abundances to within 25% in a simulation of a matter dominated \Omega_m = 1 universe. We examine the abundance of voids in the halo distribution and find fewer small, r<10 Mpc/h, voids and many more large, r>10 Mpc/h, voids compared to the dark matter. These results indicate that voids identified in the halo or galaxy distribution are related to the underlying void distribution in the dark matter in a complicated way which merits further study if voids are to be used as a precision probe of cosmology.