The Evidence of Absence: Galaxy Voids in the Excursion Set Formalism

We present an analytic model for the sizes of voids in the galaxy distribution. Peebles and others have recently emphasized the possibility that the observed characteristics of voids may point to a problem in galaxy formation models, but testing these claims has been difficult without any clear predictions for their properties. In order to address such questions, we build a model to describe the distribution of galaxy underdensities. Our model is based on the "excursion set formalism," the same technique used to predict the dark matter halo mass function. We find that, because of bias, galaxy voids are typically significantly larger than dark matter voids and should fill most of the universe. We show that voids selected from catalogs of luminous galaxies should be larger than those selected from faint galaxies: the characteristic radii range from ~5-10h^{-1} Mpc for galaxies with absolute r-band magnitudes M_r-5 log(h) < -16 to -20. These are reasonably close to, though somewhat smaller than, the observed sizes. The discrepancy may result from the void selection algorithm or from their internal structure. We also compute the halo populations inside voids. We expect small haloes (M<10^{11} Msun) to be up to a factor of two less underdense than the haloes of normal galaxies. Within large voids, the mass function is nearly independent of the size of the underdensity, but finite-size effects play a significant role in small voids (R