On an analytical framework for voids: their abundances, density profiles and local mass functions

We present a general analytical procedure for computing the number density of voids with radius above a given value within the context of gravitational formation of the large-scale structure of the Universe out of Gaussian initial conditions. To this end, we develop an accurate (under generally satisfied conditions) extension of the unconditional mass function to constrained environments, which allows us both to obtain the number density of collapsed objects of certain mass at any distance from the centre of the void, and to derive the number density of voids defined by collapsed objects. We have made detailed calculations for the spherically averaged mass density and halo number density profiles for particular voids. We also present a formal expression for the number density of voids defined by galaxies of a given type and luminosity. This expression contains the probability for a collapsed object of certain mass to host a galaxy of that type and luminosity (i.e. the conditional luminosity function) as a function of the environmental density. We propose a procedure to infer this function, which may provide useful clues as to the galaxy formation process, from the observed void densities.