Double distribution of dark matter halos with respect to mass and local overdensity

We present a double distribution function of dark matter halos, with respect to both object mass and local over- (or under-) density. This analytical tool provides a statistical treatment of the properties of matter surrounding collapsed objects, and can be used to study environmental effects on hierarchical structure formation. The size of the “local environment” of a collapsed object is defined to depend on the mass of the object. The Press-Schechter mass function is recovered by integration of our double distribution over the density contrast. We also present a detailed treatment of the evolution of overdensities and underdensities in Ω_m+Ω_Λ=1 and Ω_m=1 universes according to the spherical evolution model. We explicitly distinguish between true and linearly extrapolated overdensities and provide conversion relations between the two quantities.