The cosmological mass distribution .2. Disordered trees and the adhesion approximation

The mass distribution of cosmic structures has been connected in Paper I to the number of random trajectories (overdensities as a function of mass scale) in a Cayley tree. These may diffuse and/or branch with decreasing mass, and the two processes were related to the Press & Schechter mass distribution or to the Smoluchowski aggregation equation, respectively. Here we start from the dynamical equations for the cosmic matter field, specifically in the adhesion approximation, and we show that the distribution of the masses condensed into isotropic ''schocks'' is given by a specific instance of the above branching diffusion process, with the branching rate provided by the adhesion approximation. Such equivalence establishes a bridging between the dynamics of the cosmic fluid under gravity and the phenomenological mass distributions generated (as limiting cases) by our model based on branching diffusion. We discuss how this approach can be extended beyond the adhesion approximation, to describe the emergence of cosmic structure as a random process of the branching diffusion kind.