Density fields and halo mass functions in the geometrical adhesion toy model

In dimension 2 and above, the Burgers dynamics, the so-called "adhesion model" in cosmology, can actually give rise to several dynamics in the inviscid limit. We investigate here the statistical properties of the density field when it is defined by a "geometrical model" associated with this Burgers velocity field and where the matter distribution is fully determined, at each time step, by geometrical constructions. Our investigations are based on a set of numerical experiments that make use of an improved algorithm, for which the geometrical constructions are efficient and robust. In this work we focus on Gaussian initial conditions with power-law power spectra of slope n in the range -3<n<1, where a self-similar evolution develops, and we compute the behavior of power spectra, density probability distributions and mass functions. As expected for such dynamics, the density power spectra show universal high-k tails that are governed by the formation of pointlike masses. The two other statistical indicators however show the same qualitative properties as those observed for 3D gravitational clustering. In particular, the mass functions obey a Press-Schechter like scaling up to a very good accuracy in 1D, and to a lesser extent in 2D. Our results suggest that the "geometrical adhesion model," whose solution is fully known at all times, provides a precious tool to understand some of the statistical constructions frequently used to study the development of mass halos in gravitational clustering.