Culminating The Peak Cusp To Descry The Dark Side Of Halos

The ConflUent System of Peak trajectories (CUSP) is a rigorous formalism in the framework of the peak theory that allows one to derive from first principles and no free parameters the typical halo properties from the statistics of peaks in the filtered Gaussian random field of density perturbations. The predicted halo mass function, spherically averaged density, velocity dispersion, velocity anisotropy, ellipticity, prolateness, and potential profiles, as well as the abundance and number density profiles of accreted and stripped subhalos and diffuse dark matter, accurately recover the results of cosmological N-body simulations. CUSP is thus a powerful tool for the calculation, in any desired hierarchical cosmology with Gaussian perturbations, of halo properties beyond the mass, redshift, and radial ranges covered by simulations. More importantly, CUSP unravels the origin of the characteristic features of those properties. In this paper, we culminate its construction. We show that all halo properties but those related to subhalo stripping are independent of the assembly history of those objects, and that the Gaussian is the only smoothing window able to find the finite collapsing patches while properly accounting for the entropy increase produced in major mergers.