Dark matter ( DM ) haloes forming near the thermal cut-off scale of the density perturbations are unique , since they are the smallest objects and form through monolithic gravitational collapse , while larger haloes contrastingly have experienced mergers .
While standard cold dark matter ( CDM ) simulations readily produce haloes that follow the universal Navarro-Frenk-White ( NFW ) density profile with an inner slope , \rho \propto r ^ { - \alpha } , with \alpha = 1 , recent simulations have found that when the free-streaming cut-off expected for the CDM model is resolved , the resulting haloes follow nearly power-law density profiles of \alpha \sim 1.5 .
In this paper , we study the formation of density cusps in haloes using idealized N -body simulations of the collapse of proto-haloes .
When the proto-halo profile is initially cored due to particle free-streaming at high redshift , we universally find \sim r ^ { -1.5 } profiles irrespective of the proto-halo profile slope outside the core and large-scale non-spherical perturbations .
Quite in contrast , when the proto-halo has a power-law profile , then we obtain profiles compatible with the NFW shape when the density slope of the proto-halo patch is shallower than a critical value , \alpha _ { ini } \sim 0.3 , while the final slope can be steeper for \alpha _ { ini } \ga 0.3 .
We further demonstrate that the r ^ { -1.5 } profiles are sensitive to small scale noise , which gradually drives them towards an inner slope of -1 , where they become resilient to such perturbations .
We demonstrate that the r ^ { -1.5 } solutions are in hydrostatic equilibrium , largely consistent with a simple analytic model , and provide arguments that angular momentum appears to determine the inner slope .