  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 $\alpha\sim1.5$. 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,
$\alpha_{\rm ini} \sim 0.3$, while the final slope can be steeper for
$\alpha_{\rm ini}\ga 0.3$. 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.
