Dark Halo Cusp: Asymptotic Convergence

We propose a model for how the buildup of dark halos by merging satellites produces an inner cusp, of a density profile \ρ \ r-\αᵢ with \αᵢ \ ₐ \ 1, as seen in cosmological N-body simulations. Dekel & Devor (2002) showed that a core of \αᵢ < 1 exerts tidal compression which prevents local deposit of satellite material; the satellite sinks intact into the halo center which causes steepening to \αᵢ > 1. Using merger simulations we derive here a mass-transfer recipe in regions where the local slope is \α > 1, according to which the ratio of mean densities of halo and initial satellite within the tidal radius equals a given function \ψ(\α) that is decreasing with \α. This makes the mass transfer relatively more efficient at larger \α, which causes steepening of the profile at small \α and flattening at large \α. Given this mass-transfer recipe, linear perturbation analysis, supported by toy simulations, shows that a sequence of cosmological mergers with homologous satellites slowly leads to a fixed-point asymptotic cusp with a slope \αₛ >1. The cusp depends only weakly on the power spectrum of fluctuations, in agreement with cosmological N-body simulations. During a long interim period the profile has an NFW-like shape, with a cusp of 1 < \αᵢ < \. Thus, a cusp is enforced if enough satellite remnants make it intact into the inner halo. In order to maintain a flat core, satellites must be disrupted outside the core, e.g., as a result of puffing up due to baryonic feedback.