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 \ \ r^-\ᵢ with \ᵢ \ \ₐ \ 1, as seen in cosmological N-body simulations. Dekel & Devor (2002) showed that a core of \ᵢ 1. Using merger simulations we derive here a mass-transfer recipe in regions where the local slope is \ > 1, according to which the ratio of mean densities of halo and initial satellite within the tidal radius equals a given function \ (\) that is decreasing with \. This makes the mass transfer relatively more efficient at larger \, which causes steepening of the profile at small \ and flattening at large \. 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 \ₛ >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 < \ᵢ < \. 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.