Dark Halo Cusp: Asymptotic Convergence

We propose a model for how the buildup of dark halos by merging satellites produces a characteristic inner cusp, with a density profile ρ ∝ r, where αin → αas ≳ 1, as seen in cosmological N-body simulations of hierarchical clustering scenarios. Dekel, Devor, & Hetzroni argue that a flat core of αin < 1 exerts tidal compression that prevents local deposit of satellite material; the satellite sinks intact into the halo center, thus causing a rapid steepening to αin > 1. Using merger N-body simulations, we learn that this cusp is stable under a sequence of mergers and derive a practical tidal mass transfer recipe in regions where the local slope of the halo profile is α > 1. According to this recipe, the ratio of mean densities of the halo and initial satellite within the tidal radius equals a given function ψ(α), which is significantly smaller than unity (compared to being ~1 according to crude resonance criteria) and is a decreasing function of α. This decrease makes the tidal mass transfer relatively more efficient at larger α, which means steepening when α is small and flattening when α is large, thus causing convergence to a stable solution. 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 cusp with an asymptotic slope αas > 1. The slope depends only weakly on the fluctuation power spectrum, in agreement with cosmological simulations. During a long interim period the profile has an NFW-like shape, with a cusp of 1 < αin < αas. Thus, a cusp is enforced if enough compact satellite remnants make it intact into the inner halo. In order to maintain a flat core, satellites must be disrupted outside the core, possibly as a result of a modest puffing up due to baryonic feedback.