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 rho proportional to r(-alpha in), where alpha(in)-->alpha(as)greater than or similar to1, as seen in cosmological N-body simulations of hierarchical clustering scenarios. Dekel, Devor, & Hetzroni argue that a. at core of alpha(in)<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 alpha(in)>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 alpha>1. According to this recipe, the ratio of mean densities of the halo and initial satellite within the tidal radius equals a given function psi(alpha), which is significantly smaller than unity (compared to being similar to1 according to crude resonance criteria) and is a decreasing function of alpha. This decrease makes the tidal mass transfer relatively more efficient at larger alpha, which means steepening when alpha is small and flattening when alpha 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 alpha(as)>1. The slope depends only weakly on the fluctuation power spectrum, in agreement with cosmological simulations. During a long interim period the pro. le has an NFW-like shape, with a cusp of 1<alpha(in)<alpha(as). Thus, a cusp is enforced if enough compact satellite remnants make it intact into the inner halo. In order to maintain a. at core, satellites must be disrupted outside the core, possibly as a result of a modest puffing up due to baryonic feedback.