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 $\rho \prop r^{-\alpha_i}$ with $\alpha_i \rarrow \aalpha_a \gsim 1$, as seen in cosmological N-body simulations. Dekel & Devor (2002) showed that a core of $\alpha_i < 1$ exerts tidal compression which prevents local deposit of satellite material; the satellite sinks intact into the halo center which causes steepening to $\alpha_i > 1$. Using merger simulations we derive here a mass-transfer recipe in regions where the local slope is $\alpha > 1$, according to which the ratio of mean densities of halo and initial satellite within the tidal radius equals a given function $\psi(\alpha)$ that is decreasing with $\alpha$. This makes the mass transfer relatively more efficient at larger $\alpha$, which causes steepening of the profile at small $\alpha$ and flattening at large $\alpha$. 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 $\alpha_s >1$. 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 $1 < \alpha_i < \aas$. 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.