Explaining the cuspy dark matter halo by Landau-Ginzburg theory

Twenty three years ago, cosmological N-body simulations revealed quasi-universal NFW dark matter halos, whose physical origin is still unclear. This work tries to solve this issue by Landau-Ginzburg (LG) theory in equilibrium statistical mechanics. We replace the order parameter in LG theory by density and consider the dark matter halos as fluctuations from the equilibrium state of the background, which can be the main halos for the subhalos, or the homogeneous and isotropic Universe for the galaxy cluster halos. With the assumption that the fluctuations of Helmholtz free energy depend mainly on the density and its gradient, which is shown to be consistent with the behavior of the power spectrum near the cluster scale, we can easily obtain the universal $r^{-1}$ cusp of the equilibrium dark matte halo without other requirements. This work strongly suggests that more studies should be done for the self-gravitating systems by statistical mechanics.