Scaling relations of the slightly self-interacting cold dark matter in galaxies and clusters

Recent observations in galaxies and clusters indicate dark matter density profiles exhibit core-like structures which contradict to the numerical simulation results of collisionless cold dark matter. The idea of self-interacting cold dark matter (SICDM) has been invoked to solve the discrepancies between the observations and numerical simulations. In this article, I derive some important scaling relations in galaxies and clusters by using the long-range SICDM model. These scaling relations give good agreements with the empirical fittings from observational data in galaxies and clusters if the dark matter particles are only slightly self-interacting. Also, there may exist a universal critical optical depth $\tau_c$ that characterizes the core-like structures. These results generally support the idea of SICDM to tackle the long-lasting dark matter problem.