Radial arc statistics: A new powerful probe of the central density profile of galaxy clusters

We show that an expected number ratio of radial arcs (gravitationally lensed images whose major axes lie in the radial direction of a cluster-lens potential) to tangential arcs (gravitationally lensed images whose major axes lie in the tangential direction) has strong dependence on the central density profile of galaxy clusters and has little dependence on other parameters, e.g. cluster temperature, background source galaxy redshift etc.. A comparison of the expected number ratios with observed ratios provides a robust test to constrain the central density profile of galaxy clusters. A tentative comparison with the observational data shows that the central density profile of galaxy clusters is r^{-1} - r^{-1.5}. This result indicates that the dark matter is collisionless at least on cluster scale. Our result gives an upper limit on a collision cross-section of self-interaction of dark matter sigma_coll as sigma_coll/m < 0.1 cm^2/g where m is a dark matter particle mass.