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We develop a statistical method to measure the interaction cross-section ofDark Matter, exploiting the continuous minor merger events in which smallsubstructures fall into galaxy clusters. We find that by taking the ratio ofthe distances between the galaxies and Dark Matter, and galaxies and gas inaccreting sub-halos, we form a quantity that can be statistically averaged overa large sample of systems whilst removing any inherent line-of-sightprojections. In order to interpret this ratio as a cross-section of Dark Matterwe derive an analytical description of sub-halo infall which encompasses; theforce of the main cluster potential, the drag on a gas sub-halo, a model forDark Matter self-interactions and the resulting sub-halo drag, the force on thegas and galaxies due to the Dark Matter sub-halo potential, and finally thebuoyancy on the gas and Dark Matter. We create mock observations fromcosmological simulations of structure formation and find that collisionlessDark Matter becomes physically separated from X-ray gas by up to 20h^-1 kpc.Adding realistic levels of noise, we are able to predict achievable constraintsfrom observational data. Current archival data should be able to detect adifference in the dynamical behaviour of Dark Matter and standard modelparticles at 6 sigma, and measure the total interaction cross-section sigma/mwith 68% confidence limits of +/- 1cm2g^-1. We note that this method is notrestricted by the limited number of major merging events and is easily extendedto large samples of clusters from future surveys which could potentially pushstatistical errors to 0.1cm^2g^-1.

On the cross-section of Dark Matter using substructure infall into galaxy clusters