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Self-interacting dark matter (SIDM) has been proposed to solve small-scaleproblems in $\Lambda$CDM cosmology. In previous work, constraints on theself-interaction cross-section of dark matter have been derived assuming thatthe self-interaction cross-section is independent of velocity. However, avelocity-dependent cross-section is more natural in most theories of SIDM.Using idealized $N$-body simulations, we study merging clusters, withvelocity-dependent SIDM. In addition to the usual rare scattering in theisotropic limit, we also simulate these systems with anisotropic, small-angle(frequent) scatterings. We study the qualitative features of the mergers and wefind that the effects of velocity-dependent cross-sections are observed whencomparing early-time and late-time oscillation amplitude of the brightestcluster galaxy (BCG). Finally, we also extend the existing upper bounds on thevelocity-independent, isotropic self-interaction cross-section to the parameterspace of rare and frequent velocity-dependent self-interactions by studying thecentral densities of dark matter only isolated haloes. For these upper-boundparameters, we find that the offsets just after the first pericentre to be$\leq$ 10 kpc. On the other hand, because of BCG oscillations, we speculatethat the distribution of BCG offsets in relaxed cluster to be a statisticallyviable probe. Therefore, this motivates further studies of BCG off-centering incosmological simulations.

Simulations of galaxy cluster mergers with velocity-dependent, rare and frequent self-interactions