  If dark matter has a large self-interaction scattering cross section, then
interactions among dark-matter particles will drive galaxy and cluster halos to
become spherical in their centers. Work in the past has used this effect to
rule out velocity-independent, elastic cross sections larger than sigma/m ~
0.02 cm^2/g based on comparisons to the shapes of galaxy cluster lensing
potentials and X-ray isophotes. In this paper, we use cosmological simulations
to show that these constraints were off by more than an order of magnitude
because (a) they did not properly account for the fact that the observed
ellipticity gets contributions from the triaxial mass distribution outside the
core set by scatterings, (b) the scatter in axis ratios is large and (c) the
core region retains more of its triaxial nature than estimated before.
Including these effects properly shows that the same observations now allow
dark matter self-interaction cross sections at least as large as sigma/m = 0.1
cm^2/g. We show that constraints on self-interacting dark matter from
strong-lensing clusters are likely to improve significantly in the near future,
but possibly more via central densities and core sizes than halo shapes.
