  We investigate whether the shapes of galaxy clusters inferred from weak
gravitational lensing can be used as a test of the nature of dark matter. We
analyse mock weak lensing data, with gravitational lenses extracted from
cosmological simulations run with two different dark matter models (CDM and
SIDM). We fit elliptical NFW profiles to the shear fields of the simulated
clusters. Despite large differences in the distribution of 3D shapes between
CDM and SIDM, we find that the distributions of weak-lensing-inferred cluster
shapes are almost indistinguishable. We trace this information loss to two
causes. Firstly, weak lensing measures the shape of the projected mass
distribution, not the underlying 3D shape, and projection effects wash out some
of the difference. Secondly, weak lensing is most sensitive to the projected
shape of clusters, on a scale approaching the virial radius (~ 1.5 Mpc),
whereas SIDM shapes differ most from CDM in the inner halo. We introduce a
model for the mass distribution of galaxy clusters where the ellipticity of the
mass distribution can vary with distance to the centre of the cluster. While
this mass distribution does not enable weak lensing data to distinguish between
CDM and SIDM with cluster shapes (the ellipticity at small radii is poorly
constrained by weak lensing), it could be useful when modelling combined strong
and weak gravitational lensing of clusters.
