  We perform numerical simulations of the merging galaxy cluster 1E 0657-56
(the Bullet Cluster), including the effects of elastic dark matter scattering.
In a similar manner to the stripping of gas by ram pressure, dark matter
self-interactions would transfer momentum between the two galaxy cluster dark
matter haloes, causing them to lag behind the collisionless galaxies. The
absence of an observed separation between the dark matter and stellar
components in the Bullet Cluster has been used to place upper limits on the
cross-section for dark matter scattering. We emphasise the importance of
analysing simulations in an observationally-motivated manner, finding that the
way in which the positions of the various components are measured can have a
larger impact on derived constraints on dark matter's self-interaction
cross-section than reasonable changes to the initial conditions for the merger.
In particular, we find that the methods used in previous studies to place some
of the tightest constraints on this cross-section do not reflect what is done
observationally, and overstate the Bullet Cluster's ability to constrain the
particle properties of dark matter. We introduce the first simulations of the
Bullet Cluster including both self-interacting dark matter and gas. We find
that as the gas is stripped it introduces radially-dependent asymmetries into
the stellar and dark matter distributions. As the techniques used to determine
the positions of the dark matter and galaxies are sensitive to different radial
scales, these asymmetries can lead to erroneously measured offsets between dark
matter and galaxies even when they are spatially coincident.
