  (Abridged) We compare recent results from X-ray, strong lensing, weak
lensing, and optical observations with numerical simulations of the merging
galaxy cluster 1E0657-56. X-ray observations reveal a bullet-like subcluster
with a prominent bow shock, while lensing results show that the positions of
the total mass peaks are consistent with the centroids of the collisionless
galaxies (and inconsistent with the X-ray brightness peaks). Previous studies,
based on older observational datasets, have placed upper limits on the
self-interaction cross-section of dark matter per unit mass, sigma/m, using
simplified analytic techniques. In this work, we take advantage of new,
higher-quality observational datasets by running N-body simulations of
1E0657-56 that include the effects of self-interacting dark matter, and
comparing the results with observations. Furthermore, the recent data allow for
a new independent method of constraining sigma/m, based on the non-observation
of an offset between the bullet subcluster mass peak and galaxy centroid. This
new method places an upper limit (68% confidence) of sigma/m < 1.25 cm^2/g. If
we make the assumption that the subcluster and the main cluster had equal
mass-to-light ratios prior to the merger, we derive our most stringent
constraint of sigma/m < 0.7 cm^2/g, which comes from the consistency of the
subcluster's observed mass-to-light ratio with the main cluster's, and with the
universal cluster value, ruling out the possibility of a large fraction of dark
matter particles being scattered away due to collisions. Our limit is a slight
improvement over the previous result from analytic estimates, and rules out
most of the 0.5 - 5cm^2/g range invoked to explain inconsistencies between the
standard collisionless cold dark matter model and observations.
