Offsets between member galaxies and dark matter in clusters: a test with
  the Illustris simulation
  Dark matter with a non-zero self-interacting cross section ($\sigma_{\rm
SIDM}$) has been posited as a solution to a number of outstanding astrophysical
mysteries. Many studies of merging galaxy clusters have given constraints on
$\sigma_{\rm SIDM}$ based on the spatial offset between the member galaxy
population and the dark matter distribution. Assuming $\sigma_{\rm SIDM} = 0$,
how likely is it for us to see the galaxy-DM offset values observed in merging
clusters of galaxies? To answer this question, we formulate a hypothesis test
using data from Illustris, a $\Lambda$CDM cosmological simulation. We select 43
Illustris clusters and their galaxy members at z~0 and examine the accuracy of
commonly used galaxy summary statistics, including kernel-density-estimation
(KDE) luminosity peak, KDE number density peak, shrinking aperture, centroid
and the location of the brightest cluster galaxy (BCG). We use the dark-matter
particles to reproduce commonly adopted methods to identify dark-matter peaks
based on gravitational lensing cluster maps. By analysing each cluster in 768
projections, we determine the optimistic noise floor in the measurements of the
galaxy-DM offsets. We find that the choice of the galaxy summary statistics
affects the inferred offset values substantially, with the BCG and the
luminosity peak giving the tightest 68-th percentile offset levels, $\lesssim$
4 kpc and $\lesssim$ 32 kpc, respectively. Shrinking aperture, number density
and centroid give a large offset scatter of about 50-100 kpc at the 68-th
percentile level, even for clusters with only one dominant mass component. Out
of the 15 reported offsets from observed merging clusters that we examined, 13
of them are consistent with Illustris unrelaxed cluster offsets at the 2-sigma
(95-th percentile) level, i.e. consistent with the hypothesis that $\Lambda$CDM
is the true underlying physical model.
