In the Wake of Dark Giants: New Signatures of Dark Matter Self
  Interactions in Equal Mass Mergers of Galaxy Clusters
  Merging galaxy clusters have been touted as one of the best probes for
constraining self-interacting dark matter, but few simulations exist to back up
this claim. We simulate equal mass mergers of 10$^{15}$ M$_\odot$ halos, like
the El Gordo and Sausage clusters, with cosmologically-motivated halo and
merger parameters, and with velocity-independent dark-matter self-interactions.
Although the standard lore for merging clusters is that self-interactions lead
to large separations between the galaxy and dark-matter distributions, we find
that maximal galaxy-dark matter offsets of $\lesssim~20$~kpc form for a
self-interaction cross section of $\sigma_\text{SI}/m_\chi$ = 1 cm$^2$/g. This
is an order of magnitude smaller than those measured in observed equal mass and
near equal mass mergers, and is likely to be even smaller for lower-mass
systems. While competitive cross-section constraints are thus unlikely to
emerge from offsets, we find other signatures of self-interactions which are
more promising. Intriguingly, we find that after dark matter halos coalesce,
the collisionless galaxies (and especially the Brightest Cluster Galaxy [BGC])
oscillate around the center of the merger remnant on stable orbits of 100 kpc
for $\sigma_\text{SI}/m_\chi = 1$~cm$^2$/g for at least several Gyr, well after
the clusters have relaxed. If BCG miscentering in relaxed clusters remains a
robust prediction of SIDM under the addition of gas physics, substructure,
merger mass ratios (e.g., 10:1 like the Bullet Cluster), and complex
cosmological merger histories, the observed BCG offsets may constrain
$\sigma_\text{SI}/m_\chi \lesssim$ 0.1 cm$^2$/g---the tightest constraint yet.
