Constraining SIDM with halo shapes: revisited predictions from realistic
  simulations of early-type galaxies
  We study the effect of self-interacting dark matter (SIDM) and baryons on the
shapes of early-type galaxies (ETGs) and their dark matter haloes, comparing
them to the predictions of the standard cold dark matter (CDM) scenario. We use
a sample of five zoom-in simulations of haloes hosting ETGs ($M_{\text vir}\sim
10^{13}M_{\odot}$ and $M_{*}\sim10^{11}M_{\odot}$), simulated in CDM and a SIDM
model with constant cross-section of $\sigma_T/m_\chi = 1\ \mathrm{cm}^2
\mathrm{g}^{-1}$, with and without the inclusion of baryonic physics. We
measure the three-dimensional and projected shapes of the dark matter haloes
and their baryonic content by means of the inertia tensor and compare our
measurements to the results of gravitational lensing and X-ray observations. We
find that the inclusion of baryons greatly reduces the differences between CDM
and a SIDM and thus the ability to draw constraints on the basis of shapes. We
find that lensing measurements clearly reject the predictions from CDM
dark-matter-only simulations, whereas they show a different degree of
preference for the CDM and SIDM hydro scenarios, and cannot discard the SIDM
dark-matter-only case. The shapes of the X-ray emitting gas are also comparable
to observational results in both hydro runs, with CDM predicting higher
elongations only in the very center. Contrary to previous claims at the scale
of elliptical galaxies, we conclude that both CDM and our SIDM model with
$\sigma_T/m_\chi = 1\ \mathrm{cm}^2 \mathrm{g}^{-1}$ are able to explain
observed distributions of halo shapes, once baryons are included in the
simulations
