  Several proposed models for dark matter posit the existence of
self-interaction processes that can impact the shape of dark matter halos,
making them more spherical than the ellipsoidal halos of collisionless dark
matter. One method of probing the halo shapes, and thus the strength of the
dark matter self-interaction, is by measuring the shape of the X-ray gas that
traces the gravitational potential in relaxed elliptical galaxies. In this work
we identify a sample of 11 relaxed, isolated elliptical galaxies and measure
the ellipticity of the gravitating matter using X-ray images from the
XMM-Newton and Chandra telescopes. We explore a variety of different mass
configurations and find that the dark matter halos of these galaxies have
ellipticities around $\epsilon\approx 0.2-0.5$. While we find non-negligible
scatter in the ellipticity distribution, our results are consistent with some
degree of self-interaction at the scale of $\sigma/m \sim 1$ cm$^2$/g, yet they
also remain compatible with a cold dark matter scenario. We additionally
demonstrate how our results can be used to directly constrain specific dark
matter models and discuss implications for current and future simulations of
self-interacting dark matter models.
