We perform dark-matter-only simulations of 28 relaxed massive cluster-sized haloes for Cold Dark Matter ( CDM ) and Self-Interacting Dark Matter ( SIDM ) models , to study structural differences between the models at large radii , where the impact of baryonic physics is expected to be very limited .
We find that the distributions for the radial profiles of the density , ellipsoidal axis ratios , and velocity anisotropies ( \beta ) of the haloes differ considerably between the models ( at the \sim 1 \sigma level ) , even at \gtrsim 10 \% of the virial radius , if the self-scattering cross section is \sigma / m _ { \chi } = 1 cm ^ { 2 } gr ^ { -1 } .
Direct comparison with observationally inferred density profiles disfavours SIDM for \sigma / m _ { \chi } = 1 cm ^ { 2 } gr ^ { -1 } , but in an intermediate radial range ( \sim 3 \% of the virial radius ) , where the impact of baryonic physics is uncertain .
At this level of the cross section , we find a narrower \beta distribution in SIDM , clearly skewed towards isotropic orbits , with no SIDM ( 90 % of CDM ) haloes having \beta > 0.12 at 7 \% of the virial radius .
We estimate that with an observational sample of \sim 30 ( \sim 10 ^ { 15 } M _ { \odot } ) relaxed clusters , \beta can potentially be used to put competitive constraints on SIDM , once observational uncertainties improve by a factor of a few .
We study the suppression of the memory of halo assembly history in SIDM clusters .
For \sigma / m _ { \chi } = 1 cm ^ { 2 } gr ^ { -1 } , we find that this happens only in the central halo regions ( \sim 1 / 4 of the scale radius of the halo ) , and only for haloes that assembled their mass within this region earlier than a formation redshift z _ { f } \sim 2 .
Otherwise , the memory of assembly remains and is reflected in ways similar to CDM , albeit with weaker trends .