  Self-Interacting Dark Matter is an attractive alternative to the Cold Dark
Matter paradigm only if it is able to substantially reduce the central
densities of dwarf-size haloes while keeping the densities and shapes of
cluster-size haloes within current constraints. Given the seemingly stringent
nature of the latter, it was thought for nearly a decade that SIDM would be
viable only if the cross section for self-scattering was strongly
velocity-dependent. However, it has recently been suggested that a constant
cross section per unit mass of sigma_T/m~0.1cm^2/g is sufficient to accomplish
the desired effect. We explicitly investigate this claim using high resolution
cosmological simulations of a Milky-Way size halo and find that, similarly to
the Cold Dark Matter case, such cross section produces a population of massive
subhaloes that is inconsistent with the kinematics of the classical dwarf
spheroidals, in particular with the inferred slopes of the mass profiles of
Fornax and Sculptor. This problem is resolved if sigma_T/m~1cm^2/g at the dwarf
spheroidal scales. Since this value is likely inconsistent with the halo shapes
of several clusters, our results leave only a small window open for a
velocity-independent Self-Interacting Dark Matter model to work as a distinct
alternative to Cold Dark Matter.
