Constraining Velocity-dependent Self-Interacting Dark Matter with the Milky Way's dwarf spheroidal galaxies

The observed anti-correlation between the central dark matter (DM) densities of the bright Milky Way (MW) dwarf spheroidal galaxies (dSphs) and their orbital pericenter distances poses a potential signature of self-interacting dark matter (SIDM). In this work we investigate this possibility by analysing the range of SIDM scattering cross section per unit mass, $\sigma/m_{\chi}$, able to explain such anti-correlation. We simulate the orbital evolution of dSphs subhaloes around the MW assuming an analytical form for the gravitational potential, adopting the proper motions from the Gaia mission and including a consistent characterization of gravitational tidal stripping. The evolution of the subhaloes density profile is modelled using the gravothermal fluid formalism, where DM particle collisions induce thermal conduction that depends on $\sigma/m_{\chi}$. We find that models of dSphs, such as Carina and Fornax, reproduce the observed central DM densities with fixed $\sigma/m_{\chi}$ ranging between $30$ and $50$ cm$^{2}$g$^{-1}$, whereas other dSphs prefer larger values ranging between $70$ and $100$ cm$^{2}$g$^{-1}$. These cross sections correlate with the average collision velocity of DM particles within each subhalo's core, so that systems modelled with large cross sections have lower collision velocities. We fit the cross section-velocity correlation with a SIDM particle model, where a DM particle of mass $m_{\chi}=0.648\pm 0.154$ GeV interacts under the exchange of a light mediator of mass $m_{\phi}=0.636\pm 0.055$ MeV, with the self-interactions being described by a Yukawa potential. The outcome is a cross section-velocity relation that explains the diverse DM profiles of MW dSph satellites and is consistent with observational constraints on larger scales.