We employ isolated N-body simulations to study the response of self-interacting dark matter (SIDM) halos in the presence of the baryonic potentials. Dark matter self-interactions lead to kinematic thermalization in the inner halo, resulting in a tight correlation between the dark matter and baryon distributions. A deep baryonic potential shortens the phase of SIDM core expansion and triggers core contraction. This effect can be further enhanced by a large self-scattering cross section. We find the final SIDM density profile is sensitive to the baryonic concentration and the strength of dark matter self-interactions. Assuming a spherical initial halo, we also study evolution of the SIDM halo shape together with the density profile. The halo shape at later epochs deviates from spherical symmetry due to the influence of the non-spherical disc potential, and its significance depends on the baryonic contribution to the total gravitational potential, relative to the dark matter one. In addition, we construct a multi-component model for the Milky Way, including an SIDM halo, a stellar disc and a bulge, and show it is consistent with observations from stellar kinematics and streams.
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