We present the first cosmological simulations of dwarf galaxies, which include dark matter self-interactions and baryons. We study two dwarf galaxies within cold dark matter, and four different elastic self-interacting scenarios with constant and velocity-dependent cross sections, motivated by a new force in the hidden dark matter sector. Our highest resolution simulation has a baryonic mass resolution of $1.8times 10^2,{rm M}_odot$ and a gravitational softening length of $34,{rm pc}$ at $z=0$. In this first study we focus on the regime of mostly isolated dwarf galaxies with halo masses $sim10^{10},{rm M}_odot$ where dark matter dynamically dominates even at sub-kpc scales. We find that while the global properties of galaxies of this scale are minimally affected by allowed self-interactions, their internal structures change significantly if the cross section is large enough within the inner sub-kpc region. In these dark-matter-dominated systems, self-scattering ties the shape of the stellar distribution to that of the dark matter distribution. In particular, we find that the stellar core radius is closely related to the dark matter core radius generated by self-interactions. Dark matter collisions lead to dwarf galaxies with larger stellar cores and smaller stellar central densities compared to the cold dark matter case. The central metallicity within $1,{rm kpc}$ is also larger by up to $sim 15%$ in the former case. We conclude that the mass distribution, and characteristics of the central stars in dwarf galaxies can potentially be used to probe the self-interacting nature of dark matter.
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