We explore for the first time the effect of self-interacting dark matter (SIDM) on the dark matter (DM) and baryonic distribution in massive galaxies formed in hydrodynamical cosmological simulations, including explicit baryonic physics treatment. A novel implementation of Super-Massive Black Hole (SMBH) formation and evolution is used, as in Tremmel et al.(2015, 2016), allowing to explicitly follow SMBH dynamics at the center of galaxies. A high SIDM constant cross-section is chosen, $sigma$=10 $rm cm^2/gr$, to amplify differences from CDM models. Milky Way-like galaxies form a shallower DM density profile in SIDM than they do in CDM, with differences already at 20 kpc scales. This demonstrates that even for the most massive spirals the effect of SIDM dominates over the adiabatic contraction due to baryons. Strikingly, the dynamics of SMBHs differs in the SIDM and reference CDM case. SMBHs in massive spirals have sunk to the centre of their host galaxy in both the SIDM and CDM run, while in less massive galaxies about 80$%$ of the SMBH population is off-centered in the SIDM case, as opposed to the CDM case in which $sim$90$%$ of SMBHs have reached their hosts centre. SMBHs are found as far as $sim$9 kpc away from the centre of their host SIDM galaxy. This difference is due to the increased dynamical friction timescale caused by the lower DM density in SIDM galaxies compared to CDM, resulting in core stalling. This pilot work highlights the importance of simulating in a full hydrodynamical context different DM models combined to SMBH physics to study their influence on galaxy formation.
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