  We present a suite of baryonic cosmological zoom-in simulations of
self-interacting dark matter (SIDM) haloes within the ``Feedback In Realistic
Environment'' (FIRE) project. The three simulated haloes have virial masses of
$\sim 10^{12}\, \text{M}_\odot$ at $z=0$, and we study velocity-independent
self-interaction cross sections of 1 and 10 ${\rm cm^2 \, g^{-1}}$. We study
star formation rates and the shape of dark matter density profiles of the
parent haloes in both cold dark matter (CDM) and SIDM models. Galaxies formed
in the SIDM haloes have higher star formation rates at $z\leq1$, resulting in
more massive galaxies compared to the CDM simulations. While both CDM and SIDM
simulations show diverse shape of the dark matter density profiles, the SIDM
haloes can reach higher and more steep central densities within few kpcs
compared to the CDM haloes. We identify a correlation between the build-up of
the stars within the half-mass radii of the galaxies and the growth in the
central dark matter densities. The thermalization process in the SIDM haloes is
enhanced in the presence of a dense stellar component. Hence, SIDM haloes with
highly concentrated baryonic profiles are predicted to have higher central dark
matter densities than the CDM haloes. Overall, the SIDM haloes are more
responsive to the presence of a massive baryonic distribution than their CDM
counterparts.
