  We use hydrodynamical cosmological simulations to test the differences
between cold and self-interacting dark matter models (CDM and SIDM) in the mass
range of massive galaxies
($10^{12}M_{\odot}h^{-1}<M<10^{13.5}M_{\odot}h^{-1}$). We consider two SIDM
models: one with constant cross section $\sigma/m_{\chi}=1\mathrm{cm^2g^{-1}}$
and one where the cross section is velocity-dependent. We analyse the halo
density profiles and concentrations, comparing the predictions of
dark-matter-only and hydrodynamical simulations in all scenarios. We calculate
the best-fit Einasto profiles and compare the resulting parameters with
previous studies and define the best-fit concentration-mass relations. We find
that the inclusion of baryons reduces the differences between different dark
matter models with respect to the DM-only case. In SIDM hydro runs, deviations
from the CDM density profiles weakly depend on mass: the most massive systems
($M>10^{13}M_{\odot}h^{-1}$) show cored profiles, while the least massive ones
($M<10^{12.5}M_{\odot}h^{-1}$) have cuspier profiles. Finally, we compare the
predictions of our simulations to observational results, by looking at the dark
matter fractions and the distribution of strong lensing Einstein radii. We find
that in SIDM the DM-fractions decrease more rapidly with increasing stellar
mass than in CDM, leading to lower fractions at $M_{*}>10^{11}M_{\odot}$, a
distinctive signature of SIDM. At the same time, the distribution of Einstein
radii, derived from both CDM and SIDM hydro runs, is comparable to observed
samples of strong lenses with $M>10^{13}M_{\odot}h^{-1}$. We conclude that the
interplay between self-interaction and baryons can greatly reduce the expected
differences between CDM and SIDM models at this mass scale, and that techniques
able to separate the dark and luminous mass in the inner regions of galaxies
are needed to constrain self-interactions.
