  Self-interacting dark matter (SIDM) has the potential to significantly
influence galaxy formation in comparison to the cold, collisionless dark matter
paradigm (CDM), resulting in observable effects. This study aims to elucidate
this influence and to demonstrate that the stellar mass Tully-Fisher relation
imposes robust constraints on the parameter space of velocity-dependent SIDM
models. We present a new set of cosmological hydrodynamical simulations that
include the SIDM scheme from the TangoSIDM project and the SWIFT-EAGLE galaxy
formation model. Two cosmological simulations suites were generated: one
(Reference model) which yields good agreement with the observed $z=0$ galaxy
stellar mass function, galaxy mass-size relation, and stellar-to-halo mass
relation; and another (WeakStellarFB model) in which the stellar feedback is
less efficient, particularly for Milky Way-like systems. Both galaxy formation
models were simulated under four dark matter cosmologies: CDM, SIDM with two
different velocity-dependent cross sections, and SIDM with a constant cross
section. While SIDM does not modify global galaxy properties such as stellar
masses and star formation rates, it does make the galaxies more extended. In
Milky Way-like galaxies, where baryons dominate the central gravitational
potential, SIDM thermalises, causing dark matter to accumulate in the central
regions. This accumulation results in density profiles that are steeper than
those produced in CDM from adiabatic contraction. The enhanced dark matter
density in the central regions of galaxies causes a deviation in the slope of
the Tully-Fisher relation, which significantly diverges from the observational
data. In contrast, the Tully-Fisher relation derived from CDM models aligns
well with observations.
