We present predictions on the evolution of the Tully-Fisher (TF) relation with redshift, based on cosmological N-body/hydrodynamical simulations of disc galaxy formation and evolution. The simulations invoke star formation and stellar feedback, chemical evolution with non-instantaneous recycling, metallicity dependent radiative cooling and effects of a meta-galactic UV field, including simplified radiative transfer. At z=0, the simulated and empirical TF relations are offset by about 0.4 magnitudes (1 sigma) in the B and I bands. The origin of these offsets is somewhat unclear, but it may not necessarily be a problem of the simulations only. As to evolution, we find a brightening of the TF relation between z=0 and z=1 of about 0.85 mag in rest-frame B band, with a non-evolving slope. The brightening we predict is intermediate between the (still quite discrepant) observational estimates. This evolution is primarily a luminosity effect, while the stellar mass TF relation shows negligible evolution. The individual galaxies do gain stellar mass between z=1 and z=0, by a 50-100%; but they also correspondingly increase their characteristic circular speed. As a consequence, individually they mainly evolve ALONG the stellar mass TF relation, while the relation as such does not show any significant evolution.
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