Median evolution of the maximum circular velocity of the dark matter halo. Each column represents the variations of galactic winds (upper Fig.) and BH physics models (bottom Fig.) in order from left to right, and vertical solid lines represent the bar formation redshift. The original galaxy in TNG50 is represented by the final right column. Thinner lines correspond to the evolution of each halo. Vertical solid lines and shaded region represent the median redshift and the distribution between 20^th and 80^th percentiles of the bar formation time. No significant variation is found in the evolution of v_max among the different models.