Time evolution of the ratio of magnetic energy to compression energy in the galactic region (white circle in Fig. 2) for all the simulations during the first 3.5 Gyr. The upper and lower panels corresponds to all the models in ΛCDM and ΛCDM+PMFs settings, respectively. The green and yellow dotted lines in the top panel represent B0.05 and B0.20 models re-simulated without supernovae feedback. The more rapid increase in E_mag compared to E_comp demonstrates the amplification of magnetic fields, even with a week seed (e.g., B_λ = 0.05 nG), through gas accretion, halo collapse and mergers. This amplification occurs in models with the fiducial value of supernovae feedback E_SN = 10⁵¹ erg and with zero feedback.