The magnetic fields observed in the Milky~Way and nearby galaxies appear to be in equipartition with the turbulent, thermal, and cosmic ray energy densities, and hence are expected to be dynamically important. However, the origin of these strong magnetic fields is still unclear, and most previous attempts to simulate galaxy formation from cosmological initial conditions have ignored them altogether. Here, we analyse the magnetic fields predicted by the simulations of the Auriga Project, a set of 30 high-resolution cosmological zoom simulations of Milky~Way-like galaxies, carried out with a moving-mesh magneto-hydrodynamics code and a detailed galaxy formation physics model. We find that the magnetic fields grow exponentially at early times owing to a small-scale dynamo with an e-folding time of roughly $100,rm{Myr}$ in the center of halos until saturation occurs around $z=2-3$, when the magnetic energy density reaches about $10%$ of the turbulent energy density with a typical strength of $10-50,rm{mu G}$. In the galactic centers the ratio between magnetic and turbulent energy remains nearly constant until $z=0$. At larger radii, differential rotation in the disks leads to linear amplification that typically saturates around $z=0.5$ to $z=0$. The final radial and vertical variations of the magnetic field strength can be well described by two joint exponential profiles, and are in good agreement with observational constraints. Overall, the magnetic fields have only little effect on the global evolution of the galaxies as it takes too long to reach equipartition. We also demonstrate that our results are well converged with numerical resolution.
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