Magnetic fields are observed on all astrophysical scales of the modern Universe, from planets and stars to galaxies and galaxy clusters. There are also hints of Mpc-scale magnetic fields in the intergalactic medium. Such large-scale fields, if confirmed, have most likely been generated shortly after the Big Bang and might hold a key for solving one of the greatest mysteries in cosmology: the origin of cosmic magnetic fields.
In my talk, I will give an overview of the current status of modeling primordial magnetic fields and their evolution during cosmic history. I will also discuss an extended theory of magnetohydrodynamics (MHD), so-called chiral MHD, which is suitable for modeling the primordial relativistic plasma. Here, effects from the chiral chemical potential, i.e. the asymmetry between left- and right-handed fermions, are included which can give rise to an additional electric current that, in turn, can trigger an instability in the magnetic field. Using direct numerical simulations, the efficient amplification of weak magnetic seed fields via such chiral MHD dynamos can be demonstrated.  In my talk, I will discuss different scenarios of the evolution of magnetic fields in chiral MHD and make predictions for the properties of intergalactic magnetic fields. The future generation of radio telescopes might be able to detect these cosmological magnetic fields and thereby constrain fundamental physical processes less than a second after the Big Bang.