Axion driven cosmic magneto-genesis during the QCD crossover

We propose a mechanism for the generation of a magnetic field in the early universe during the QCD crossover assuming that dark matter is made of axions. Thermoelectric fields arise at pressure gradients in the primordial plasma due to the difference in charge, energy density and equation of state between the quark and lepton components. The axion field is coupled to the EM field, so when its spatial gradient is misaligned with the thermoelectric field, an electric current is driven. Due to the finite resistivity of the plasma an electric field appears that is generally rotational. For a QCD axion mass consistent with observational constraints and a conventional efficiency for turbulent dynamo amplification --- driven by the same pressure gradients responsible for the thermoelectric fields --- a magnetic field is generated on subhorizon scales. After significant Alfv\'enic unwinding it reaches a present day strength of $B \sim 10^{-13 }$ G on a characteristic scale $L_B \sim $ 20 pc. The resulting combination of $BL_B^{1/2}$ is significantly stronger than in any astrophysical scenario, providing a clear test for the cosmological origin of the field through $\gamma$-ray observations of distant blazars. The amplitude of the pressure gradients may be inferred from the detection of concomitant gravitational waves, while several experiments are underway to confirm or rule out the existence of axions.