We present a detailed analysis of an astrophysical mechanism that generates cosmological magnetic fields during the Epoch of Reionization. It is based on the photoionization of the Intergalactic Medium by the first sources formed in the Universe. First the induction equation is derived, then the characteristic length and time scales of the mechanism are identified, and finally numerical applications are carried out for first stars, primordial galaxies and distant powerful quasars. In these simple examples, the strength of the generated magnetic fields varies between the order of $10^{-23}$ G on hundreds of kiloparsecs to $10^{-19}$ G on hundreds of parsecs in the neutral Intergalactic Medium between the Stromgren spheres of the sources. Thus this mechanism contributes to the premagnetization of the whole Universe before large scale structures are in place. It operates with any ionizing source, at any time during the Epoch of Reionization. Finally, the generated fields possess a characteristic spatial configuration which may help discriminate these seeds from those produced by different mechanisms.
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