In the standard cosmological model , magnetic fields and vorticity are generated during the radiation era via second-order density perturbations .
In order to clarify the complicated physics of this second-order magnetogenesis , we use a covariant approach and present the electromagneto-dynamical equations in the nonlinear regime .
We use the tight-coupling approximation to analyze Thomson and Coulomb scattering .
At the zero-order limit of exact tight-coupling , we show that the vorticity is zero and no magnetogenesis takes place at any nonlinear order .
We show that magnetogenesis also fails at all orders if either protons or electrons have the same velocity as the radiation , and momentum transfer is neglected .
Then we prove a key no-go result : at first-order in the tight-coupling approximation , magnetic fields and vorticity still can not be generated even via nonlinear effects .
The tight-coupling approximation must be broken at first order , for the generation of vorticity and magnetic fields , and we derive a closed set of nonlinear evolution equations that governs this generation .
We estimate that the amplitude of the magnetic field at recombination on the horizon scale is \sim 10 ^ { -27 } G .