If vector type perturbations are present in the primordial plasma before recombination , the generation of magnetic fields is known to be inevitable through the Harrison mechanism .
In the context of the standard cosmological perturbation theory , nonlinear couplings of first-order scalar perturbations create second-order vector perturbations , which generate magnetic fields .
Here we reinvestigate the generation of magnetic fields at second-order in cosmological perturbations on the basis of our previous study , and extend it by newly taking into account the time evolution of purely second-order vector perturbations with a newly developed second-order Boltzmann code .
We confirm that the amplitude of magnetic fields from the product-terms of the first-order scalar modes is consistent with the result in our previous study .
However , we find , both numerically and analytically , that the magnetic fields from the purely second-order vector perturbations partially cancel out the magnetic fields from one of the product-terms of the first-order scalar modes , in the tight coupling regime in the radiation dominated era .
Therefore , the amplitude of the magnetic fields on small scales , k \gtrsim 10 ~ { } h { Mpc } ^ { -1 } , is smaller than the previous estimates .
The amplitude of the generated magnetic fields at cosmological recombination is about B _ { rec } = 5.0 \times 10 ^ { -24 } ~ { } { Gauss } on k = 5.0 \times 10 ^ { -1 } ~ { } h { Mpc } ^ { -1 } .
Finally , we discuss the reason for the discrepancies that exist in estimates of the amplitude of magnetic fields among other authors .