Magnetic field spectrum at cosmological recombination revisited

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, non-linear 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, $k \gtrsim 10~h{\rm Mpc}^{-1}$, is smaller than the previous estimates. The amplitude of the generated magnetic fields at cosmological recombination is about $B_{\rm rec} =5.0\times 10^{-24}~{\rm Gauss}$ on $k = 5.0 \times 10^{-1}~h{\rm Mpc}^{-1}$. Finally, we discuss the reason of the discrepancies that exist in estimates of the amplitude of magnetic fields among other authors.