Precise numerical estimation of the magnetic field generated around recombination

We investigate the generation of magnetic fields from nonlinear effects around recombination. As tight-coupling is gradually lost when approaching $z\simeq 1100$, the velocity difference between photons and baryons starts to increase, leading to an increasing Compton drag of the photons on the electrons. The protons are then forced to follow the electrons due to the electric field created by the charge displacement; the same field, following Maxwell’s laws, eventually induces a magnetic field on cosmological scales. Since scalar perturbations do not generate any magnetic field as they are curl-free, one has to resort to second-order perturbation theory to compute the magnetic field generated by this effect. We reinvestigate this problem numerically using the powerful second-order Boltzmann code SONG. We show that: (i) all previous studies do not have a high enough angular resolution to reach a precise and consistent estimation of the magnetic field spectrum; (ii) the magnetic field is generated up to $z\simeq 10$; (iii) it is in practice impossible to compute the magnetic field with a Boltzmann code for scales smaller than 1 Mpc. Finally we confirm that for scales of a few Mpc, this magnetic field is of order $2\times {10}^{-29}\mathrm{G}$, many orders of magnitude smaller than what is currently observed on intergalactic scales.

Figure 1

The averaged magnetic field $B(\lambda )$ as a function of the averaging scale for various redshifts. We apply a factor $(1+z{)}^2$ to cancel the adiabatic decay of the magnetic field. The curves at $1+z=10$ and $1+z=1$ overlap since there are no sources for the magnetic field at late times. In contrast, there is a significant difference between the magnetic field after recombination at $1+z=1000$ and $1+z=100$ demonstrating the continuation of the sources after recombination.

Figure 2

Stability of the magnetic field with respect to the angular resolution in SONG, represented by the maximum multipole ${\ell }_{\max }$ considered in the Boltzmann hierarchy. Large scales are not sensitive to this cut, whereas small scales need a cut at ${\ell }_{\max }\approx 50$ to reach percent level accuracy.

Figure 3

The time evolution of the magnetic field power spectrum for several k-modes. Solid lines are from bottom to top $k/k_{\mathrm{eq}}=0.1$, 1, 10, dashed lines are $k/k_{\mathrm{eq}}=0.4$, 4 and dotted lines $k/k_{\mathrm{eq}}=2$, 20. All modes grow at a constant rate while superhorizon and begin to decay once they enter the horizon. The peak is due to the generation of magnetic fields during recombination, and subsequently the evolution reverts back to constant decay once the photon density is negligible.