Kinetic Theory of Cosmological Magnetogenesis at Second Order: A New Density-Gradient Source and Comparison with the Harrison Mechanism

Abstract

We derive and compare three mechanisms of cosmological magnetogenesis: the Thomson-scattering velocity-difference mechanism of Takahashi et al.\ (2005), a new density-gradient source identified here for the first time, and the Harrison bulk-flow mechanism of Cembranos et al.\ (2020). Starting from the coupled Maxwell-Boltzmann equations, the complete kinetic theory chain is derived in a single document -- from the BBGKY hierarchy and Thomson collision term, through the generalised Ohm&#39;s law, to the second-order magnetic induction equation. The Ohm&#39;s law correction terms are each bounded by $m_e/m_p\approx5.4\times10^{-4}$, confirming the standard single-fluid approximation to better than $0.1\%$. At second order in cosmological perturbations, products of first-order scalar source vorticity, we identify a coupling between the photon density contrast $δ_γ\equiv δρ^{(1)}_γ/ \barρ_γ$ and the electron-photon velocity difference $(u_e-u_γ)^{(1)}$ that was implicitly present in previous treatments but never isolated. Numerical evaluation with CAMB~v1.6.6 at $z=1100$ shows that this term contributes at ${\approx}0.97\times B_{\rm Tak}$, giving a scattering-mechanism total ${\approx}1.4\times$ the Takahashi result. The Harrison mechanism at the Planck bulk-flow limit ($β<8.5\times10^{-4}$) yields $B\approx5.7\times10^{-24}$~G at 1~Mpc today and dominates for $β\gtrsim2\times10^{-3}$, mildly above the Planck limit. All seed fields exceed the galactic dynamo threshold by many orders of magnitude.