Cosmic Neutrino Background as a Ferromagnet

If cosmic background neutrinos interact very weakly with each other, through spin-spin interactions, then they may have experienced a phase transition, leading to a ferromagnetic ordering. The small magnetic field resulting from ferromagnetic ordering -- if present before galaxy formation -- could act as a primordial seed of the magnetic fields observed in several galaxies. Our findings suggest that the magnetization could occur in the right epoch, if the exchange boson of neutrino-neutrino interaction is a massless boson beyond the Standard Model, with a coupling constant of $2.2\times 10^{-13} \left(\frac{m_\nu}{10^{-4}\,\rm{eV}}\right)^2<g<2.3\times 10^{-7}$. The estimation of the magnetic seed is $2.3 \times 10^{-27}\,\rm{G}\lesssim B_{\rm CNB}\lesssim 6.8\times 10^{-10}G$.