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Primordial magnetic fields from self-ordering scalar fields

A symmetry-breaking phase transition in the early universe could have led to the formation of cosmic defects. Because these defects dynamically excite not only scalar and tensor type cosmological perturbations but also vector type ones, they may serve as a source of primordial magnetic fields. In this study, we calculate the time evolution and the spectrum of magnetic fields that are generated by a type of cosmic defects, called global textures, using the non-linear sigma (NLSM) model. Based on the standard cosmological perturbation theory, we show, both analytically and numerically, that a vector-mode relative velocity between photon and baryon fluids is induced by textures, which inevitably leads to the generation of magnetic fields over a wide range of scales. We find that the amplitude of the magnetic fields is given by $B\sim{10^{-9}}{((1+z)/10^3)^{-2.5}}({v}/{m_{\rm pl}})^2({k}/{\rm Mpc^{-1}})^{3.5}/{\sqrt{N}}$ Gauss in the radiation dominated era for $k\lesssim 1$ Mpc$^{-1}$, with $v$ being the vacuum expectation value of the O(N) symmetric scalar fields. By extrapolating our numerical result toward smaller scales, we expect that $B\sim {10^{-17}}((1+z)/1000)^{-1/2}({v}/{m_{\rm pl}})^2({k}/{\rm Mpc^{-1}})^{1/2}/{\sqrt{N}}$ Gauss on scales of $k\gtrsim 1$ Mpc$^{-1}$ at redshift $z\gtrsim 1100$. This might be a seed of the magnetic fields observed on large scales today.