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.
