Cosmology in the Einstein-Electroweak Theory and Magnetic Fields

In the SU(2)_{L} x U(1)_{Y} standard electroweak theory coupled with the Einstein gravity, new topological configurations naturally emerge, if the spatial section of the universe is globally a three-sphere(S^3) with a small radius. The SU(2)_L gauge fields and Higgs fields wrap the space nontrivially, residing at or near a local minimum of the potential. As the universe expands, however, the shape of the potential rapidly changes and the local minimum eventually disappears. The fields then start to roll down towards the absolute minimum. In the absence of the U(1)_Y gauge interaction the resulting space is a homogeneous and isotropic S^3, but the U(1)_Y gauge interaction necessarily induces anisotropy while preserving the homogeneity of the space. Large magnetic fields are generically produced over a substantial period of the rolling-over transition. The magnetic field configuration is characterized by the Hopf map.