Ingredients and Equations for making a Magnetic Field in the Early Universe

The ingredients required to create a magnetic field in the early Universe are identified, and compared with Sakharov’s conditions for baryogenesis. It is also shown that a long range coherent magnetic field is not generated by the classical rolling Higgs vacuum expectation value during the electroweak phase transition. The Universe is observed to have coherent magnetic fields over a wide variety of scales, extending from the earth to possibly galaxy clusters [1]. These observations are puzzling, because it is not clear how the large-scale fields are made. Our galactic disk locally has a coherent magnetic field of strength ∼ 10 Gauss, approximately in the direction of the galactic arm [1, 2]. Observations of the Faraday rotation of radio waves from extragalactic sources by various spiral galaxies suggest that these galaxies today have aximuthal magnetic fields of strength ∼ 3 × 10 Gauss. There are also recent data suggesting micro Gauss fields are common in galaxy clusters. Measuring magnetic fields at high redshift is difficult, but it appears that there is a galaxy at z ∼ .4 with a microGauss field, and that the Lyman α forest may also include clouds with fields of about this strength [1]. An extensive review of the observations and astrophysics of magnetic field preservation and amplification can be found in [1]. An aximuthal field is a natural configuration inside a differentially rotating spheroid, and suggests that the present magnetic field in spiral galaxies may have been amplified by the dynamo mechanism [1, 3]. However, the timescale during which the dynamo grows the field, and by how much the field is magnified, are unclear, so although one knows that a seed field is required on galactic scales, its magnitude is not well defined ( ~ B ∼ 10 —10 Gauss). There are various astrophysical [1] and cosmological [4, 5, 6, 7, 8] scenarios for making this seed field. This paper addresses two distinct issues related to the generation of magnetic fields. In the first section, the symmetries, or properties, of the early