The Nucleosynthesis Bound on Primordial Fields, Stated at the Bounce

Primordial nucleosynthesis is sensitive to the expansion rate when the light elements formed, and a magnetic field contributes to that rate exactly as radiation does. This paper converts the resulting bound into bounce-frame quantities. Requiring the magnetic share of the radiation density at a temperature of one megaelectronvolt to be below ten per cent caps the field then at 1.86e+09 tesla, equivalent to 1.02e+03 nanogauss today; a one per cent share caps it at 3.24e+02 nanogauss. Both are far weaker than the microwave-background limit for megaparsec-scale fields. The nucleosynthesis bound earns its place elsewhere: it applies to the total magnetic energy on all scales, including small scales where the microwave limit has no reach, and in a bounced cosmology it applies twice in spirit, since the contracting phase passed through the same temperature on its way down. A field generated during contraction had to satisfy the nucleosynthesis condition during the hot contracting epoch as well as in the expanding one, a double application that singular cosmologies do not face. Computed from stated assumptions, not measured.