Implications of a Primordial Magnetic Field for Magnetic Monopoles,
  Axions, and Dirac Neutrinos
  We explore some particle physics implications of the growing evidence for a
helical primordial magnetic field (PMF). From the interactions of magnetic
monopoles and the PMF, we derive an upper bound on the monopole number density,
$n(t_0) < 1 \times 10^{-20}~{\rm cm}^{-3}$, which is a "primordial" analog of
the Parker bound for the survival of galactic magnetic fields. Our bound is
weaker than existing constraints, but it is derived under independent
assumptions. We also show how improved measurements of the PMF at different
redshifts can lead to further constraints on magnetic monopoles. Axions
interact with the PMF due to the $\varphi {\bf E}\cdot {\bf B}$ interaction.
Including the effects of the cosmological plasma, we find that the helicity of
the PMF is a source for the axion field. Although the magnitude of the source
is small for the PMF, it could potentially be of interest in astrophysical
environments. Finally we apply constraints on the neutrino magnetic dipole
moment that arise from requiring successful big bang nucleosynthesis in the
presence of a PMF, and using the suggested strength $\sim 10^{-14}~{\rm G}$ and
coherence length $\sim 10~{\rm Mpc}$ we find $\mu_\nu \lesssim 10^{-16} \mu_B$.
