Statistical Properties of Scale-Invariant Helical Magnetic Fields and
  Applications to Cosmology
  We investigate the statistical properties of isotropic, stochastic, Gaussian
distributed, helical magnetic fields characterized by different shapes of the
energy spectra at large length scales and study the associated realizability
condition. We discuss smoothed magnetic fields that are commonly used when the
primordial magnetic field is constrained by observational data. We are
particularly interested in scale-invariant magnetic fields that can be
generated during the inflationary stage by quantum fluctuations. We determine
the correlation length of such magnetic fields and relate it to the infrared
cutoff of perturbations produced during inflation. We show that this scale
determines the observational signatures of the inflationary magnetic fields on
the cosmic microwave background. At smaller scales, the scale-invariant
spectrum changes with time. It becomes a steeper weak-turbulence spectrum at
progressively larger scales. We show numerically that the critical length scale
where this happens is the turbulent-diffusive scale, which increases with the
square root of time.
