Statistical Properties of Scale-Invariant Helical Magnetic Fields and Applications to Cosmology

TL;DR

This paper analyzes the statistical properties of scale-invariant helical magnetic fields generated during inflation and explores their observational signatures in the cosmic microwave background, highlighting their evolution over time.

Contribution

It provides a detailed characterization of the statistical and spectral properties of inflationary magnetic fields and their impact on cosmological observations.

Findings

Correlation length relates to the inflationary infrared cutoff.

Scale-invariant spectrum evolves into a steeper turbulence spectrum over time.

Turbulent-diffusive scale increases with the square root of time.

Abstract

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.