Primordial Magnetic Fields and Causality

TL;DR

This work analyzes how causality constrains primordial magnetic fields and the implications for their role as seeds of cosmic magnetism and for CMB signatures. By deriving the most general isotropic, homogeneous correlation in Fourier space for a causal, divergence-free field, the authors show the large-scale magnetic energy spectrum must be blue with index $n=2$, and the helicity component must have $m=3$, invalidating a white-noise assumption. The resulting scale dependence implies extremely strong suppression of magnetic-field amplitudes on Mpc scales, yielding $B_ lambda \,\sim B_0 (L/\lambda)^{5/2}$ for the non-helical part and $\mathcal{B}_\u001b lambda \,\sim (L/\lambda)^3 \mathcal{B}_0$ for helicity, with typical electroweak-era seeds producing only $\sim 10^{-39}$–$10^{-40}$ G on 1 Mpc scales. Consequently, such fields are unlikely to imprint detectable CMB anisotropies or seed cluster magnetic fields, favoring either sub-horizon seed scales or inflationary generation with a red spectrum as viable alternatives.

Abstract

We discuss the implications of causality on a primordial magnetic field. We show that the residual field on large scales is much more suppressed than usually assumed and that a helical component is even more reduced. Due to this strong suppression, even maximal primordial fields generated at the electroweak phase transition can just marginally seed the fields in clusters, but they cannot leave any detectable imprint on the cosmic microwave background.