Damping of Cosmic Magnetic Fields

TL;DR

This paper addresses how cosmic magnetic fields evolve in an expanding relativistic fluid by deriving the propagation velocities and damping rates of Alfvén, fast magnetosonic, and slow magnetosonic MHD waves in both radiation-diffusion and free-streaming regimes. It develops a relativistic imperfect-fluid framework with viscosity, bulk viscosity, and heat conduction tied to particle mean free paths, and computes dispersion relations and overdamped limits across epochs spanning neutrino decoupling to recombination. The main contributions are explicit damping scales for fast, slow, and Alfvén modes, including diffusion-dominated damping before neutrino decoupling and photon diffusion before recombination, and the identification of surviving slow/Alfvén modes on sub-Silk scales that could influence small-scale structure formation. These results imply that primordial magnetic energy is enhanced or suppressed in different eras, impacting Big Bang nucleosynthesis constraints and the seeding of galactic fields while offering potential avenues for early structure formation.

Abstract

We examine the evolution of magnetic fields in an expanding fluid composed of matter and radiation with particular interest in the evolution of cosmic magnetic fields. We derive the propagation velocities and damping rates for relativistic and non-relativistic fast and slow magnetosonic, and Alfvén waves in the presence of viscous and heat conducting processes. The analysis covers all MHD modes in the radiation diffusion and the free-streaming regimes. When our results are applied to the evolution of magnetic fields in the early universe, we find that cosmic magnetic fields are damped from prior to the epoch of neutrino decoupling up to recombination. Our findings have multifold implications for cosmology. The dissipation of magnetic field energy into heat during the epoch of neutrino decoupling ensures that most magnetic field configurations generated in the very early universe satisfy big bang nucleosynthesis constraints. Further dissipation before recombination constrains models in which primordial magnetic fields give rise to galactic magnetic fields or density perturbations. Finally, the survival of Alfvén and slow magnetosonic modes on scales well below the Silk mass may be of significance for the formation of structure on small scales (abridged).

Figures (2)

• Figure 1: Scales relevant for the evolution of Alfvén and slow magnetosonic waves before recombination, calculated for modes that propagate at $\cos\theta=1$ and a background magnetic field of $B_0 = 3\times 10^{-9} {\rm Gauss}$ today. All length scales are given in comoving units. Any mode with fixed comoving wavelength $\lambda_c$ will at cosmic temperature $T$ be in the photon diffusion regime if it is to the left of the dotted line, or in the photon free-streaming regime if it is to the right. Modes with wavelength $\lambda_c$ will at temperature $T$ be non-oscillatory (overdamped) if they are between the two solid lines. The two dashed lines indicate the temperature at which a mode of given wavelength is damped by one e-fold, either during its oscillatory evolution in photon diffusion or its overdamped evolution in photon free-streaming. The figure assumes $\Omega_b=0.0125$ and $h=1$, and equality between radiation and matter energy density at $T_{\rm EQ}=5.5$eV. See § 4.2 for a more detailed explanation.
• Figure 2: The evolution of the Fourier amplitude as a function of cosmic temperature calculated numerically (solid line), and analytically using the WKB approximation (dashed line), for Alfven waves with three different comoving wavelengths indicated on the figure. For the calculation we assume $B_0 = 3\times 10^{-9} {\rm Gauss}$, $\cos\theta=1$, $\Omega_b=0.0125$, and $h=1$, and we have fixed the ionization fraction at $X_e = 1$ even for temperatures below the approximate temperature of recombination $T\approx 0.25 eV$. The largest scale which the numerical calculation shows to be damped by one e-fold before recombination (at ), corresponds to the analytically calculated length scale of $2\times 10^{23}\,{\rm cm}$.