Microwave Background Anisotropies from Alfven waves

TL;DR

This paper examines microwave background anisotropies in the presence of primordial magnetic fields, focusing on Alfvén vector perturbations sourced by a homogeneous field with speed $v_A$ defined by $v_A^2 = B_0^2/[4\pi(\rho_r+p_r)]$. It derives the vector perturbation dynamics, computes the resulting CMB temperature fluctuations, and demonstrates that a fixed field direction induces off-diagonal correlations between $a_{\ell-1,m}$ and $a_{\ell+1,m}$, signaling statistical anisotropy. Using a power-law spectrum for the initial vorticity and accounting for damping, the authors obtain expressions for $C_\ell(m)$ and $D_\ell(m)$ and derive observational constraints on $B_0$ as a function of the spectral index $n$, with strong limits for $-7<n<-3$ and weaker limits for $n>-3$; causally generated fields with $n=2$ are not constrained by these CMB signatures. The work highlights the potential of MAP/PLANCK to bound large-scale vector fields through measurement of off-diagonal CMB correlations, offering a model-independent probe beyond scalar or tensor perturbations.

Abstract

We investigate microwave background anisotropies in the presence of primordial magnetic fields. We show that a homogeneous field with fixed direction can amplify vector perturbations. We calculate the correlations of $δT/T$ explicitly and show that a large scale coherent field induces correlations between $a_{\ell-1,m}$ and $a_{\ell+1,m}$. We discuss constraints on amplitude and spectrum of a primordial magnetic field imposed by observations of CMB anisotropies.

Figures (4)

• Figure 1: An Aitoff projection of the function $f({\bf n})$ for a homogeneous magnetic field pointing in the $\theta=\pi/4$, $\phi=\pi/2$ direction and the reference vector ${\bf n}_0$ pointing in the $z$-direction ($\theta=0)$ (see equation (\ref{['f_eqn']})).
• Figure 2: The function $f({\bf n})$ for ${\bf B_0}$ pointing in the $\theta=\pi/2$, $\phi=0$ direction.
• Figure 3: The function $f({\bf n})$ for ${\bf B_0}$ pointing in the $\theta=0$ direction ( i.e., parallel to ${\bf n}_0$). The gray scale scheme has enhanced the variation in $f$.
• Figure 4: The upper limit on the magnetic field amplitude $B_0$ due to CMB anisotropies caused by Alfvén waves, shown as a function of the magnetic field spectral index $n$. We assume $D_\ell \le 0.1 C_\ell$. Allowed values of the field must lie in the dashed region.