CMB anisotropies from primordial inhomogeneous magnetic fields

TL;DR

This study analyzes how primordial inhomogeneous magnetic fields with $B_\lambda \sim 10^{-9}\,\rm G$ and a nearly scale-invariant spectrum imprint CMB anisotropies, focusing on tensor and vector perturbations and their transfer functions. By solving the full covariant perturbation equations and including neutrino anisotropic stress, the authors show a neutrino compensation mechanism on super-horizon scales that suppresses gravitational wave and vorticity sourcing after decoupling, while pre-decoupling generation can yield an observable tensor component with non-Gaussian, potentially blue-tilted signatures. Vector modes produce small-scale B-modes, with BB power peaking near $\ell\sim1000$ and amplitude scaling as $B_\lambda^4$ for $n\approx-2.9$, potentially detectable for $B_\lambda\sim3\times10^{-9}$ G. The results clarify how to distinguish magnetic-field-induced B-modes from inflationary gravitational waves via non-Gaussianity and small-scale power and provide a publicly available CAMB-based tool for computing these spectra.

Abstract

Primordial inhomogeneous magnetic fields of the right strength can leave a signature on the CMB temperature anisotropy and polarization. Potentially observable contributions to polarization B-modes are generated by vorticity and gravitational waves sourced by the magnetic anisotropic stress. We compute the corresponding CMB transfer functions in detail including the effect of neutrinos. The shear rapidly causes the neutrino anisotropic stress to cancel the stress from the magnetic field, suppressing the production of gravitational waves and vorticity on super-horizon scales after neutrino decoupling. A significant large scale signal from tensor modes can only be produced before neutrino decoupling, and the actual amplitude is somewhat uncertain. Plausible values suggest primordial nearly scale invariant fields of ~ 10^(-10)G today may be observable from their large scale tensor anisotropy. They can be distinguished from primordial gravitational waves by their non-Gaussianity. Vector mode vorticity sources B-mode power on much smaller scales with a power spectrum somewhat similar to that expected from weak lensing, suggesting amplitudes ~ 10^(-9)G may be observable on small scales for a spectral index of n ~ -2.9. In the appendix we review the covariant equations for computing the vector and tensor CMB power spectra that we implement numerically.

Figures (2)

• Figure 1: Typical CMB temperature TT (top solid), polarization EE (bottom solid), BB (dashed thick) and cross-correlation TE (dotted; absolute value) power spectra from vector modes with $B_\lambda=3\times 10^{-9} \text{G}$, $n=-2.9$. The thin dashed line shows the scalar adiabatic mode TT spectrum (without magnetic fields). The increase in power at $\ell\alt 10$ is due to reionization at redshift $z\sim 13$.
• Figure 2: Typical CMB B-mode polarization power spectra spectra for vector modes (thick dashed), tensor modes (dotted) and total (solid) for $B_\lambda=3\times 10^{-9} \text{G}$, $n=-2.9$, $\eta_*^\nu/\eta_{\text{in}}=10^6$. The bottom dotted line is from the negligible tensor modes sourced after neutrino decoupling (the compensated mode). The top dotted line is from tensors sourced after magnetic field generation until neutrino decoupling (and should be regarded as an estimate correct to a few orders of magnitude). The thin dashed lines show the $B$-mode spectrum from weak lensing (peaking at $\ell\sim 1000$), and scale invariant primordial tensors with initial power ratio $\sim 10^{-1}$ (peaking at $\ell \sim 100$). The magnetic field spectra scale as $B_\lambda^4$.