CMB anisotropies from primordial inhomogeneous magnetic fields

Primordial inhomogeneous magnetic fields of the right strength can leave a signature on the cosmic microwave background (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 superhorizon 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 with B_10^-10G 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 B_10^-9G may be observable on small scales for a spectral index n-2. 9. In the Appendix we review the covariant equations for computing the vector and tensor CMB power spectra that we implement numerically.