Microwave Background Signals from Tangled Magnetic Fields

An inhomogeneous cosmological magnetic field will create Alfvén-wave modes that induce a small rotational velocity perturbation on the last scattering surface of the microwave background radiation. The Alfvén-wave mode survives Silk damping on much smaller scales than the compressional modes. This, in combination with its rotational nature, ensures that there will be no sharp cutoff in anisotropy on arc min scales. We estimate that a magnetic field which redshifts to a present value of $3\times {10}^{-9\mathrm{}}\mathrm{G}$ produces temperature anisotropies at the $10\mu \mathrm{K}$ level at and below 10 arc min scales. A tangled magnetic field, which is large enough to influence the formation of large scale structure, is therefore potentially detectable by future observations.