  Primordial magnetic fields and massive neutrinos can leave an interesting
signal in the CMB temperature and polarization. We perform a systematic
analysis of general perturbations in the radiation-dominated universe,
accounting for any primordial magnetic field and including leading- order
effects of the neutrino mass. We show that massive neutrinos qualitatively
change the large- scale perturbations sourced by magnetic fields, but that the
effect is much smaller than previously claimed. We calculate the CMB power
spectra sourced by inhomogeneous primordial magnetic fields, from before and
after neutrino decoupling, including scalar, vector and tensor modes, and
consistently modelling the correlation between the density and anisotropic
stress sources. In an appendix we present general series solutions for the
possible regular primordial perturbations.
