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Primordial magnetic fields could provide an explanation for the galactic magnetic fields observed today; in which case, they may leave interesting signals in the CMB and the small-scale matter power spectrum. We discuss how to approximately calculate the important nonlinear magnetic effects within the guise of linear perturbation theory and calculate the matter and CMB power spectra including the Sunyaev-Zel'dovich contribution. We then use various cosmological data sets to constrain the form of the magnetic field power spectrum. Using solely large-scale CMB data (WMAP7, QUaD, and ACBAR) we find a 95% C. L. on the variance of the magnetic field at 1 Mpc of B_<6. 4 nG. When we include South Pole Telescope data to constrain the Sunyaev-Zel'dovich effect, we find a revised limit of B_<4. 1 nG. The addition of Sloan Digital Sky Survey Lyman- data lowers this limit even further, roughly constraining the magnetic field to B_<1. 3 nG.
Shaw et al. (2012) studied this question.