Limit on Primordial Small-Scale Magnetic Fields from Cosmic Microwave Background Distortions

Spatially varying primordial magnetic fields may be efficiently dissipated prior to the epoch of recombination due to the large viscosity of the baryon-photon fluid. We show that this dissipation may result in observable chemical potential $\mu$ and Compton $y$ distortions in the cosmic microwave background spectrum. Current upper limits on $\mu$ and $y$ from FIRAS constrain magnetic fields to have strength $B_0<3\mathrm{}\times {10}^{-8\mathrm{}}$ G (scaled to the present) between comoving coherence length $\approx 400$ pc and $\approx 0.6$ Mpc. These represent the strongest upper limits on small-scale primordial magnetic fields to date.