Constraining primordial magnetic fields with distortions of the black-body spectrum of the cosmic microwave background: pre- and post-decoupling contributions

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Observational limits on y- and μ-type distortions can constrain properties of magnetic fields in the early universe. For a Gaussian, random, and non-helical field, μ and y are calculated a function of the present-day strength of the field, B₀, smoothed over a certain Gaussian width, kc⁻¹, and spectral index, nB, defined by PB(k)∝ kⁿᵇ. For nB=-2.9 and kc⁻¹=1 Mpc, the COBE/FIRAS limit on μ yields B₀<40 nG, whereas the projected PIXIE limit on μ would yield B₀<0.8 nG. For non-scale-invariant spectra, constraints can be stronger. For B₀=1 nG with kc⁻¹=1 Mpc, the COBE/FIRAS limit on μ excludes a wide range of spectral indices given by nB>-2.6. After decoupling, energy dissipation is due to ambipolar diffusion and decaying MHD turbulence, creating a y-type distortion. The distortion is completely dominated by decaying MHD turbulence, and is of order y≈ 10⁻⁷ for a few nG field smoothed over the damping scale at the decoupling epoch, kd,dec≈ 290 (B₀/1 nG)⁻¹ Mpc⁻¹. This contribution is as large as those of the known contributions such as reionization and virialized objects at lower redshifts. The projected PIXIE limit on y would exclude B₀>1.0 and 0.6 nG for nB=-2.9 and -2.3, respectively, and B₀>0.6 nG for nB≥ 2. The current limits on the optical depth to Thomson scattering restrict the predicted y-type distortion to be y 10⁻⁸. (Abridged)