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We investigate the impact of a stochastic background of Primordial MagneticFields (PMF) generated before recombination on the ionization history of theUniverse and on the Cosmic Microwave Background radiation (CMB).Pre-recombination PMFs are dissipated during recombination and reionization viadecaying MHD turbulence and ambipolar diffusion. This modifies the local matterand electron temperatures and thus affects the ionization history and Thomsonvisibility function. We use this effect to constrain PMFs described by aspectrum of power-law type, extending our previous study (based on ascale-invariant spectrum) to arbitrary spectral index. We derive upper boundson the integrated amplitude of PMFs due to the separate effect of ambipolardiffusion and MHD decaying turbulence and their combination. We show thatambipolar diffusion is relevant for $n_{\rm B}>0$ whereas for $n_{\rm B}<0$ MHDturbulence is more important. The bound marginalized over the spectral index onthe integrated amplitude of PMFs with a sharp cut-off is $\sqrt{\langle B^2\rangle}<0.83$ nG. We discuss the quantitative relevance of the assumptions onthe damping mechanism and the comparison with previous bounds.

Improved CMB anisotropy constraints on primordial magnetic fields from the post-recombination ionization history