Constraints on the Primordial Magnetic Field from $\sigma_8$
  A primordial magnetic field (PMF) can affect the evolution of density field fluctuations in the early universe.In this paper we constrain the PMF amplitude $B_\lambda$ and power spectral index $n_\mathrm{B}$ by comparing calculated density field fluctuations with observational data, i.e. the number density fluctuation of galaxies.We show that the observational constraints on cosmological density fluctuations, as parameterized by $\sigma_8$, lead to strong constraints on the amplitude and spectral index of the PMF. 
Dai G. Yamazaki, Kiyotomo Ichiki, Toshitaka Kajino, Grant. J. Mathews