We performed detailed MHD simulation in the post-recombination universe to quantify how primordial magnetic fields affect density perturbations. We found that several previous studies based on analytical estimates over-estimated the matter power spectrum. In the figure below we compare the result from our simulations with previous studies.
In the above figure we stop plotting the matter power spectrum on small scales where pre-recombination evolution becomes important. On those small scales we expect further enhancement of matter power spectrum, and this was discussed in detail in previous project
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