In this work we report a numerical study of the cosmic magnetic field amplification due to collisionless plasma instabilities. The collisionless magnetohydrodynamic equations derived account for the pressure anisotropy that leads, in specific conditions, to the firehose and mirror instabilities. We study the time evolution of seed fields in turbulence under the influence of such instabilities. An approximate analytical time evolution of magnetic field is provided. The numerical simulations and the analytical predictions are compared. We found that i) amplification of magnetic field was efficient in firehose unstable turbulent regimes, but not in the mirror unstable models, ii) the growth rate of the magnetic energy density is much faster than the turbulent dynamo, iii) the efficient amplification occurs at small scales. The analytical prediction for the correlation between the growth timescales with pressure anisotropy ratio is confirmed by the numerical simulations. These results reinforce the idea that pressure anisotropies - driven naturally in a turbulent collisionless medium, e.g. the intergalactic medium -, could efficiently amplify the magnetic field in the early Universe (post-recombination era), previous to the collapse of the first large-scale gravitational structures. This mechanism, though fast for the small scale fields ($sim$kpc scales), is however unable to provide relatively strong magnetic fields at large scales. Other mechanisms that were not accounted here (e.g., collisional turbulence once instabilities are quenched, velocity shear, or gravitationally induced inflows of gas into galaxies and clusters) could operate afterwards to build up large scale coherent field structures in the long time evolution.
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