Amplification of Primordial Magnetic Fields by Anisotropic Gravitational Collapse

If a magnetic field is frozen into a plasma that undergoes spherical compression then the magnetic field B varies with the plasma density \rho according to B \propto \rho^{2/3}. In the gravitational collapse of cosmological density perturbations, however, quasi-spherical evolution is very unlikely. In anisotropic collapses the magnetic field can be a much steeper function of gas density than in the isotropic case. We investigate the distribution of amplifications in realistic gravitational collapses from Gaussian initial fluctuations using the Zel'dovich approximation. Representing our results using a relation of the form B\propto \rho^{\alpha}, we show that the median value of \alpha can be much larger than the \alpha=2/3 resulting from spherical collapse, even if there is no initial correlation between magnetic field and principal collapse directions. These analytic arguments go some way towards understanding the results of numerical simulations.