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Recent numerical studies suggest that magnetic fields play an important rolein primordial star formation in the early universe. However, the detailedevolution of the magnetic field in the collapse phase still has uncertaintiesbecause of the complicated physics associated with turbulence in a collapsingmagnetized system. Here, we perform a suite of numerical MHD simulations thatfollow the collapse of magnetized, turbulent primordial gas clouds toinvestigate the evolution of the magnetic field associated with the turbulence,assuming a polytropic equation of state with exponent $\gamma_{\rm eff}$ andwith various numerical resolutions. In addition, we generalize the analytictheory of magnetic field growth/saturation so that it can deal with variousexponents $\gamma_{\rm eff}$ and turbulence energy spectra. We find that thenumerical results are well reproduced by the theory for various $\gamma_{\rmeff}$ through the collapse phase during the formation of the first stars. Themagnetic field is eventually amplified by a factor of $10^{12}$ -- $10^{15}$due to kinematic and non-linear turbulent dynamo effects and reaches 3% -- 100%of the equipartition level, depending on $\gamma_{\rm eff}$. We also find thatthe transition between the kinematic and non-linear stages can be analyticallyestimated. These results indicate that the strong magnetic field accompanied bysupersonic turbulence is a general property and suggest that it can play acrucial role in the formation of the first stars.

Amplification and saturation of turbulent magnetic field in collapsing primordial gas clouds