Collapse-accelerated small-scale dynamos in the first stars and galaxies

Here, we explore magnetic field amplification by the small-scale dynamo (SSD) during the formation of the first stars and galaxies. Gravitational collapse continuously modifies turbulent velocities and length scales, making the SSD fundamentally different from a dynamo operating in stationary turbulence. In arXiv:2503.19131, we developed a supercomoving formulation of turbulent dynamos and showed that the increasing dynamo growth rate during collapse leads to super-exponential (SE) magnetic field amplification in the kinematic stage. Here, we show that collapse also accelerates the slower nonlinear stage, allowing magnetic energy to reach progressively larger scales. We apply this framework to first-star and first-galaxy formation and find that the impact of collapse depends strongly on the dynamo efficiency. If the dynamo is sufficiently fast, the field reaches saturation before substantial collapse occurs, leaving little room for further acceleration. At the opposite extreme, if the dynamo is too slow, dynamo amplification remains weak and flux freezing dominates. Between these limits, collapse can substantially accelerate the dynamo, with speed-ups of >10 in the parameter space explored, enabling dynamically significant fields to develop within the available collapse time. Nonlinear scale-by-scale saturation further produces fields correlated on a non-negligible fraction of the system. We provide estimates and scaling relations for the final field strength and correlation scale and their dependence on halo, turbulence, dynamo, and collapse parameters. The earlier emergence of dynamically important fields can affect fragmentation, angular-momentum transport, and the subsequent evolution of the first stars and galaxies.