Turbulent Amplification During Contraction: Growth E-Folds and the Saturation Threshold

A contracting plasma is a stirred plasma: as the volume shrinks, flows develop, and a small-scale dynamo can in principle amplify a seed field exponentially. This paper integrates the dynamo growth rate through the contracting phase of a bounced cosmology, from a scale factor of 1e-6 to the turnaround, and compares the total amplification with the energy budget that present-day limits permit. For eddy velocities of a tenth of light speed on a tenth of the horizon scale, the accumulated growth is 3.69e+01 e-folds; combined with adiabatic compression over the same interval, a seed at the start of contraction is amplified by a factor of about 1e48 in field strength by the turnaround. Saturation of the permitted budget, a magnetic fraction of about 1.9e-6 of the turnaround density, therefore requires a seed of only 4.40e-17 tesla at the start of contraction. Whether a contraction dynamo delivers a seed-limited field or a budget-limited one is decided by that threshold, and the threshold is low enough that battery-class seeds plausibly exceed it. The limiting resource is not time but the budget itself, and any excess above it becomes a disposal obligation for the expanding phase. The integration is computed from stated assumptions, not measured.