Angular momentum transfer in primordial discs and the rotation of the first stars

We investigate the rotation velocity of the first stars by modelling the angular momentum transfer in the primordial accretion disc.Assessing the impact of magnetic braking, we consider the transition in angular momentum transport mode at the Alfv$\acute{\rm e}$n radius, from the dynamically dominated free-fall accretion to the magnetically dominated solid-body one.The accreting protostar at the centre of the primordial star-forming cloud rotates with close to breakup speed in the case without magnetic fields.Considering a physically-motivated model for small-scale turbulent dynamo amplification, we find that stellar rotation speed quickly declines if a large fraction of the initial turbulent energy is converted to magnetic energy ($\gtrsim 0.14$). Alternatively, if the dynamo process were inefficient, for amplification due to flux-freezing, stars would become slow rotators if the pre-galactic magnetic field strength is above a critical value, $\simeq 10^{-8.2}$G, evaluated at a scale of $n_{\rm H} = 1 {\rm cm}^{-3}$, which is significantly higher than plausible cosmological seed values ($\sim 10^{-15}$G). Because of the rapid decline of the stellar rotational speed over a narrow range in model parameters, the first stars encounter a bimodal fate: rapid rotation at almost the breakup level, or the near absence of any rotation.