We explore the co-evolution of the specific angular momentum of dark matter haloes and the cold baryons that comprise the galaxies within. We study over two thousand central galaxies within the reference cosmological hydrodynamical simulation of the Evolution and Assembly of GaLaxies and their Environments (EAGLE) project. We employ a methodology within which the evolutionary history of a system is specified by the time-evolving properties of the Lagrangian particles that define it at z=0. We find a strong correlation between the evolution of the specific angular momentum of todays stars (cold gas) and that of the inner (whole) dark matter halo they are associated with. This link is particularly strong for the stars formed before the epoch of maximum expansion and subsequent collapse of the central dark matter halo (turnaround). Spheroids are typically assembled primarily from stars formed prior to turnaround, and are therefore destined to suffer a net loss of angular momentum associated with the strong merging activity during the assembly of the inner dark matter halo. Stellar discs retain their specific angular momentum since they are comprised of stars formed mainly after turnaround, from gas that mostly preserves the high specific angular momentum it acquired by tidal torques during the linear growth of the halo. Since the specific angular momentum loss of the stars is tied to the galaxys morphology today, it may be possible to use our results to predict, statistically, the assembly history of a halo given the morphology of the galaxy it hosts.
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