  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 today's 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 galaxy's 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.
