  Recent observational evidence for initial mass function (IMF) variations in
massive quiescent galaxies at $z = 0$ challenges the long-established paradigm
of a universal IMF. While a few theoretical models relate the IMF to birth
cloud conditions, the physical driver underlying these putative IMF variations
is still largely unclear. Here we use post-processing analysis of the Illustris
cosmological hydrodynamical simulation to investigate possible physical origins
of IMF variability with galactic properties. We do so by tagging stellar
particles in the simulation (each representing a stellar population of
$\approx10^{6}~\mathrm{M}_{\odot}$) with individual IMFs that depend on various
physical conditions, such as velocity dispersion, metallicity, or SFR, at the
time and place the stars are formed. We then follow the assembly of these
populations throughout cosmic time, and reconstruct the overall IMF of each
$z=0$ galaxy from the many distinct IMFs it is comprised of. Our main result is
that applying the observed relations between IMF and galactic properties to the
conditions at the star-formation sites does not result in strong enough IMF
variations between $z = 0$ galaxies. Steeper physical IMF relations are
required for reproducing the observed IMF trends, and some stellar populations
must form with more extreme IMFs than those observed. The origin of this result
is the hierarchical nature of massive galaxy assembly, and it has implications
for the reliability of the strong observed trends, for the ability of
cosmological simulations to capture certain physical conditions in galaxies,
and for theories of star-formation aiming to explain the physical origin of a
variable IMF.
