  We exploit the pioneering cosmological hydrodynamical simulation, EAGLE, to
study how the connection between halo mass (M_halo), stellar mass (M*) and
star-formation rate (SFR) evolves across redshift. Using Principal Component
Analysis we identify the key axes of correlation between these physical
quantities, for the full galaxy sample and split by satellite/central and
low/high halo mass. The first principal component of the z=0 EAGLE galaxy
population is a positive correlation between M_halo, M* and SFR. This component
is particularly dominant for central galaxies in low mass haloes. The second
principal component, most significant in high mass haloes, is a negative
correlation between M_halo and SFR, indicative of environmental quenching. For
galaxies above M*~10^10M_solar, however, the SFR is seen to decouple from the
M_halo-M* correlation; this result is found to be independent of environment,
suggesting that mass quenching effects are also in operation. We find extremely
good agreement between the EAGLE principal components and those of SDSS
galaxies; this lends confidence to our conclusions. Extending our study to
EAGLE galaxies in the range z=0-4, we find that, although the relative numbers
of galaxies in the different subsamples change, their principal components do
not change significantly with redshift. This indicates that the physical
processes that govern the evolution of galaxies within their dark matter haloes
act similarly throughout cosmic time. Finally, we present halo occupation
distribution model fits to EAGLE galaxies and show that one flexible
6-parameter functional form is capable of fitting a wide range of different
mass- and SFR-selected subsamples.
