On the prospect of using the maximum circular velocity of halos to
  encapsulate assembly bias in the galaxy-halo connection
  We investigate a conceptual modification of the halo occupation distribution
approach, using the halos' present-day maximal circular velocity, $\vmax$, as
an alternative to halo mass. In particular, using a semi-analytic galaxy
formation model applied to the Millennium WMAP7 simulation, we explore the
extent that switching to $\vmax$ as the primary halo property incorporates the
effects of assembly bias into the formalism. We consider fixed number density
galaxy samples ranked by stellar mass and examine the variations in the halo
occupation functions with either halo concentration or formation time. We find
that using $\vmax$ results in a significant reduction in the occupancy
variation of the central galaxies, particularly for concentration. The
satellites occupancy variation on the other hand increases in all cases. We
find effectively no change in the halo clustering dependence on concentration,
for fixed bins of $\vmax$ compared to fixed halo mass. Most crucially, we
calculate the impact of assembly bias on galaxy clustering by comparing the
amplitude of clustering to that of a shuffled galaxy sample, finding that the
level of galaxy assembly bias remains largely unchanged. Our results suggest
that while using $\vmax$ as a proxy for halo mass diminishes some of occupancy
variations exhibited in the galaxy-halo relation, it is not able to encapsulate
the effects of assembly bias potentially present in galaxy clustering. The use
of other more complex halo properties, such as $\vpeak$, the peak value of
$\vmax$ over the assembly history, provides some improvement and warrants
further investigation.
