# On the prospect of using the maximum circular velocity of halos to encapsulate assembly bias in the galaxy-halo connection

## TL;DR

This study examines whether replacing halo mass with maximum circular velocity ($ ext{V}_{max}$) in galaxy-halo models captures assembly bias effects, finding limited success and suggesting more complex properties like $ ext{V}_{peak}$ may be needed.

## Contribution

The paper evaluates the effectiveness of using $ ext{V}_{max}$ instead of halo mass to incorporate assembly bias into galaxy-halo connection models, revealing its limitations.

## Key findings

- Using $ ext{V}_{max}$ reduces central galaxy occupancy variation.
- Satellite occupancy variation increases with $ ext{V}_{max}$.
- Galaxy assembly bias remains largely unaffected by switching to $ ext{V}_{max}$.

## Abstract

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.