Evaluating the origins of the secondary bias based on the correlation of halo properties with the linear density field

TL;DR

This study investigates the origins of secondary bias in dark matter halos by analyzing correlations between halo properties and the linear density field across different scales, revealing internal and external correlation effects.

Contribution

It introduces a detailed analysis of how internal and external linear density correlations contribute to secondary bias, providing a clearer understanding of its physical origins.

Findings

Secondary bias is linked to correlations with the linear density field.

Internal and external correlations have opposite effects on halo properties.

The combined effect of these correlations explains the complex halo-mass dependence.

Abstract

Using two sets of large $N$-body simulations, we study the origin of the correlations of halo assembly time ($z_{\rm f}$), concentration ($v_{\rm max}/v_{\rm 200}$) and spin ($\lambda$) with the large-scale evolved density field at given halo mass, i.e. the secondary bias. We find that the secondary bias is the secondary effect of the correlations of halo properties with the linear density estimated at the same comoving scale. Using the linear density on different scales, we find two types of correlations. The internal correlation, which reflects the correlation of halo properties with the mean linear over-density $\delta_{\rm L}$ within the halo Lagrangian radius $R_{\rm L}$, is positive for both $z_{\rm f}$ and $v_{\rm max}/v_{\rm 200}$, and negative for $\lambda$. The external correlation, which describes the correlation of halo properties with linear overdensity at $R>R_{\rm L}$ for given $\delta_{\rm L}$, shows trends opposite to the internal correlation. Both of the external and internal correlations depend only weakly on halo mass, indicating a similar origin for halos of different masses. Our findings offer a transparent perspective on the origin of the secondary bias. The secondary bias can be largely explained by the competition of the external and internal correlations together with the correlation of the linear density field on different scales. These two types of correlations combined can establish the complex halo-mass dependence of the secondary bias observed in the simulations.