Secondary bias of dark matter halos: Perspective in the linear density field.

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 density field around halos. We find that the correlations of the three halo properties with the large-scale density at $z=0$ are the secondary effects of their correlations with the initial linear density field. Using the linear density on different scales, we find two types of correlations. The $L$-correlation, which describes the correlation of halo properties with the mean linear over-density $\delta^{\rm i}_{\rm L}$ 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 $E$-correlation, which describes the correlations of halo properties with $\delta^{\rm i}(R/R_{\rm L}>1)$ for given $\delta^{\rm i}_{\rm L}$, shows trends opposite to the $L$-correlation. Both of the $E$- and $L$-correlations depend only weakly on halo mass, indicating a similar origin for halos of different masses. The dependence of the halo bias on the the three halo properties can be well explained by the competition of the $E$- and $L$-correlations and the correlationof the linear density field on different scales. These two types of correlation together can establish the complex halo-mass dependence of the clustering bias produced by the three halo properties.