Anisotropic Secondary Bias of Dark Matter Haloes in a $\Lambda$CDM Universe

Secondary bias is the dependence of halo clustering on properties beyond halo mass. Using the $z=0$ TNG300-1-Dark simulation, we study anisotropic secondary bias (ASB): the variation of secondary bias with direction relative to the halo major axis. We first use ordinary, orientation-averaged secondary bias (OSB) as a baseline to compare three environmental manifestations: halo-environment alignment, outer matter anisotropy, and tidal anisotropy. Matching tidal anisotropy suppresses much of the OSB, whereas matching halo-environment alignment or outer matter anisotropy does not. ASB behaves differently. It is weak for formation time, concentration, and triaxiality, but strong for both spin definitions and minor-to-major axis ratio; slowly rotating and more elongated haloes are more strongly aligned with filamentary structure. Matching halo-environment alignment substantially reduces the spin- and shape-dependent ASB signals, whereas matching tidal anisotropy or the outer matter axis ratio leaves them largely intact. Halo definition has little impact on ASB, yet strongly affects low-mass spin bias: including unbound particles can move dense-environment haloes with low bound-particle spin into the high all-particle-spin sample. These results clarify which clustering signals are associated with halo-environment alignment, matter anisotropy, or tidal anisotropy, and which are sensitive to halo definition.