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

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

Problem Statement
In the $\Lambda$CDM framework, dark matter halo clustering is primarily governed by mass, but it also exhibits "secondary bias" (or assembly bias), where clustering depends on properties such as formation time, concentration, spin, and shape. While the physical origin of this bias is often linked to the tidal field, the specific environmental mechanisms driving these signals remain debated. Furthermore, halo shapes are not randomly oriented relative to the cosmic web; major axes preferentially align with filaments. This paper addresses the gap in understanding how secondary bias varies with direction relative to the halo's major axis. The authors define Anisotropic Secondary Bias (ASB) as the dependence of secondary bias on the angle between the pair-separation vector and the halo major axis. The study seeks to distinguish between three distinct environmental manifestations—halo-environment alignment, outer matter anisotropy, and tidal anisotropy—to determine which controls the overall (orientation-averaged) secondary bias (OSB) and which controls the directional (ASB) signal.

Methodology
The study utilizes the $z=0$ TNG300-1-Dark simulation (a dark-matter-only counterpart of IllustrisTNG) containing $2500^3$ particles in a $205 h^{-1}$ Mpc box.

• Halo Definitions: The analysis primarily uses the SObound definition (spherical overdensity host containing only gravitationally bound particles) as a reference. For spin analysis, the authors compare SObound spin ($\lambda_b$) against SOall spin ($\lambda_a$, including unbound particles) to test halo-definition systematics.
• Halo Properties: The study measures virial mass ($M_{vir}$), formation redshift ($z_{1/2}$), concentration ($c_{vmax}$), spin ($\lambda$), shape (triaxiality $T$ and axis ratio $c/a$).
• Environmental Descriptors: Three categories of environmental quantities are defined:
  1. Halo-environment alignment: The alignment of the halo major axis with the surrounding density ($n_\parallel/n_\perp$) or tidal field ($A \cdot e_3$).
  2. Outer matter anisotropy: The shape of the matter distribution in the shell $2R_{vir} < r < 4R_{vir}$, quantified by the axis ratio $c/a(4R_{vir})$ and the alignment of the outer matter major axis.
  3. Tidal anisotropy ($\alpha$): The anisotropy of the smoothed tidal field derived from the deformation tensor.
• Clustering Measurements:
  • OSB: Measured as the relative bias $b_{rel}^S$ of secondary-property-selected subsamples (upper/lower quartiles) relative to the full sample at fixed mass.
  • ASB: Measured using the alignment correlation function (ACF), separating pairs into "parallel" ($\theta < 45^\circ$) and "perpendicular" ($\theta > 45^\circ$) relative to the halo major axis.
• Conditioning Tests: To isolate causal links, the authors perform matched-control tests. They construct samples where the distribution of a specific environmental descriptor (e.g., tidal anisotropy) is matched between the upper and lower quartiles of a halo property. If the secondary bias signal disappears after matching, that environmental descriptor is identified as the driver.

Key Contributions and Results

1. OSB Baseline and Environmental Drivers:

   • The study establishes that tidal anisotropy ($\alpha$) is the dominant driver of OSB. Matching $\alpha$ substantially suppresses the secondary bias signals for all six halo properties across most mass ranges.
   • In contrast, matching halo-environment alignment or outer matter anisotropy does not suppress OSB, indicating these are not the primary drivers of the orientation-averaged clustering dependence.

2. Characterization of ASB:

   • ASB is weak or absent for formation time, concentration, and triaxiality.
   • ASB is strong for both spin definitions ($\lambda_b$ and $\lambda_a$) and the minor-to-major axis ratio ($c/a$).
   • Specifically, slowly rotating haloes and more elongated haloes exhibit stronger alignment with the filamentary structure.
   • A key finding is that the low-mass spin-bias inversion (where low-spin haloes are more clustered) is anisotropic: it appears only for pairs parallel to the halo major axis for the bound-particle spin ($\lambda_b$), but not for the all-particle spin ($\lambda_a$) or in the perpendicular direction.

3. Decoupling OSB and ASB Mechanisms:

   • Halo-environment alignment is the primary tracer of ASB. Matching the alignment measures ($A \cdot e_3$ or $n_\parallel/n_\perp$) substantially reduces the spin- and shape-dependent ASB signals. However, these same measures do not explain OSB.
   • Outer matter anisotropy (measured independently of the halo axis) does not erase ASB, suggesting ASB is not merely a result of haloes residing in more anisotropic environments, but rather a coupling between halo properties and orientation.
   • Tidal anisotropy explains a large portion of OSB but has little effect on ASB. Matching $\alpha$ leaves the orientation-dependent signals for spin and shape largely intact.

4. Halo Definition Systematics:

   • Halo definition significantly impacts spin OSB at low masses. Including unbound particles (SOall) shifts dense-environment haloes with low bound-particle spin into the high all-particle spin sample, altering the low-mass spin-bias inversion.
   • However, halo definition has minimal impact on ASB. The qualitative trends of orientation-dependent clustering for spin and shape remain consistent across SOall, SObound, and SubFind definitions.

Significance and Claims
The paper claims to clarify the distinct physical origins of orientation-averaged and orientation-dependent secondary bias. It argues that OSB and ASB are not simply different projections of the same environmental mechanism:

• OSB is largely controlled by the scalar tidal anisotropy, which regulates the overall clustering amplitude.
• ASB is controlled by the specific coupling between halo orientation and secondary properties (spin and shape) within the cosmic web, traced by halo-environment alignment.

The authors emphasize that these distinctions are critical for observational cosmology. Ignoring ASB could contaminate redshift-space distortion (RSD) measurements (where orientation-dependent selection mimics velocity effects) and weak lensing analyses (where intrinsic alignments and orientation-dependent halo-matter profiles introduce systematics). Furthermore, the results suggest that galaxy-halo models linking galaxy properties to spin or shape must account for halo definition (specifically particle membership) to avoid systematic errors in inferred secondary bias. The study concludes that ASB provides a necessary framework to identify which halo properties generate orientation-dependent clustering systematics.