Halo Occupation Distribution estimation performance for LSST data

1. Problem Statement

The Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) and other upcoming wide-field surveys will provide unprecedented depth and sky coverage, observing billions of galaxies. However, these surveys will primarily rely on photometric data rather than spectroscopic data.

• The Challenge: Accurately determining the Halo Occupation Distribution (HOD)—the statistical relationship between galaxies and their host dark matter haloes—is crucial for cosmology and galaxy formation studies. Traditional HOD estimation often relies on spectroscopic redshifts to identify galaxy groups and their centers.
• The Gap: Existing methods, such as the Background Subtraction Technique (BST), have been validated using spectroscopic data. However, applying these methods to purely photometric data is difficult due to:
  • Photometric redshift uncertainties ($z_{photo}$) leading to projection effects.
  • The inability to directly identify group centers and masses without spectroscopy.
  • The need to estimate group sizes without prior knowledge of halo mass.
  • The challenge of distinguishing central galaxies from satellites in dense environments using only apparent magnitudes and positions.

2. Methodology

The authors propose an extended framework to apply the BST to photometric data, validated using the cosmoDC2 mock galaxy catalogue (derived from the Outer Rim N-body simulation). The methodology involves four key components:

A. Background Subtraction Technique (BST) Extension

The core BST assumes that galaxy groups are local overdensities. The method estimates the number of true group members ($N$) by subtracting the estimated background density ($n_0$) from the total count within a characteristic radius ($r_c$):
$$N = n_i - n_0 \frac{A_i}{A_0}$$

• Adaptation: The authors adapt this to use only photometric inputs (angular positions, apparent magnitudes, and photometric redshifts) rather than spectroscopic group catalogs.

B. Central Galaxy Finder (CGF)

Since photometric data lacks pre-defined group centers, the authors implement an iterative algorithm to identify candidate central galaxies:

1. Selection: Select bright galaxies ($M_r < -19.5$) as candidates.
2. Discard Radius: Define a search radius ($r_{dsc}$) based on the candidate's luminosity.
3. Iteration: Identify all galaxies within the search radius and redshift bin as members of that group, then remove them from the candidate list.
4. Repeat: Move to the next brightest remaining galaxy.

• Parameters: The algorithm uses a scaling factor ($f_r$) for the radius and a redshift bin width ($f_z$). The authors optimize these to maximize purity (fraction of identified centrals that are true centrals) over completeness, as high purity is critical for unbiased HOD estimation.

C. Luminosity-Based Group Size Proxy

In real data, the characteristic halo radius ($r_c$, typically $r_{200}$) is unknown. The authors derive a linear relationship between the central galaxy's luminosity ($L$) and the halo radius:
$$r_c^{lum} = C_1 \cdot L + C_2$$
This allows the BST to estimate the group size directly from observable photometry, removing the need for prior halo mass modeling.

D. Systematic Testing

The pipeline is tested under increasingly realistic conditions:

1. Idealized: Using true redshifts, true group centers, and true halo masses.
2. Luminosity Proxy: Replacing true $r_c$ with $r_c^{lum}$.
3. Photometric Redshifts: Replacing true redshifts with estimates from FlexZBoost (machine learning) and BPZ (template fitting).
4. Halo Mass Uncertainty: Adding Gaussian noise (0.1 and 0.2 dex) to halo masses to simulate estimation errors.
5. Full Pipeline: Applying the CGF to identify centers, using photometric redshifts, and incorporating mass uncertainties.

3. Key Contributions

• Methodological Extension: Successfully extended the BST to function entirely on photometric data, removing the dependency on spectroscopically confirmed group catalogs.
• Central Galaxy Identification: Developed and validated the Central Galaxy Finder (CGF), an iterative centroiding approach that achieves high purity (>70%) in identifying central galaxies from photometric data alone.
• Radius Proxy: Established a robust luminosity-to-radius scaling relation ($r_c^{lum}$) that serves as a viable proxy for halo size in the absence of mass estimates.
• Systematic Error Quantification: Provided a comprehensive analysis of how photometric redshift errors, halo mass uncertainties, and group-finder purity/completeness trade-offs impact HOD recovery.

4. Results

The performance was evaluated by comparing the recovered HOD ($\langle N | M_h \rangle$) against the true HOD from the simulation across absolute magnitude limits ($M_r = -17$ to $-20$) and halo masses up to $10^{15} M_\odot$.

• Idealized Case: The BST recovered the true HOD with excellent agreement (median ratio $\approx 1.02$, dispersion $\approx 0.15$).
• Luminosity Proxy: Using $r_c^{lum}$ instead of true $r_{200}$ resulted in minimal degradation (median ratio $\approx 0.93 - 0.98$), confirming the proxy's validity.
• Photometric Redshifts:
  • FlexZBoost: Performed well, with median ratios close to unity (0.90–0.98).
  • BPZ: Showed slightly higher scatter and lower performance at the faint end ($M_r = -17$) due to larger redshift dispersion, but remained robust for brighter samples.
• Halo Mass Uncertainty: Adding 0.1–0.2 dex scatter to halo masses had a negligible impact on the recovered HOD, indicating the method is robust against typical mass estimation errors.
• Full Pipeline (CGF + Photo-z):
  • When using the CGF to identify centers, performance remained high for bright samples ($M_r = -19, -20$), with median ratios of 0.93–0.95.
  • Performance degraded for the faintest sample ($M_r = -17$) due to low statistics and lower completeness of the CGF at faint magnitudes.
  • Purity vs. Completeness: The chosen CGF parameters prioritized purity (>70%) over completeness (~60%). This trade-off was deemed necessary to avoid biasing the HOD with misidentified satellite galaxies.

5. Significance

• Readiness for LSST: This work demonstrates that the HOD can be reliably estimated from future photometric surveys (LSST, Euclid, Roman) without waiting for complete spectroscopic follow-up.
• Statistical Power: By enabling the use of the full depth and area of photometric surveys, this method significantly enhances the statistical power of HOD studies, allowing for the analysis of satellite populations at fainter magnitudes than previously possible.
• Cosmological Constraints: Robust HOD estimation is essential for constraining cosmological parameters (e.g., $\sigma_8$, $\Omega_m$) and understanding galaxy formation physics (e.g., assembly bias, galactic conformity).
• Scalability: The method is computationally efficient and adaptable, providing a pipeline ready for the massive datasets expected from the LSST Dark Energy Science Collaboration.

In conclusion, the paper validates a robust, observationally motivated pipeline that bridges the gap between theoretical HOD models and the reality of large-scale photometric surveys, paving the way for precision cosmology with the next generation of telescopes.