Exploring non-Poisson satellite occupation in HOD models and its impact on 2- and 3-point galaxy clustering

Technical Summary: Exploring Non-Poisson Satellite Occupation in HOD Models and its Impact on 2- and 3-Point Galaxy Clustering

Problem Statement
The connection between galaxies and dark matter halos is a cornerstone of modern cosmology, typically modeled using the Halo Occupation Distribution (HOD) framework. A standard assumption in HOD models is that the number of satellite galaxies within a halo of fixed mass follows a Poisson distribution. However, numerical simulations and observational analyses suggest that satellite populations may exhibit deviations from strict Poisson statistics due to effects such as halo assembly bias, tidal stripping, and correlated infall histories. These deviations manifest as sub-Poissonian or super-Poissonian scatter in satellite counts, which alters the second factorial moment of the occupation number. Since this moment directly controls the amplitude of the one-halo term in clustering statistics, ignoring non-Poisson behavior could introduce systematic errors in small-scale clustering analyses and cosmological constraints derived from Stage-IV surveys like DESI and Euclid.

Methodology
To address this, the authors introduce the Conway–Maxwell–Poisson (CMP) distribution as a minimal, flexible extension of the standard Poisson model. The CMP distribution is defined by a probability mass function $P(N_{sat}=k | \lambda, \nu) = \frac{\lambda^k}{(k!)^\nu Z(\lambda, \nu)}$, where $\nu$ is a dispersion parameter: $\nu=1$ recovers the Poisson distribution, $\nu < 1$ yields super-Poissonian behavior (variance > mean), and $\nu > 1$ yields sub-Poissonian behavior (variance < mean).

The methodology involves three primary technical components:

1. Analytical Approximations and Numerical Schemes: Because the normalization constant $Z(\lambda, \nu)$ has no closed form, the authors derive analytical approximations for the mean and variance in two regimes:
   • Small $\lambda$ ($\lambda \to 0$): A series expansion up to $O(\lambda^4)$ is derived to approximate the mean and variance.
   • Large $\lambda$ ($\lambda \to \infty$): Asymptotic expansions based on saddle-point methods are utilized.
   • Global Scaling: A smooth interpolation using an error function is constructed to connect these regimes, ensuring a continuous mapping between the target mean occupation $\langle N_{sat}(M) \rangle$ and the CMP parameter $\lambda$ with $\sim 5\%$ accuracy for $0.5 < \nu < 2$.
2. Mock Catalog Generation: Using the HODDIES package and AbacusSummit N-body simulations (Planck 2018 $\Lambda$CDM cosmology), the authors generate 16 mock galaxy catalogues representing a Luminous Red Galaxy (LRG) sample. The standard HOD parameters are fixed, while the CMP dispersion parameter $\nu$ is varied from 0.5 to 2.0.
3. Clustering Statistics Analysis: The impact of varying $\nu$ is quantified using:
   • 2-Point Statistics: Projected correlation function ($w_p$), and monopole ($\xi_0$) and quadrupole ($\xi_2$) of the redshift-space correlation function.
   • 3-Point Statistics: The galaxy bispectrum (monopole $B_{000}$ and quadrupole $B_{202}$) computed using the Sugiyama estimator in the $k$-range $[0, 0.3] \, h/\text{Mpc}$.
   • Higher-Order Moments: Count-in-Cylinders (CiC) statistics to probe the local galaxy environment and higher-order moments of the distribution.

Key Contributions and Results

• Implementation of CMP in HOD: The paper provides a robust numerical scheme to implement the CMP distribution within HOD frameworks, successfully decoupling the mean satellite occupation from the variance via the parameter $\nu$ while maintaining the physical mean $\langle N_{sat} \rangle$.
• Impact on Occupation Statistics: Varying $\nu$ significantly alters the satellite count distribution. For $\nu < 1$, the distribution broadens (super-Poissonian), increasing the probability of high-occupancy halos; for $\nu > 1$, it narrows (sub-Poissonian). The mean number of satellites varies by at most $\pm 5\%$ relative to the Poisson case, a deviation attributed to the asymmetry of the CMP distribution at low $\lambda$.
• 2-Point Clustering Impact: Non-Poisson statistics primarily affect the one-halo term (small scales, $r_p \lesssim 1 \, h^{-1}\text{Mpc}$).
  • The projected correlation function $w_p$ shows deviations of up to 10% relative to the Poisson case.
  • The monopole and quadrupole show deviations of up to 5%.
  • These changes correlate with the variance: larger variance (lower $\nu$) enhances clustering power, while smaller variance suppresses it.
• 3-Point Clustering (Bispectrum) Impact: The tree-level galaxy bispectrum (in the Sugiyama basis) is found to be robust against non-Poisson assumptions in the range $k_{\text{max}} = 0.3$.
  • Variations in the bispectrum monopole are less than 2%.
  • The authors conclude that for current and near-future large-scale analyses ($k < 0.3$), non-Poisson satellite statistics are unlikely to bias cosmological constraints derived from the bispectrum, as the signal is smaller than expected data errors.
• Count-in-Cylinders (CiC) Sensitivity: CiC statistics exhibit high sensitivity to the variance of satellite occupation.
  • Deviations from the Poisson case reach up to 30%.
  • This confirms that CiC is a powerful probe for breaking degeneracies in HOD parameters and detecting non-Poisson behavior, outperforming standard 2-point statistics in this regard.

Significance and Claims
The paper claims that the CMP distribution offers a "minimal and controlled mechanism" to test the Poisson hypothesis within the HOD framework. The authors assert that while non-Poisson statistics are critical for accurate modeling of small-scale clustering (affecting the one-halo term significantly), their impact on cosmological constraints derived from power spectrum and bispectrum measurements at large scales ($k < 0.3$) is limited.

The work emphasizes that as observational precision increases, understanding the galaxy-halo connection must evolve beyond the Poisson assumption to control systematic uncertainties. The authors suggest that while the bispectrum may be robust, higher-order statistics like CiC and galaxy multiplets are essential for constraining the dispersion parameter $\nu$. The paper concludes that incorporating such extensions is necessary for deriving robust systematic error budgets for future cosmological measurements, though it modestly notes that a detailed analysis of the impact on cosmological parameter inference using the bispectrum remains a subject for future work.