Halo Occupation Model (HOMe)

• HOMe is a theoretical framework that defines the probability of galaxy occupancy in dark matter halos using statistical distributions.
• It differentiates between central and satellite galaxies to model both small- and large-scale clustering via 1-halo and 2-halo terms.
• The model is implemented through simulations and Bayesian inference to create mock catalogs and constrain galaxy formation efficiency.

The Halo Occupation Model (HOMe) is a theoretical and computational framework in large-scale structure and galaxy formation research. It relates the distribution of galaxies to the underlying distribution of dark matter halos, providing a statistical connection between galaxy populations and the hierarchical structure of dark matter in the universe.

1. Definition and Scope

The Halo Occupation Model (HOMe) specifies the probability distribution $P(N | M)$ that a halo of virial mass $M$ contains $N$ galaxies of a given type (e.g., above a luminosity threshold). This mapping, often called the Halo Occupation Distribution (HOD), statistically characterizes the way galaxies populate dark matter halos and forms the basis for predicting galaxy clustering on various scales. The model is essential for interpreting observed galaxy clustering, weak lensing, and for constructing mock galaxy catalogs from cosmological simulations.

The formalism typically distinguishes between central and satellite galaxies within each halo, enabling more accurate modeling of small- and large-scale galaxy correlations.

2. Mathematical Formalism

Halo Occupation Distribution

Let $n_g$ be the mean galaxy number density, and $n(M)$ the halo mass function. The average number of galaxies per halo of mass $M$ is:

$$\langle N(M) \rangle = \sum_{N=0}^{\infty} N \, P(N|M)$$

The total galaxy number density is then:

$n_g = \int \langle N(M) \rangle n(M) dM$

A commonly adopted parameterization is:

• The central galaxy follows a nearest-integer distribution (often Bernoulli)
• The satellite galaxy number is often Poisson-distributed, conditional on a central galaxy being present.

For example,

$$\langle N_{cen}(M) \rangle = \frac{1}{2} \left[1 + \text{erf} \left( \frac{\log M - \log M_{min}}{\sigma_{\log M}} \right) \right]$$

$$\langle N_{sat}(M) \rangle = \langle N_{cen}(M) \rangle \left( \frac{M - M_0}{M_1'} \right)^{\alpha}$$

for $M > M_0$, with $M_0, M_1', \alpha$ as parameters.

Galaxy Correlation Function Decomposition

The galaxy two-point correlation function is decomposed into:

1. 1-halo term: pairs within the same halo, sensitive to the satellite HOD and the halo density profile.
2. 2-halo term: pairs in distinct halos, governed by the large-scale bias and central occupation.

The total correlation $\xi_{gg}(r)$ is:

$\xi_{gg}(r) = \xi_{gg}^{1h}(r) + \xi_{gg}^{2h}(r)$

with each term calculated as an integral over the halo mass function, occupation moments, and spatial distribution.

3. Physical Assumptions and Interpretation

The HOMe assumes galaxy formation is governed primarily by halo mass, with additional dependencies sometimes introduced (e.g., assembly bias, environmental dependence). It abstracts away the complex baryonic physics of galaxy formation into simple, statistically motivated functions.

Central-satellite decomposition reflects the differing formation and evolution processes experienced by the most massive galaxy in a halo ("central") and its less massive companions ("satellites"). Occupation function parameters are constrained by fitting observed clustering statistics.

4. Computational Implementation and Inference

HOMe is applied by populating halos from $N$-body simulations or analytic halo catalogs using the specified HOD. The implementation steps are:

1. Sample halo catalog from a simulation with known masses and positions.
2. For each halo, randomly assign galaxies following $P(N_{cen}|M)$ and $P(N_{sat}|M)$.
3. Place centrals at the halo center; sample satellite positions from the halo's NFW or other assumed density profile.
4. Calculate mock observable statistics (e.g., projected correlation function $w_p(r_p)$) and compare to data.
5. Calibrate HOD parameters via likelihood maximization or Bayesian inference.

5. Extensions and Related Models

While the basic HOMe ties galaxy occupancy solely to halo mass, extensions include:

• Conditional Luminosity Function (CLF): replaces occupation number with the conditional distribution of galaxy luminosities.
• Subhalo Abundance Matching (SHAM): matches galaxies to simulated subhalos by luminosity or stellar mass.
• Decorated HODs: incorporate secondary halo properties (concentration, formation history).
• Environmental HODs: explicitly parameterize occupation dependence on the large-scale environment.

These models address observed phenomena such as "assembly bias", where halo properties beyond mass affect galaxy content.

6. Applications and Research Impact

HOMe has been the standard toolset for:

• Interpreting observed galaxy clustering across SDSS, DES, and other surveys.
• Making robust forecasts for galaxy surveys and cosmological parameter estimation.
• Constructing synthetic mock catalogs for large-scale structure analyses.
• Weak lensing and galaxy-galaxy lensing signal interpretation by linking matter and galaxy distributions.
• Informing empirical models of galaxy formation and evolution.

HOD modeling results constrain the efficiency of galaxy formation as a function of halo mass and provide insights into feedback processes, satellite quenching, and the link between dark matter and luminous tracers.

7. Limitations and Challenges

While powerful, HOMe relies on simplifying assumptions:

• The absence of strong non-mass dependencies (i.e., neglect of assembly bias can bias results).
• Limited physical insight into the detailed baryonic processes, relegating it primarily to a phenomenological mapping.
• Calibration depends strongly on the accuracy of the underlying simulation, halo mass function, and cosmological parameters.

Recent work focuses on incorporating more physics-motivated parameters, improving the predictive power in the non-linear regime, and connecting occupation models with hydrodynamical simulation results.