Halo Occupation Distribution¶

A model of galaxy-count probabilities conditional on dark-matter halo mass.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9792
Domain group: Natural Sciences
Origin domain: Astronomy & Astrophysics
Subdomains: Cosmology and Galaxy Clustering, Large Scale Structure → Astronomy & Astrophysics
Aliases: HOD

Core Idea¶

The halo occupation distribution asks how many galaxies of a specified sample occupy a dark-matter halo of mass M. Its minimum object is P(N|M), a conditional probability law over galaxy count N, not one universal scalar parameter. Clustering models often refine it with central/satellite components and prescriptions for position or velocity inside halos. Changing the sample or cosmology changes the fitted relation.

A two-mass-bin toy HOD can give distinct probabilities for zero, one or two galaxies at each mass; the scatter matters as well as the mean. Abazajian and colleagues inferred HOD parameters from SDSS clustering of a selected galaxy sample. This is a real model fit, not direct enumeration of invisible halos. The current Probability Distribution node includes named-family machinery not required by every HOD, so it is related but not asserted as a strict parent.

Structural Signature¶

Sig role-phrases:

Halo mass M — Conditions occupancy on a dark-matter halo mass in a declared cosmological model. It is constitutive. Counterfactual: A galaxy count with no halo-mass condition is not HOD.
Defined galaxy population — Fixes which luminosity-selected or other sample galaxies are counted. It is constitutive. Counterfactual: Changing the sample changes the HOD.
Galaxy occupancy count N — Counts sample galaxies associated with a halo. It is constitutive. Counterfactual: A mass-only relation with no galaxy count omits occupancy.
Conditional count law P(N|M) — Distributes probability across possible occupancies at each mass. It is constitutive. Counterfactual: A mean alone can omit scatter and zero-versus-multiple occupancy.
Within-halo and inference convention — Optionally supplies central/satellite or spatial and velocity detail under survey and cosmological assumptions. It is boundary. Counterfactual: A fitted relation is not a direct census of invisible halos.

What It Is Not¶

Not one parameter. P(N|M) is a conditional law, sometimes represented by several fitted parameters.
Not just a mean. Count scatter matters to clustering predictions.
Not a direct halo census. Dark-halo occupancy is inferred from model and data.
Not sample-independent. Galaxy selection changes what N counts.
Closest near-miss. A mean galaxy-count curve by mass is the nearest miss when its scatter and count distribution are unspecified.

Scope of Application¶

Galaxy clustering. Connect correlations to modeled halo occupancy.
Galaxy formation. Compare predicted central and satellite populations.
Survey inference. Fit parameters for a declared observed galaxy sample.
Cosmological model tests. Assess joint halo and occupation assumptions against clustering.

Clarity¶

Name the galaxy sample, halo mass M and count N, then specify P(N|M) or its declared approximation. A mean curve with no scatter is the nearest miss if the distribution is at issue. Separate observed clustering from inferred halo occupancy and state cosmology and selection assumptions.

Manages Complexity¶

Galaxy surveys record luminous positions and redshifts but do not identify every dark halo directly. HOD compresses the uncertain galaxy–halo connection into conditional count probabilities, with optional within-halo prescriptions. It allows clustering comparison without narrating each latent halo. The simplification remains model-dependent: sample selection, cosmology and parameterization may be entangled.

Abstract Reasoning¶

Define the galaxy population and selection.
Specify halo mass and occupancy count.
Assign or fit normalized P(N|M), including scatter.
Add central/satellite and spatial/velocity details when the prediction needs them.
Compare predicted and observed clustering while marking model dependence.

Knowledge Transfer¶

Conditional count modeling can transfer when the conditioning variable, count and law remain explicit. But without dark halos and a selected galaxy population it is not HOD. The SDSS fit supports one inferred model, not a universal count rule. The live Probability Distribution entry imposes parametric-family machinery beyond every allowed HOD, so strict ancestry is withheld.

Examples¶

Canonical¶

For an illustrative sample, low-mass halos have probabilities 0.7 and 0.3 of hosting zero and one selected galaxy. Higher-mass halos have probabilities 0.1, 0.5 and 0.4 of hosting zero, one and two. Each row is a normalized P(N|M), not a measured survey result; spatial and velocity prescriptions would add information for clustering predictions.

Mapped back: Halo mass M → low- and high-mass halo bins; Defined galaxy population → one declared toy galaxy sample; Galaxy occupancy count N → zero, one or two selected galaxies; Conditional count law P(N|M) → normalized mass-conditioned rows; Within-halo and inference convention → illustrative law without a direct survey claim.

Applied / In Practice¶

Abazajian and collaborators analyzed projected clustering in a volume-limited Sloan Digital Sky Survey sample and fitted an HOD with halo-mass and satellite-occupation parameters. The observed object is galaxy clustering; the occupation of model dark-matter halos is inferred, conditional on cosmology and parameterization, not counted halo by halo.

Mapped back: Halo mass M → inferred mass of model halos; Defined galaxy population → volume-limited SDSS galaxy sample; Galaxy occupancy count N → modeled galaxy numbers including satellites; Conditional count law P(N|M) → fitted occupation relation; Within-halo and inference convention → projected correlation data under declared cosmology.

Structural Tensions¶

T1 — Visible Galaxies versus Latent Halos. Galaxy correlations are observed while halo occupancy is inferred under a model.

Diagnostic: Which parts were directly measured?

T2 — Mean Count versus Full Distribution. An average may hide zero-, one- and many-galaxy probabilities.

Diagnostic: What scatter was assumed?

Structural–Framed Character¶

HOD is formally structural in P(N|M) but astronomically bounded in its variables. Evaluative weight: fit quality is evaluated, not intrinsic to the law. Human-practice-bound: sample choice and parameterization are analyst decisions. Institutional origin: a survey supplies data, not the definition. Vocabulary travels: conditional distributions recur widely; galaxy and halo roles do not. Import versus recognize: a galaxy-clustering fit is literal; organizations metaphorically occupying groups is analogy.

A general conditional-count model is an explicitly future-prime candidate; the current Probability Distribution node includes extra named-family commitments. Its character: a formal astrophysical population model conditional on halo mass.

Structural Core vs. Domain Accent¶

The portable relation is a conditional law of counts; astronomy gives the variables.

What is skeletal. Count N is distributed conditional on M, and aggregate observations can constrain that law. This conditional-count structure is a future-prime candidate, not an accepted parent.

What is domain-bound. M is dark-matter halo mass, N counts a selected galaxy population, and central/satellite plus position/velocity rules connect occupation to clustering.

Why this does not clear the prime bar. Removing halos and galaxies leaves a generic conditional law. Treating a fitted HOD as a direct census also erases its astronomical inference boundary.

This entry is a kind of Formal Model.

Related — probability distribution. HOD uses a conditional count law without necessarily a named family.
Related — galaxy clustering. Correlations constrain the law rather than equal it.
Related — halo model. It supplies the dark-matter framework.

Relationships to Other Abstractions¶

Current abstraction Halo Occupation Distribution Domain-specific

Parents (1) — more general patterns this builds on

Halo Occupation Distribution is a kind of Formal Model Domain-specific

It is a parameterized formal model connecting galaxies and dark-matter halos.

Hierarchy path (1) — routes to 1 parentless root

Halo Occupation Distribution → Formal Model → Representation → Abstraction

Neighborhood in Abstraction Space¶

Halo Occupation Distribution sits in a sparse region of the domain-specific corpus (64^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Domain-Specific Indicators & Measurement Methods (26 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Mean occupation. Tell: Is count scatter also specified?
Galaxy catalogue. Tell: Were halo masses directly observed?
Luminosity function. Tell: Is galaxy count conditioned on host halo mass?
Universal HOD. Tell: Which selected sample and cosmology define it?

References¶

Berlind and Weinberg, The Halo Occupation Distribution, Astrophysical Journal 575 (2002): https://arxiv.org/abs/astro-ph/0109001
Zheng et al., Theoretical Models of the Halo Occupation Distribution, Astrophysical Journal 633 (2005): https://arxiv.org/abs/astro-ph/0408564
Abazajian et al., Cosmology and the Halo Occupation Distribution from Small-Scale Galaxy Clustering in the Sloan Digital Sky Survey: https://arxiv.org/abs/astro-ph/0408003
Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Halo_occupation_distribution (revision 1203597022).