Quasar Halo Occupation Distribution

• Quasar Halo Occupation Distribution (HOD) is a statistical framework that characterizes how quasars populate dark matter haloes using conditional probabilities and mean occupation functions.
• It employs methods such as two-point clustering, direct counts, and simulation-based abundance matching to infer host halo masses, duty cycles, and the split between central and satellite quasars.
• Recent studies converge on a narrow central halo-mass scale of a few 10^12 h⁻¹ M☉ for luminous quasars while uncertainties in the one-halo satellite term remain.

Searching arXiv for recent and foundational papers on quasar halo occupation distribution. Quasar Halo Occupation Distribution (HOD) is the halo-model description of how quasars populate dark matter haloes, usually formulated through the conditional probability $P(N\mid M)$ that a halo of mass $M$ hosts $N$ quasars and its first moment, the mean occupation function $\langle N(M)\rangle$. In quasar applications, the HOD is used to infer the characteristic host-halo mass of luminous accretion, the split between central and satellite quasars, the duty cycle encoded by central occupation, and the dependence of these quantities on redshift, luminosity, and obscuration. The subject has developed through a combination of projected two-point clustering, direct counts of quasars in groups and clusters, simulation-based abundance matching, and lensing-based halo-mass inference, with broad convergence on a characteristic central-host scale of order a few $10^{12}\,h^{-1}M_\odot$ for many luminous quasar samples, but with continuing uncertainty in the one-halo term and the satellite population (Richardson et al., 2012, Rodríguez-Torres et al., 2016, Eltvedt et al., 2024).

1. Formal definitions and common parameterizations

In the standard HOD framework, the basic quantity is

$P(N\mid M),$

the probability that a halo of mass $M$ contains $N$ quasars. The corresponding mean occupation function is

$\langle N(M)\rangle = \sum_N N\,P(N\mid M),$

and is commonly decomposed into central and satellite components,

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$

In quasar work, $M$0 is often interpreted as a halo-mass-dependent duty cycle, because the central term represents the fraction of haloes of mass $M$1 whose central black hole is in an active luminous phase (Chakraborty et al., 2018, Richardson et al., 2013).

A widely used quasar parameterization is the Chatterjee et al. form, adopted in optical-quasar, X-ray AGN, infrared-quasar, and obscured-quasar studies: $M$2

$M$3

Here $M$4 and $M$5 control the softened central threshold, while $M$6, $M$7, and $M$8 control the satellite normalization, slope, and low-mass cutoff. Alternative forms are also used. A deliberately compact model for $M$9 quasars assumes a lognormal total occupation centered on a characteristic halo mass $N$0, with a mass-independent satellite fraction $N$1, while recent ATLAS work uses the Zheng et al. five-parameter central-plus-satellite form to derive an effective halo mass function from the angular autocorrelation (Eftekharzadeh et al., 2018, Eltvedt et al., 2024).

The distinction between the full HOD and the directly measured mean occupation function is important. Some direct-count studies do not fit separate central and satellite terms; instead they measure the total $N$2 empirically in massive haloes and then compare it to clustering-based HOD models. This methodological split is central to the literature because clustering alone leaves strong degeneracies in the high-mass tail and in the satellite contribution (Chakraborty et al., 2018, Chatterjee et al., 2013).

2. Observational and modeling routes to the quasar HOD

The traditional route is projected two-point clustering. Richardson et al. combined SDSS DR7 quasars with the Hennawi et al. binary-quasar sample to measure $N$3 from $N$4 to $N$5, then fit a central-plus-satellite HOD constrained by both clustering and number density. Related analyses applied the same formalism to X-ray AGN, WISE-selected quasars, and obscured quasar populations, usually with MCMC over $N$6 after fixing or effectively removing $N$7 (Richardson et al., 2012, Richardson et al., 2013, Mitra, 2016, Mitra et al., 2018).

A second route uses very small-scale clustering to isolate the one-halo term. The $N$8 analysis combining KDE-complete quasar pairs with eBOSS large-scale clustering treated $N$9 and $\langle N(M)\rangle$0 as the two principal HOD parameters and showed explicitly that the satellite fraction is driven by the $\langle N(M)\rangle$1 regime. This approach is especially valuable because large scales primarily constrain bias and characteristic mass, while the one-halo term constrains central-satellite and satellite-satellite pairs (Eftekharzadeh et al., 2018).

A third route is simulation-based rather than analytic. The first-year eBOSS analysis used a modified SHAM on BigMultiDark-Planck, parameterizing the host-halo $\langle N(M)\rangle$2 distribution of quasars as a Gaussian and then stochastically assigning quasars to halos and subhalos. In that framework the occupancy can be read off empirically as

$\langle N(M)\rangle$3

yielding a sharply peaked quasar HOD in halo mass even without adopting a conventional analytic central-plus-satellite formula (Rodríguez-Torres et al., 2016).

A fourth route is direct halo assignment. Matching quasars to MaxBCG or Planck SZ clusters measures the mean occupation function without first inferring it from the 2PCF. These studies count quasars in cluster-centered cylinders in projected radius and redshift, correct statistically for interlopers, and obtain $\langle N(M)\rangle$4 by dividing the total quasar count in each halo-mass bin by the number of clusters in that bin. The direct MOF is then used to break clustering degeneracies, especially those affecting the high-mass tail (Chatterjee et al., 2013, Chakraborty et al., 2018).

Most recently, halo-model fits to quasar angular clustering have been cross-checked with CMB-lensing or galaxy-cross-correlation information. ATLAS combined the quasar angular autocorrelation with quasar-CMB lensing, while HSC-SSP III used LRG–AGN cross-correlations to infer separate HODs for Type I and Type II quasars at $\langle N(M)\rangle$5 (Eltvedt et al., 2024, Rosado et al., 13 Oct 2025).

3. Characteristic halo masses, duty cycles, and the baseline quasar population

For optically selected SDSS quasars at median redshift $\langle N(M)\rangle$6, the canonical HOD result is a median host-halo mass for central quasars of

$\langle N(M)\rangle$7

with satellite quasars confined to much rarer, cluster-scale haloes,

$\langle N(M)\rangle$8

The same analysis inferred an average duty cycle around the median host mass of

$\langle N(M)\rangle$9

at $10^{12}\,h^{-1}M_\odot$0, and tentative evidence that the median central-host mass rises to

$10^{12}\,h^{-1}M_\odot$1

with $10^{12}\,h^{-1}M_\odot$2 at $10^{12}\,h^{-1}M_\odot$3, implying a steeper and narrower central cutoff at high redshift (Richardson et al., 2012).

Later clustering analyses preserved the same characteristic mass scale while differing in the one-halo sector. The eBOSS Y1 modified-SHAM analysis found quasars at $10^{12}\,h^{-1}M_\odot$4 in haloes of $10^{12}\,h^{-1}M_\odot$5, with bias evolving from $10^{12}\,h^{-1}M_\odot$6 at $10^{12}\,h^{-1}M_\odot$7 to $10^{12}\,h^{-1}M_\odot$8 at $10^{12}\,h^{-1}M_\odot$9, and with data insufficient to distinguish between models with different satellite fractions. By contrast, the small-scale KDE+eBOSS analysis preferred

$P(N\mid M),$0

showing that the inferred satellite fraction depends sensitively on whether very small-scale pair counts are included and on the luminosity range of the sample (Rodríguez-Torres et al., 2016, Eftekharzadeh et al., 2018).

More recent ATLAS clustering-plus-lensing work again favored a narrow host-halo distribution. A five-parameter HOD fit at effective $P(N\mid M),$1 yielded $P(N\mid M),$2, or $P(N\mid M),$3, and found that $P(N\mid M),$4 of ATLAS quasars lie in

$P(N\mid M),$5

in $P(N\mid M),$6. The same study emphasized that this narrow halo-mass distribution coexists with a wide quasar luminosity range, suggesting that luminosity variation is driven mainly by accretion rate rather than halo mass (Eltvedt et al., 2024).

Sample	Redshift	Representative HOD inference
SDSS optical quasars	$P(N\mid M),$7	$P(N\mid M),$8, $P(N\mid M),$9
eBOSS Y1 quasars	$M$0	$M$1, $M$2 to $M$3 across redshift bins
KDE+eBOSS quasars	$M$4	$M$5, $M$6
ATLAS quasars	$M$7	$M$8; $M$9 in $N$0

These results support a common baseline picture: luminous quasars at $N$1–2 usually occupy a relatively narrow halo-mass window centered on a few $N$2, but the one-halo term is not yet uniquely fixed.

4. Satellite occupation, one-halo clustering, and direct mean-occupation measurements

The principal ambiguity in quasar HOD work is the satellite component. In the original SDSS DR7 HOD fit, the small-scale clustering required only a tiny satellite fraction,

$N$3

so bright quasars were overwhelmingly central objects. In that model the satellite population lived almost entirely in $N$4 haloes and served mainly to generate the observed one-halo excess. Yet later small-scale analyses of somewhat different samples found $N$5, and WISE and obscured-quasar analyses found intermediate values, indicating strong model and selection dependence in the one-halo term (Richardson et al., 2012, Eftekharzadeh et al., 2018, Mitra, 2016, Mitra et al., 2018).

Direct MOF measurements were developed precisely to resolve this degeneracy. The MaxBCG-based measurement at $N$6 found that the mean occupation in massive haloes rises monotonically with halo mass, with a power-law slope

$N$7

The number distribution of quasars per halo was close to Poisson, and the radial distribution within haloes was adequately described by a power law with 3D slope

$N$8

The same study found no evidence that quasar luminosity depends on host halo mass and only a slight indication of downsizing in the conditional black-hole mass function, so cluster quasars at low redshift appeared to have characteristic luminosity and black-hole mass scales rather than a strong halo-mass trend (Chatterjee et al., 2013).

The high-redshift extension using Planck SZ clusters and SDSS DR12 quasars pushed this direct approach to $N$9. From $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$0 quasars matched to $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$1 PSZ1 clusters, the measured MOF was fit by

$\langle N(M)\rangle = \sum_N N\,P(N\mid M),$2

with $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$3 in $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$4 after converting $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$5 to $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$6 via $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$7. This steep slope implies a strong monotonic increase of quasar occupation across the cluster-scale regime and no evidence for a downturn or saturation at the high-mass end. The paper noted that the abstract quoted an intercept of $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$8, whereas the main text and fitted equation gave $\langle N(M)\rangle = \sum_N N\,P(N\mid M),$9; the latter was identified as the final best-fit result (Chakraborty et al., 2018).

The direct MOF work is therefore central to the field’s internal controversy. It supports HODs with a rising high-mass tail and disfavors models in which quasar occupation declines in $$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$0–$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$1 haloes. At the same time, because these direct measurements are confined to massive groups and clusters and involve only small quasar counts, they do not by themselves determine the lower-mass central cutoff that dominates the global quasar population.

5. Dependence on selection, obscuration, and AGN class

Selection effects are not secondary in quasar HOD studies; they are one of the main observables. For WISE-selected unobscured quasars at median $$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$2, four-parameter HOD modeling gave a central host scale of

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$3

and a satellite host scale of

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$4

with a projected satellite fraction

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$5

The central mass is consistent with optically selected SDSS quasars, but the satellite halo mass is lower and the nominal satellite fraction is about an order of magnitude higher, suggesting that infrared selection probes a somewhat different phase or environment than purely optical selection (Mitra, 2016).

A closely related obscured-versus-unobscured analysis of WISE quasars used the same Chatterjee-type HOD and found a sequence in median central host mass: $$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$6 for optically selected quasars,

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$7

for IR-bright unobscured quasars, and

$$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$8

for obscured quasars. The projected satellite fractions increased in the same order, and the study concluded that these trends tend to disfavor a strict “orientation only” unification picture, although the uncertainties in the WISE one-halo regime remained large because the angular 2PCF could not be measured below $$\langle N(M)\rangle = \langle N_{\rm cen}(M)\rangle + \langle N_{\rm sat}(M)\rangle.$$9 kpc (Mitra et al., 2018).

The strongest current environmental split comes from HSC-SSP III. Using $M$00 LRGs cross-correlated with $M$01 HSC+WISE AGN at $M$02, the derived HODs implied that Type I AGN reside, on average, in substantially more massive haloes than Type II AGN: $M$03 a difference reported at $M$04. The same analysis found that Type I AGN are significantly less likely to be satellites,

$M$05

whereas Type II AGN have

$M$06

and the combined reddened+obscured sample has

$M$07

The shallower one-halo slope for Type I and more galaxy-like one-halo term for Type II were presented as a significant challenge to a strict unified model in which Type I and Type II quasars differ only by torus orientation (Rosado et al., 13 Oct 2025).

6. Physical interpretation and relation to other AGN populations

Within the HOD framework, the central quasar occupation is an instantaneous duty cycle. This permits direct comparison between optical quasars and other AGN classes. Modeling of X-ray-bright AGN at $M$08 gave a median central host mass

$M$09

an upper limit $M$10, an average central duty cycle

$M$11

and a characteristic X-ray AGN lifetime of

$M$12

Compared with the SDSS-quasar duty cycle

$M$13

and

$M$14

this implies that the X-ray AGN phase lasts about two orders of magnitude longer than the optically bright quasar phase (Richardson et al., 2013).

This comparison has been used to advance an evolutionary picture in which luminous quasars preferentially occupy a halo-mass “sweet spot” around a few $M$15, while X-ray AGN inhabit somewhat more massive group haloes and radio AGN even more massive systems. A plausible implication is that the quasar phase corresponds to a short-duty-cycle, high-Eddington episode in central galaxies, whereas the X-ray phase corresponds to longer-duty-cycle, more moderate accretion in more massive environments. The direct MOF evidence for a rising high-mass tail further suggests that once one reaches rich groups and clusters, satellite activation can contribute materially to the total occupation even though the absolute occupation per halo remains well below unity (Richardson et al., 2013, Chakraborty et al., 2018).

The quasar HOD has also become an input to other observables. In an HOD-based calculation of the Sunyaev–Zeldovich signal from quasar hosts, the mean occupation function from Richardson et al. was combined with the halo mass function and a halo $M$16–$M$17 relation to compute the quasar-weighted mean SZ signal. Under the simplifying assumption that the quasar HOD is redshift-independent and specified at all redshifts by the same parameters as those at $M$18, the calculation found that the average virialized-halo SZ signal decreases with redshift and that the stacked signal reported by Ruan et al. could be explained by halo gas alone within model uncertainties, without requiring an additional feedback-induced SZ component (Chowdhury, 2015).

Recent ATLAS lensing-plus-clustering work adds a related interpretation. Because $M$19 of quasars were inferred to lie in $M$20 while clustering showed little strong luminosity dependence, the study argued that quasar luminosity is largely independent of host halo mass over the probed range and is instead more closely tied to accretion rate. It also suggested that the approximately constant quasar space density with redshift may be linked to gravitational growth of the preferred halo population, while the luminosity evolution of quasars reflects depletion of available gas and stars rather than a strong shift in host-halo occupation amplitude (Eltvedt et al., 2024).

7. Degeneracies, systematics, and current outlook

The main limitation of quasar HOD inference remains degeneracy in the one-halo sector. Clustering analyses alone permit very different satellite prescriptions to fit similar 2PCFs, especially when redshift errors or fiber collisions suppress the observable signal below $M$21. The first-year eBOSS analysis explicitly found that it could not distinguish between models with different satellite fractions; the very small-scale KDE+eBOSS analysis showed that $M$22 changes materially when more precise $M$23 information is added; and WISE studies emphasized that the absence of reliable small-scale points leaves $M$24 and $M$25 weakly constrained (Rodríguez-Torres et al., 2016, Eftekharzadeh et al., 2018, Mitra et al., 2018).

Direct MOF measurements have a complementary but narrower limitation. They are currently confined to massive haloes and small quasar counts: $M$26 quasars in $M$27 PSZ1 clusters in the Planck-based study, and low-occupation statistics in MaxBCG clusters at $M$28. These data robustly constrain whether the high-mass tail rises or falls, but they do not independently determine the characteristic halo mass of the dominant central-quasar population around $M$29–$M$30. They are also sensitive to cluster mass definitions, halo-radius conventions, interloper subtraction, and redshift-window choices, although both MaxBCG and Planck analyses reported that plausible variations in these choices were smaller than the statistical errors (Chatterjee et al., 2013, Chakraborty et al., 2018).

A further source of heterogeneity is parameterization. Some studies use Chatterjee-type central-plus-satellite models, others use a narrow lognormal occupation with a global $M$31, others infer the HOD through modified SHAM, and recent lensing work adopts Zheng-type five-parameter forms. Agreement on the characteristic mass scale does not imply identical assumptions about the high-mass tail, satellite statistics, or the width of the host-halo distribution. This is one reason that apparently discrepant satellite fractions can coexist with broadly similar two-halo masses (Richardson et al., 2012, Eftekharzadeh et al., 2018, Eltvedt et al., 2024).

The observational direction is nevertheless clear. Several studies argue that larger and more homogeneous cluster or galaxy samples, especially when combined with small-scale clustering, will permit explicit central-satellite decomposition, redshift-dependent HOD fits, and stronger control of selection functions. The Planck-cluster MOF study proposed future combinations of MaxBCG with high-redshift RedMapper, and identified larger forthcoming halo catalogs from eROSITA, Euclid, LSST, and CMB-S4 as the route to better statistics. The X-ray AGN forecasting analysis stated that

$M$32

is sufficient to constrain HOD parameters at the $M$33 level, while noting that small-scale clustering remains crucial for satellite constraints (Chakraborty et al., 2018, Richardson et al., 2013).

The current state of the subject can therefore be summarized as follows. The quasar HOD is now well established as a central-dominated occupation with a preferred halo-mass scale near a few $M$34 for many luminous quasar samples, a low overall duty cycle, and a high-mass tail that rises rather than declines in massive groups and clusters. What is not yet settled is the detailed one-halo physics: how often quasars are satellites, how that fraction varies with luminosity and obscuration, and whether Type I and Type II quasars represent primarily orientation classes or genuinely distinct halo environments. The newest cross-correlation HOD analyses point strongly toward the latter.