Halo occupation numbers and galaxy bias

J.A. Peacock; R.E. Smith (University of Edinburgh)

REVIEW

Halo occupation numbers and galaxy bias

J.A. Peacock, R.E. Smith (University of Edinburgh)

Astrophysics · Submitted 29 Apr 2000 · 4 references

Pith's verdict: A clean, honest heuristic that gets the galaxy bias recipe roughly right; the one number the amplitude claim leans on is a residual from an extrapolation the authors never quantify.

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arXiv IDastro-ph/0005010 (version 2)

Pith IDML27UIWQ, Pith's permanent record of this paper and its review

CitationJ.A. Peacock, R.E. Smith (University of Edinburgh). "Halo occupation numbers and galaxy bias." arXiv:astro-ph/0005010 (2000). https://pith.science/paper/ML27UIWQ

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Editor's note

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Worth your time, and worth citing. The lasting contribution isn't the density-clump picture — that's Neyman, Scott & Shane 1953 and they say so — it's the two-part recipe: fix the occupation number N(M) by matching the observed group luminosity function and the N^-2.7 richness distribution onto the halo mass function, then put one galaxy at each halo centre with the rest tracing the mass. That combination is what everyone now calls HOD plus central-satellites, and this is one of the papers that made it standard. The velocity-bias result falls out of the same down-weighting of massive haloes, which is a nice bonus and holds up against the later semianalytic work they compare to (Benson et al.). Credit where due: Fig. 2 is a real check, not a hand-wave — randomly-placed haloes plus linear power reproduces the Jenkins et al. nonlinear spectra across all scales, and that's an external benchmark the model didn't get to tune. The galaxy-side prediction is a forward test against APM, which was not used in the calibration. That's the strongest thing in the paper and it's genuinely strong. On the soft spots, the stress-test note is right and sharper than the reader's framing. The reader worried about the high-N end, but that end is actually anchored — Section 3.3 explicitly falls back on the AGS occupation numbers above N~20 and blends with N^0.92. The weak link is the low end. The isolated-galaxy fraction F~0.5 is a residual: total galaxy density minus group members, with the N=2 contribution supplied by extrapolating a power law fitted only at N>=3. Since dilution enters the amplitude roughly as 1/F^2, moving F to 0.4 or 0.6 shifts the prediction by order the residual offset they already admit to ('still lies slightly above the APM data'). They flag the monotonic scatter-free mass map as something that 'must be wrong in detail', which is honest, but they don't put an error bar on the low-N extrapolation, and that's the piece the 'close to APM' sentence actually rests on. So read the agreement as consistency, not as a parameter-free win. None of this breaks the central claim, which is framed as heuristic from the first sentence. Who should read it: anyone building mock catalogues, anyone thinking about scale-dependent bias, and anyone who wants the pedagogical version before diving into the semianalytics. Send it out.

Referee report

Significance

The paper is a compact, well-organized statement of what became the halo occupation distribution framework, and it earns credit on several counts. The mass-side calculation is checked against an external benchmark: Fig. 2 shows the random-halo plus linear spectrum reproduces the Jenkins et al. (1998) nonlinear spectra across all scales, which is a genuine validation rather than a self-comparison. The galaxy-side prediction is a forward test against APM clustering, which is not used in the calibration of N(M), so the exercise is not circular in the strict sense. The paper also makes a falsifiable prediction: the observed small-scale slope of 1.6-1.8 is argued to require cuspy (M99-like, gamma=1.5) rather than NFW (gamma=1) cores, which is a testable statement. The analytic argument that cluster-normalized haloes of fixed mass have smaller core radii at higher Omega (Section 4.2) is clean and independent of the empirical recipe. The authors are candid about the weak points, stating that the monotonic N(M) map 'must be wrong in detail' and that the high-mass occupation numbers 'will not be completely reliable'.

Major objections (5)

[Section 3.3, Fig. 5, Eq. (17)] The intermediate-scale amplitude rests on the isolated-galaxy fraction F, which Section 3.3 obtains as a residual: total galaxy number density minus group members, with the N=2 contribution supplied by extrapolating rho(N) proportional to N^-2.7 fitted only at N>=3. Because the surviving clustered fraction enters the correlation amplitude roughly as 1/F^2 (Eq. 17), a change in F from 0.5 to 0.4 or 0.6 shifts the predicted amplitude by of order 40-50%, which is comparable to or larger than the residual offset above APM acknowledged in Section 4.2. The paper reports 'almost exactly 0.5' for both CfA and ESP and treats the coincidence as support, but this is a derived residual, not a direct count. The authors should quantify the sensitivity of the Fig. 8 amplitude to the low-N extrapolation, for example by varying the N^-2.7 index or the N=2 bin within plausible bounds and reporting the resulting band on the predicted power spectrum.
[Section 3.3, final paragraph] The substitution of N by N^0.92 to reconcile the Fig. 6 occupation numbers with the AGS analysis at high mass is introduced as 'a convenient approximation' without derivation. Since this adjustment directly suppresses the contribution of the most massive haloes, and Section 4.2 states that a further factor-of-two reduction of N at 10^15 Msun would be needed to bring the Lambda-CDM prediction onto the APM data, the reader needs to know how much of the final amplitude and shape in Fig. 8 depends on this exponent. Please state the mass range over which the substitution is active and show the predicted spectrum with and without it.
[Section 4.2, Fig. 8] The abstract's claim that the predictions are 'close to' the observed galaxy correlations for Lambda-CDM should be qualified against the text, which states the prediction 'still lies slightly above the APM data' and then proposes that a factor-of-two reduction of N at 10^15 Msun would fix it. This is a post-hoc tuning rather than a prediction, and the size of the residual should be expressed in units of the APM error bars so the reader can judge whether 'close' is defensible. As written, the qualitative contrast between Lambda-CDM and tau-CDM is the robust result, while the absolute amplitude agreement is marginal.
[Section 4.2, Fig. 8] The tau-CDM conclusion needs to be separated into two parts. The analytic argument that cluster-normalized haloes of fixed mass have r_v proportional to Omega^-1/3, and hence enhanced small-scale clustering at high Omega, is clean and independent of the empirical calibration. But N(M) is itself derived per-cosmology from the same group data (Figs. 3 and 6), so part of the tau-CDM over-prediction is built into the recipe rather than being a property of the cosmology. The paper notes that Benson et al. could match the AGS luminosity function only for Lambda-CDM; the authors should state explicitly how much of the factor-of-two excess in tau-CDM survives if the occupation numbers are held fixed at the Lambda-CDM values.
[Section 2.2, Eq. (13)] The prescription of adding the linear power spectrum to the random-halo power is asserted rather than derived, and the paper itself calls it 'a simple guess'. It works remarkably well in Fig. 2, but the success should be presented as an empirical finding to be checked, not as a justified decomposition. A short justification, or at least a statement of the scales over which the additivity is expected to hold and where it breaks down, would strengthen the mass-side result on which the rest of the paper builds.

Minor objections (7)

[Appendix A, Eqs. (2) and (24)] Equations (2) and (24) state the same isothermal-sphere result from Peebles (1974); the duplication could be removed or cross-referenced.
[Section 3.3, Fig. 5] The two group catalogues have different limiting magnitudes (MB < -19.4 for CfA and MB < -18.5 for ESP), yet both yield an isolated fraction of approximately 0.5. Please comment on whether this agreement is expected or fortuitous, since it is used as supporting evidence.
[Section 4.1, Fig. 7] The figure caption states that density contrast is log-encoded from 0.5 to 10, but the visual comparison between panels (c) and (d) is qualitative; a brief quantitative statement of how the galaxy and sparse-mass catalogues differ in their two-point statistics would help the reader connect the panels to Fig. 8.
[Section 4.2] The claim that 'the amplitude of the correlations is mainly sensitive to Omega, rather than to the detailed shape of the underlying mass power spectrum' is drawn from two models only. It should be softened or supported with at least one additional spectral shape.
[Section 4.2, small-scale slope discussion] The inference that the observed slope of 1.6-1.8 argues for M99-like cuspy cores over NFW is a strong conclusion drawn from a heuristic placement rule. It is a legitimate falsifiable prediction, but the wording should acknowledge that the central-galaxy-plus-satellites ansatz drives the result and that scatter in galaxy positions within haloes could soften it.
[Section 4, general] All comparisons use a single 239.5 h^-1 Mpc box per cosmology with no estimate of cosmic variance or box-to-box scatter. A brief statement of the expected sampling error on the measured power spectra would help calibrate the significance of the Lambda-CDM versus tau-CDM difference.
[Section 2.3] The two cosmological parameter sets are stated without the Hubble parameter h; since several quantities (masses, radii, wavenumbers) are quoted in h-dependent units, stating the assumed h explicitly would avoid ambiguity.

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The paper

The paper claims

Galaxy clustering can be predicted from a dark-matter simulation alone, once you decide how many galaxies each halo holds and where inside it they sit.

The reading

The paper asks whether the observed clustering of galaxies can be produced from a dark-matter simulation without simulating the baryons at all. Its answer is a two-part recipe: decide how many galaxies sit in a halo of a given mass (the halo occupation number), and decide where inside that halo they sit. The first part is not guessed — it is read off the observed luminosity function of galaxy groups and the observed distribution of group richnesses, so a halo's galaxy content is pinned by data rather than by a model of star formation. The second part assumes one galaxy sits at the halo centre and the rest orbit as satellites tracing the halo mass. With those two ingredients, a flat $Omega=0.3$ cold dark matter model reproduces the observed galaxy correlations and a realistic pairwise velocity dispersion, while an $Omega=1$ model with the same spectral shape over-predicts small-scale power by about a factor of two.

What carries the argument

The halo occupation number, $N(M)$: the mean number of galaxies brighter than a sample limit inside a halo of mass $M$. It carries the argument because it converts an observed group catalogue into a mass-dependent weighting of haloes, and that weighting is what suppresses the contribution of massive haloes and produces both the clustering amplitude and the reduced velocity dispersion. It is paired with a placement rule — one central galaxy plus satellites tracing the halo mass profile — which sets the small-scale slope, and with an additive decomposition of the power spectrum into randomly placed haloes plus the linear spectrum, $Delta^2_{rm tot} = Delta^2_{rm random haloes} + Delta^2_{rm linear}$, which is what lets the mass clustering be reproduced before any galaxy assignment is made.

Extended reading notes

Core claim

Galaxy bias — the mismatch between where galaxies are and where the mass is — need not be a free function fitted to the data. The authors argue it follows from two quantities: the mean number of galaxies a halo of mass $M$ contains above a luminosity limit, $N(M)$, and the radial distribution of those galaxies inside the halo. They first show that the nonlinear mass clustering itself is almost fully accounted for by randomly placed haloes with realistic density profiles plus the linear spectrum, which fixes the mass side of the comparison. They then fix $N(M)$ empirically by matching cumulative group counts to cumulative halo counts, and place one galaxy at each halo centre with the remainder following the mass. Applied to the Virgo simulations, this yields galaxy power spectra close to a single power law over nearly three decades in wavenumber; the $Lambda$CDM model lands near the APM data, while the $tau$CDM model sits about a factor of two high at $k gtrsim 1,hmathrm{Mpc}^{-1}$. The same down-weighting of massive haloes lowers the predicted pairwise velocity dispersion below that of the dark matter, addressing the long-standing complaint that cluster-normalized models produce too few high-speed pairs.

Load-bearing premise

The whole construction rests on reading a mean occupation number off the observed group counts by assuming that group richness maps monotonically onto halo mass with no scatter — an assumption the authors themselves say must be wrong in detail but adopt because it is the simplest scheme available.

Editorial extensions

If this is right

If $N(M)$ is fixed by group data, the observed galaxy correlation amplitude becomes mainly a measure of $Omega$ rather than of the shape of the primordial spectrum, because cluster-normalized haloes of a given mass have smaller core radii at high $Omega$ and therefore cluster more strongly.
The small-scale galaxy correlation should follow the halo density profile rather than flattening as the mass correlation does, since pairs there are dominated by central–satellite pairs; a profile with $rho propto r^{-1.5}$ gives approximately the observed slope.
Haloes with occupation number one contribute no correlated pairs and merely dilute the signal; with roughly half of all galaxies isolated, this can produce antibias on intermediate scales even though censoring low-mass haloes alone would give positive bias.
The galaxy pairwise velocity dispersion should fall below that of the mass, mostly because massive haloes are down-weighted and partly because the central galaxy carries no velocity dispersion from its parent halo.
Given a group luminosity function, the same prescription can generate mock galaxy catalogues from any $N$-body halo catalogue, without a star-formation model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The recipe is a machine for turning any group catalogue into a prediction for galaxy clustering, so the natural test is whether the same $N(M)$ inferred at $M_B=-19$ continues to hold for fainter and for luminosity-selected samples; the paper notes only that empirically the amplitude varies little for $M_B < -19$ and says nothing about what happens below it.
Because the small-scale slope of the galaxy correlation inherits the inner slope of the dark-matter profile, this argument runs backwards as a probe of halo cores: a confirmed $r^{-1}$ core would leave the observed steep small-scale clustering unexplained within this picture, which is a sharper constraint than cluster lensing usually provides.
The authors state that the richness power law $rho(N) propto N^{-2.7}$ has no supporting evidence beyond $N approx 20$ and that they replace it there with an ad hoc $N^{0.92}$ adjustment to match the group luminosity function; the predicted clustering of the richest systems is therefore the least constrained part of the model, and the part most likely to move.
The predicted velocity bias is separable from the amplitude fit, since it shows up in redshift-space distortions at fixed real-space clustering; a measurement of the pairwise dispersion at a few Mpc for a sample with a well-measured group luminosity function would test the down-weighting of massive haloes independently of the power-spectrum normalization.

Original abstract

We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $\Omega=0.3$ model ($\Lambda$CDM), but not for an $\Omega=1$ model with the same power spectrum ($\tau$CDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $\Omega$, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the $\Lambda$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,698 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.54 · compares with prior results 0.61 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $\Omega=0.3$ model ($\Lambda$CDM), but not for an $\Omega=1$ model with the same power spectrum ($\tau$CDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $\Omega$, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the $\Lambda$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a heuristic model that displays the main features of realistic theories for galaxy bias.	contribution 0.96	0.89	0.10	asserted 0.92
s1	We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes.	result 0.92	0.79	0.14	asserted 1.00
s2	A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo.	method 0.92	0.30	0.07	asserted 0.98
s3	The first factor is constrained by the empirical luminosity function of groups.	method 0.95	0.16	0.03	asserted 1.00
s4	For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass.	method 0.99	0.45	0.03	asserted 0.80
s5	These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density.	interpretation 0.70	0.67	0.10	asserted 0.99
s6	We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $\Omega=0.3$ model ($\Lambda$CDM), but not for an $\Omega=1$ model with the same power spectrum ($\tau$CDM).	result 1.00	0.96	0.55	asserted 1.00
s7	This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $\Omega$, and hence display enhanced small-scale clustering.	interpretation 0.97	0.50	0.12	asserted 1.00
s8	Finally, the pairwise velocity dispersion of galaxies in the $\Lambda$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field.	result 0.99	0.83	0.09	asserted 0.99
s9	This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.	interpretation 0.98	0.80	0.07	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from simulation or computation 0.80.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The window function for the halo density profile given by Moore et al. (1999). The profile has ρ ∝ r−3/2 inside a core radius rc, with ρ ∝ r−3 at larger radii, truncated at the virial radius, rv. The shape of the window function is determined by the ratio rv/rc; the plotted lines correspond to rv/rc = 2, 4, 8, 16, 32 & 64. characterized by a shape parameter Γ, which is nearly but not quite Ωh when the primordial index is n = 1 (see e.g. Peacock & Dodds 1994 for a discussion of different possible definitions for Γ). The normalization is specified via the variance in fractional…

**Figure 2.** Density power spectrum computed for the τCDM (top) and ΛCDM (bottom) models. The linear spectrum is shown dashed; the solid line shows the nonlinear spectrum, calculated according to the approximation of Peacock & Dodds (1996). The spectrum according to randomly-placed haloes is denoted by open circles; if the linear power spectrum is added, the main features of the nonlinear spectrum are well reproduced (filled circles). For reference, the dot-dashed line shows the APM power spectrum (Maddox, Efstathiou & Sutherland 1996). power spectrum on all scales. This is a novel way of lookin

**Figure 3.** The empirical luminosity–mass relation required to reconcile the observed AGS luminosity function with two variants of CDM. L∗ is the characteristic luminosity in the AGS luminosity function (a Zwicky absolute magnitude of −21.42 for h = 1). Note the rather flat slope around M = 1013 to 1014h−1M⊙, especially for ΛCDM. can be converted to an estimate of the total stellar luminosity in a group. This allows a determination of the All Galaxy Systems (AGS) luminosity function: the distribution of virialized clumps of galaxies as a function of their total luminosity, from small systems…

**Figure 4.** The effective luminosity associated with galaxies of luminosity > Lmin (i.e. total luminosity density divided by the number density of galaxies with L > Lmin). A Schechter function with α = 1.28 is assumed, following Folkes et al. (1999); the dashed line shows the effect of α = 0. In this plot, L∗ is the characteristic luminosity in the galaxy luminosity function, as distinct from the AGS L∗

**Figure 5.** The number density of groups as a function of the number of galaxies they contain, for approximately volume-limited subsamples of the CfA and ESO Slice Project group catalogues. The dashed lines show the relation ρ(N) ∝ N−2.7 . The open points show the estimated number density of ‘isolated’ galaxies, i.e. those that reside in groups with N = 1. This was derived from the total galaxy number density, minus those in groups (with an extrapolated contribution for N = 2). The solid lines show the prediction of the AGS luminosity function, as discussed in the text. 3.3 Discreteness issues…

**Figure 6.** N is plotted as a continuous variable, but integer arithmetic should be applied in practice to obtain the occupation number corresponding to a halo of given mass. To be explicit, the occupation number for a halo of given mass is to be deduced from

**Figure 7.** Stages in generating a mock galaxy sample for a ΛCDM universe. The slices are 120 h−1 Mpc on a side, with a thickness of 1/5 of the side. Density contrast is log encoded from a value of 0.5 (white) to 10 (black). The panels show, clockwise from top left, (a) the initial mass distribution; (b) the mass distribution reconstructed from spherical M99 haloes, excluding the non-halo particles; (c) the galaxy catalogue; (d) a random sampling of the mass, with the same number density as the galaxies. the mass function. The voids will not be truly empty, but the halo mass function is shifted…

**Figure 8.** The power spectra for galaxy catalogues constructed from (a) the τCDM model; (b) the ΛCDM model, using the empirical recipe for the occupation number as a function of mass. The model predictions are shown as filled circles; open circles show only the effects of censoring: the power spectrum of the mass, excluding particles in haloes below the lower mass limit. The linear spectrum is shown dashed; the solid line shows the nonlinear spectrum, calculated according to the approximation of Peacock & Dodds (1996). The dot-dashed line shows the APM power spectrum (Maddox, Efstathiou &…

**Figure 9.** The line-of-sight pairwise velocity dispersion for the ΛCDM model, plotted against projected pair separation.. The top curve shows the results for all the mass (stars); the lower pair of curves shows the predicted galaxy results, with (filled circles) and without (open circles) assuming that one galaxy occupies the halo centre. Assuming one central galaxy lowers the dispersion significantly, but the main effect comes from the lower efficiency of galaxy formation in high-mass haloes. pairwise peculiar velocity dispersion (σ12, defined in equation 16 of Jenkins et al. 1998) is shown…

Figures reproduced from arXiv:astro-ph/0005010 by the authors.

References

Works cited (4)

[1]

Benson A.J., Cole S., Frenk C.S., Baugh C.M., Lacey C.G., 2000a, MNRAS, 311, 793 Benson A.J., Baugh C.M., Cole S., Frenk C.S., Lacey C.G., 2000b, MNRAS, in press, astro-ph/9910488 Bond J.R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440 Bullock J.S., Kolatt T.S., Sigad Y., Somerville R.S., Kravtsov A.V., Klypin A.A., Primack J.R., Dekel A., 1999,...

View on Pith Pith review arXiv 1991
[2]

Neyman, Scott & Shane 1953, ApJ, 117, 92 Ostriker J.P., Suto Y., 1990, ApJ, 348, 378 Peacock J.A., Dodds S.J., 1994, MNRAS, 267, 1020 Peacock J.A., Dodds S.J., 1996, MNRAS, 280, L19 Peacock J.A., 1997, MNRAS, 284, 885 Pearce F.R., et al., 1999, ApJ, 521, L99 Peebles P.J.E., 1974, A&A, 32, 197 Peebles P.J.E., 1980, The Large-Scale Structure of the

View on Pith 1953
[3]

Universe, Princeton Univ. Press, Princeton, NJ Press W.H., Schechter P., 1974, ApJ, 187, 425 Ramella M., Pisani A., Geller M.J., 1997, AJ, 113, 483 Ramella M. et al., 1999, A&A, 342, 1 Seljak U., 2000, astro-ph/0001493 Sheth R.K., Jain B., 1997, MNRAS, 285, 231 Sheth R.K., Tormen G., 1999, MNRAS, 308, 119 Somerville R.S., Primack J.R., 1999, MNRAS, 310, 1...

View on Pith Pith review arXiv 1974
[4]

Navarro, Frenk & White (1996; NFW): ρ/ρb = ∆c y(1 + y)2 ; ( r < r v); y ≡ r/rc. (32) The parameter ∆ c is related to the core radius and the virial radius via ∆c = 200c3/3ln(1 + c) − c/(1 + c) ; c ≡ rv/rc (33) (we change symbol from NFW’s δc to avoid confusion with the linear-theory density threshold for collapse, and also be- cause our deﬁnition of densi...

View on Pith 1996

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astro-ph.GA 2025-06 conditional novelty 6.0 of 10

Using mock gravitational wave catalogs built from the SDSS DR7 galaxy survey, the clustering bias of GW sources is most sensitive to host stellar mass, secondarily to star formation rate, and nearly insensitive to met...
The Gravitational Wave Bias Parameter from Angular Power Spectra: Bridging Between Galaxies and Binary Black Holes
astro-ph.GA 2024-11 conditional novelty 6.0 of 10

The gravitational wave bias of mock binary black holes increases with the pivot mass of the assumed host galaxy mass selection, and can exceed the galaxy bias by up to about 30 percent only for steep host probability slopes.
Ultra large-scale cosmology in next-generation experiments with single tracers
astro-ph.CO 2015-05 accept novelty 6.0 of 10

Full-sky Fisher forecasts across four future survey types show that single-tracer power spectra cannot detect general-relativistic clustering corrections, but can reach sigma(f_NL) of order 1-2 and clearly detect lens...
Super-Sample Covariance in Simulations
astro-ph.CO 2014-01 accept novelty 6.0 of 10

Separate-universe simulations calibrate the power spectrum response to a background density mode, reproducing super-sample covariance in survey sub-volumes at the few percent level and revealing a dilation effect that...
The connection between massive galaxies and their dark matter haloes from $z=0.25$ to $z \sim 4$ from the combined COSMOS, XMM-LSS and ECDF-S fields
astro-ph.GA 2026-10 conditional novelty 5.0 of 10

The stellar-to-halo mass ratio for massive galaxies peaks at z~1 and declines afterward, indicating that massive galaxy formation becomes less efficient below z~1, with halos growing but central galaxies remaining quiescent.
Weighing Galaxies Inside-Out: Small-Scale Lensing and the Stellar Mass Problem
astro-ph.GA 2026-10 conditional novelty 5.0 of 10

Euclid small-scale galaxy-galaxy lensing could constrain the stellar IMF mismatch parameter α to roughly 8.5 percent, competitive with strong lensing plus dynamics.
Searching for primordial features with radio surveys: synergy between the power spectrum and bispectrum
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

Joint power spectrum and bispectrum analysis from future HI intensity mapping surveys improves constraints on primordial feature amplitudes by 30-40% and achieves percent-level precision on oscillation frequencies whe...
Probing inflationary features with galaxy ultraviolet luminosity function observables
astro-ph.CO 2026-02 conditional novelty 5.0 of 10

Galaxy UV luminosity function data at z=6–9 give upper limits on bump-like inflationary features at k≈0.3–20 Mpc^-1, similar to but not stronger than optical-depth constraints.
Low-redshift constraints on structure growth from CMB lensing tomography
astro-ph.CO 2025-10 conditional novelty 5.0 of 10

Low-redshift galaxy clustering and CMB lensing tomography with hybrid effective field theory gives S8=0.79±0.06, consistent with Planck, while data alone prefer Ωm=0.245±0.024.
Evidence for suppression of structure growth in the concordance cosmological model
astro-ph.CO 2023-02 conditional novelty 5.0 of 10

Cosmological data prefer a growth index γ = 0.633 ± 0.025, excluding the general-relativistic flat-ΛCDM value γ = 0.55 at 3.7σ, indicating suppressed late-time structure growth.
Hyper Suprime-Cam Year 3 Results: Cosmology from Cosmic Shear Power Spectra
astro-ph.CO 2023-04 conditional novelty 4.0 of 10

Tomographic cosmic shear power spectra from 416 square degrees of HSC Year 3 imaging give S8 = 0.776 (+0.032/-0.033), about 2 sigma below Planck 2018 and consistent with DES-Y3 and KiDS-1000.
Reconstructing the history of structure formation using redshift distortions
astro-ph 2008-07 conditional novelty 4.0 of 10

Redshift-space distortions measure the combination f times sigma8 directly, a bias-independent growth statistic that the paper argues can separate dark energy models from modified gravity.
Modelling baryonic feedback for survey cosmology
astro-ph.CO 2019-05 conditional novelty 2.0 of 10

A community review of baryonic feedback modelling on the matter power spectrum, with recommendations and illustrative Fisher forecasts for the next decade of survey cosmology.
Parametrizations for tests of gravity
astro-ph.CO 2019-08 accept novelty 1.0 of 10

A review of parametrized frameworks (PPF, EFT, PPN, ppE) for testing gravity across cosmological and astrophysical scales, with an outlook toward unification.

How later papers used it

289 later papers cite it

Evidence: substantial. 278 of the citing papers share no author with it, and 51 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 331 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 of 51 papers with this label · 1 by the same authors

Used the Halo Occupation Distribution as described in another work to model how observed galaxies populate the underlying dark matter halo distribution.

Axion-Induced Patchy Screening of the Cosmic Microwave Background (2024) · 0.99 · paraphrased: the licence does not allow quoting
Used a halo model prescription based on the halo occupation distribution model that another work used to describe the 2MRS sample.

Detecting ultra-high energy cosmic ray anisotropies through cross-correlations (2020) · 0.99 · paraphrased: the licence does not allow quoting

By the same authors

Used the halo model.

An accurate halo model for fitting non-linear cosmological power spectra and baryonic feedback models (2015) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

94 arXiv papers cite this in 108 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 43 sentences without a confident label

Uses its method or tool 27 papers

Therefore, we make use of the halo model formalism [ref] to make theoretical predictions for comparison with the simulated data.

Fast Simulation of Cosmological Neutral Hydrogen based on the Halo Model · Analytical Methods · 0.95
In this section, following the method developed in [ref] and [ref] [ref] , we use the halo model to calculate the cross-correlation between the SZ effect and the distribution of galaxy clusters, and give the covariance matrix for its measurement.

Cross-correlating the Thermal Sunyaev-Zel'dovich Effect and the Distribution of Galaxy Clusters · Theoretical Calculation · 1.00
To resolve the modeling difficulties of clustering observables on small scales, we use the halo model approach [ref] .

Hyper Suprime-Cam Year 3 Results: Cosmology from Galaxy Clustering and Weak Lensing with HSC and SDSS using the Emulator Based Halo Model · Introduction · 0.97

Compares with it 2 papers

It also yields fits on the same level of accuracy as the numerical HALOFIT package [ref] , based on a more solid physical picture of structure formation.

Amending the halo model to satisfy cosmological conservation laws · Amending the Halo Model · 0.98
In this appendix we compare the PT approach to a simplified, analytical halo model [ref] with the goal of understanding the expected size of the stochastic terms (see also the discussion in ref.

An analysis of parameter compression and full-modeling techniques with Velocileptors for DESI 2024 and beyond · Connection to the halo model · 0.97

Cites it as background 32 papers

The halo model [ref] makes the assumption that all matter resides in virialized halos of mass $M_vir = 4/3 πR_vir^3 Δ_vir ρ̅_m$, and splits the two-point statistics of the matter field into correlations between halos (two-halo term) and correlations within a single halo (one-halo term): [equation] The ingredients of the halo model are: the halo mass function $dn(M)$ with $n(M)$ the number density of halos at fixed mass $M$, along with halo bias $b(M)$, and the spherically averaged halo profile $ρ_M (r)$(e.g. NFW [ref] ).

An Analytic Hybrid Halo + Perturbation Theory Model for Small-scale Correlators: Baryons, Halos, and Galaxies · The halo model · 0.91
The halo model is based on the assumption that the non-linear density field can be described as a superposition of haloes [ref] .

CHEFT: A Hybrid Effective Field Theory halo model · The halo-model decomposition · 0.91
The halo model was developed by a number of authors [ref] , and for a review see [ref] .

Nonlinear clustering in models with primordial non-Gaussianity: the halo model approach · The halo model approach · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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Benson A.J., Cole S., Frenk C.S., Baugh C.M., Lacey C.G., 2000a, MNRAS, 311, 793 Benson A.J., Baugh C.M., Cole S., Frenk C.S., Lacey C.G., 2000b, MNRAS, in press, astro-ph/9910488 Bond J.R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440 Bullock J.S., Kolatt T.S., Sigad Y., Somerville R.S., Kravtsov A.V., Klypin A.A., Primack J.R., Dekel A., 1999,...

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Universe, Princeton Univ. Press, Princeton, NJ Press W.H., Schechter P., 1974, ApJ, 187, 425 Ramella M., Pisani A., Geller M.J., 1997, AJ, 113, 483 Ramella M. et al., 1999, A&A, 342, 1 Seljak U., 2000, astro-ph/0001493 Sheth R.K., Jain B., 1997, MNRAS, 285, 231 Sheth R.K., Tormen G., 1999, MNRAS, 308, 119 Somerville R.S., Primack J.R., 1999, MNRAS, 310, 1...

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[4] [4]

Navarro, Frenk & White (1996; NFW): ρ/ρb = ∆c y(1 + y)2 ; ( r < r v); y ≡ r/rc. (32) The parameter ∆ c is related to the core radius and the virial radius via ∆c = 200c3/3ln(1 + c) − c/(1 + c) ; c ≡ rv/rc (33) (we change symbol from NFW’s δc to avoid confusion with the linear-theory density threshold for collapse, and also be- cause our deﬁnition of densi...

View on Pith 1996