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Halo Models of Large Scale Structure

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Asantha Cooray (Caltech), Ravi Sheth (University of Pittsburgh)

Astrophysics · Submitted 28 Jun 2002 · 300 references

Pith's verdict: The standard halo-model reference, genuinely useful and mostly honest, with one acknowledged internal inconsistency (the k→0 integral constraint) that it flags but never resolves.

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arXiv IDastro-ph/0206508 (version 1)
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CitationAsantha Cooray (Caltech), Ravi Sheth (University of Pittsburgh). "Halo Models of Large Scale Structure." arXiv:astro-ph/0206508 (2002). https://pith.science/paper/OTKFA4RC
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The paper

The paper claims

The halo model claims that all nonlinear clustering statistics follow from three halo ingredients: how many halos of each mass exist, how they cluster, and how mass is arranged inside them.

The reading

This review argues that the messy outcome of nonlinear gravity can be organized by one bookkeeping choice: put every particle into a virialized dark matter halo, then describe the halo population statistically. If the abundance of halos by mass, their large-scale clustering bias, and the mass profile inside each halo are known, the same three ingredients can generate the dark matter power spectrum, bispectrum, trispectrum, and their covariance. The paper's evidence is a sequence of comparisons with numerical simulations, followed by applications that swap dark matter particles for galaxies, virial motions, gas pressure, or projected mass. A sympathetic reader should care because this would turn a computationally expensive nonlinear problem into a fast, physically interpretable calculation that connects galaxy surveys, weak lensing, and secondary CMB fluctuations within one language.

What carries the argument

The load-bearing device is the one-halo/two-halo decomposition. A halo of mass $m$ contributes a normalized Fourier profile $u(k|m)$; the number density of such halos is $n(m)$; and halos of different masses are correlated through a bias factor $b(m)$ times the linear density field. The one-halo term integrates $n(m)(m/��bar)^2|u(k|m)|^2$ and controls small scales, where pairs sit inside a single halo. The two-halo term multiplies the linear power spectrum by $[�int dm�,n(m)b(m)(m/��bar)u(k|m)]^2$ and controls large scales, where pairs sit in separate halos. Because convolutions of real-space profiles become products in Fourier space, the same integrals generate the bispectrum, trispectrum, and covariance with extra powers of $u$ and bias. This is what lets the review move from dark matter to galaxies, velocities, lensing, and CMB secondary effects without changing the underlying bookkeeping.

Extended reading notes

Core claim

The central claim is that the statistics of nonlinear gravitational clustering are dominated by two regimes that can be summed: pairs of particles inside the same halo and pairs in different halos. In Fourier space the dark matter power spectrum is written as $P(k)=P^{1h}(k)+P^{2h}(k)$, with $P^{1h}(k)$ integrating the halo mass function times the square of the normalized halo profile, and $P^{2h}(k)$ built from the linear power spectrum times halo bias factors and profile transforms. The review shows that this decomposition, with a simulation-calibrated mass function, a two-parameter cuspy profile, and a calibrated mass-concentration relation, tracks simulation measurements of the power spectrum, equilateral bispectrum, square trispectrum, and power-spectrum covariance. It then argues that the same weighting can be replaced by galaxy occupation, virial velocity, gas pressure, or lensing projection, so that galaxies, velocity and momentum fields, weak lensing, and thermal and kinetic Compton scattering of CMB photons become applications of one statistical template.

Load-bearing premise

The load-bearing premise is that every halo can be treated as a smooth, spherical object whose internal density profile depends only on its mass, and that halo pairs cluster according to linear theory times a deterministic mass-dependent bias, even where the single-halo term dominates.

Editorial extensions

If this is right

  • Galaxy clustering can be inverted for the mean number of galaxies per halo mass and for the low-mass cutoff below which halos host no galaxies.
  • Weak-lensing power spectra, bispectra, and non-Gaussian covariance matrices can be computed analytically from the same halo ingredients, making survey error forecasts faster.
  • Thermal and kinetic Compton scattering of CMB photons by hot gas, and the nonlinear integrated Sachs-Wolfe effect, can be predicted from halo pressure and velocity profiles rather than simulated case by case.
  • Redshift-space distortions and velocity and momentum statistics can be modeled by adding virial motions inside halos to the linear motions of halo centers.
  • The halo model can serve as a fast interpolation between perturbation theory on large scales and numerical simulations on small scales for survey forecasting.

Reading between the lines

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

  • A hybrid scheme that uses compensated profiles only for the one-halo term while keeping the two-halo term from perturbation theory could cure the excess large-scale power without losing the small-scale success; the review discusses both pieces but does not combine them this way.
  • Residuals between halo-model lensing and thermal-Compton predictions and real data would isolate gas physics such as cooling, feedback, or non-thermal pressure from the gravitational clustering template.
  • A useful cross-check would be to require the same mass-concentration relation to fit galaxy-galaxy lensing, CMB lensing, and cluster pressure profiles; disagreement would reveal which probe is contaminated by baryonic physics.
  • Because the model's large-scale limit reduces to linear theory with bias, it offers a clean way to separate scale-dependent galaxy bias from nonlinear growth when comparing surveys at different redshifts.
Original abstract

We review the formalism and applications of the halo-based description of nonlinear gravitational clustering. In this approach, all mass is associated with virialized dark matter halos; models of the number and spatial distribution of the halos, and the distribution of dark matter within each halo, are used to provide estimates of how the statistical properties of large scale density and velocity fields evolve as a result of nonlinear gravitational clustering. We first describe the model, and demonstrate its accuracy by comparing its predictions with exact results from numerical simulations of nonlinear gravitational clustering. We then present several astrophysical applications of the halo model: these include models of the spatial distribution of galaxies, the nonlinear velocity, momentum and pressure fields, descriptions of weak gravitational lensing, and estimates of secondary contributions to temperature fluctuations in the cosmic microwave background.

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, 3,682 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We review the formalism and applications of the halo-based description of nonlinear gravitational clustering. In this approach, all mass is associated with virialized dark matter halos; models of the number and spatial distribution of the halos, and the distribution of dark matter within each halo, are used to provide estimates of how the statistical properties of large scale density and velocity fields evolve as a result of nonlinear gravitational clustering. We first describe the model, and demonstrate its accuracy by comparing its predictions with exact results from numerical simulations of nonlinear gravitational clustering. We then present several astrophysical applications of the halo model: these include models of the spatial distribution of galaxies, the nonlinear velocity, momentum and pressure fields, descriptions of weak gravitational lensing, and estimates of secondary contributions to temperature fluctuations in the cosmic microwave background.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We review the formalism and applications of the halo-based description of nonlinear gravitational clustering. contribution 0.86 0.53 0.10 asserted 0.56
s1 In this approach, all mass is associated with virialized dark matter halos; models of the number and spatial distribution of the halos, and the distribution of dark matter within each halo, are used to provide estimates of how the statistical properties of large scale density and velocity fields evolve as a result of nonlinear gravitational clustering. method 0.88 0.13 0.19 asserted 0.97
s2 We first describe the model, and demonstrate its accuracy by comparing its predictions with exact results from numerical simulations of nonlinear gravitational clustering. method 0.74 0.91 0.11 asserted 1.00
s3 We then present several astrophysical applications of the halo model: these include models of the spatial distribution of galaxies, the nonlinear velocity, momentum and pressure fields, descriptions of weak gravitational lensing, and estimates of secondary contributions to temperature fluctuations in the cosmic microwave background. contribution 0.98 0.74 0.13 asserted 0.98

What kind of paper the classifier reports this is: review 1.00 · evidence from simulation or computation 0.85.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
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 complex distribution of dark matter (a) found in n
Figure 2. Distribution of galaxies (in color) superposed on th
Figure 3. The halo mass function in numerical simulations of th
Figure 4. Large scale bias relation between halos and mass (fro
Figure 5. Higher order moments of the halo distribution if the i
Figure 6. Distribution of dark matter around halo centers (fro
Figure 7. Mean concentration at fixed mass, c∆ = rvir/rs, for dark matter halos as a function of halo mass (from results presented in [78]). Different panels show the same cosmological models and power spectra as
Figure 8. Distribution of concentrations at fixed mass for dark
Figure 9. Fourier transforms of normalized NFW profiles
Figure 10. Density profiles and correlation functions associa
Figure 11. Power spectrum of the dark matter density field at the
Figure 12. The (a) equilateral bispectrum and (b) square trisp
Figure 13. (a) Qeq(k) and (b) Qsq at z = 0. Different lines styles show the different contributions to the total (bold dashed) in the halo model description. Thin solid lines show the second order perturbation theory (PT) and HEPT values. In (a), the thick solid line shows the fitting formula for Qeq from [239]. Notice that on linear scales, the halo model prediction is about twenty percent larger than the PT value in (a), and about a factor of two larger than the PT value in (b). converge to the HEPT value at small scales and the PT value at large scales. Notice that the HEPT prediction is…
Figure 14. Qeq(k) as a function of scale, (a) measured in numerical simulations and (b) compared to halo model predictions. In (a), the triangles are measurements in a box of size 100h −1 Mpc while squares denote measurements in box sizes of 300h −1 Mpc. The linear perturbation theory (PT) and hyperextended perturbation theory (HEPT) values are show as solid lines. In (b), the halo model predictions associated with Press-Schechter and Sheth-Tormen mass functions generally bracket the measurements. The dashed lines show the result of only including contributions from halos less massive than…
Figure 15. The correlations in dark matter power spectrum betw
Figure 16. Real-space moments with p = 3 (skewness), 4 (kurtosis) and 5 as a func￾tion of smoothing scale. Squares and triangles show measurements in high and low resolution simulations, illustrating how difficult it is to make the measurement. Solid lines show the predictions based on the NFW profile but with Press-Schechter (lower) and Sheth-Tormen (upper) mass functions. Dashed line shows the HEPT prediction. The figure is from [236]. with those measured in the simulations. Agreement in the off–diagonal ele￾ments is typically better than ±0.1, even in the region where non-Gaussian effects…
Figure 17. Ratio of the single halo term contribution to the tot
Figure 18. Dark matter power spectrum (top) and reduced bispec
Figure 19. The angular two-point correlation function of gala
Figure 20. The PSCz galaxy power spectrum (symbols, from [105]) compared to the dark matter power spectrum in a ΛCDM model (solid curve). We have fixed the amplitude of the dark matter power spectrum so that it matches the data on large scales. The discrepancy on smaller nonlinear scales suggests that the bias between the galaxies and the dark matter must be scale dependent. the two-point correlation function of the dark matter, and its Fourier trans￾form, the dark matter power spectrum, are not power laws (see, e.g., the solid curve in
Figure 21. Projected two-point correlation function of galax
Figure 22. The total luminosity in galaxies brighter than
Figure 23. The average number of galaxies as a function of dark m
Figure 24. The number of subclumps in a halo as a function of pare
Figure 25. The mean, hNi, and second factorial moment, hN(N −1)i of the distribution of the number of galaxies per halo as a function of the halo mass m. Symbols show measurements in the semianalytic model of [157] and we have selected objects which are predicted to have absolute magnitudes between −19 and −20 in the SDSS r ∗−band. Results for absolute magnitudes in the range −17 to −18, and −18 to −19 are qualitatively similar, although the peak for the lower luminosity bins shifts to lower masses. for ξn depends on the n−th moment of p(Ngal|m). For example, the bi- and trispectra require…
Figure 26. Correlation functions of different tracers of the da
Figure 27. (a) Contributions by halo mass to the one-halo term i
Figure 28. The PSCz galaxy power spectrum (symbols) and the result of tuning the first two moments of p(Ngal|m) so as to produce the power law like behavior (solid curve). mation models are likely to increase with ongoing wide-field surveys such as the Sloan Digital Sky Survey (SDSS) and the 2dFGRS. The halo approach to galaxy clustering has already become helpful for interpreting the SDSS two￾point galaxy correlation function [69] and the lensing-mass correlation [99]. 7 Velocities One of the great strengths of the halo-based approach is that it provides a clear prescription for identifying…
Figure 29. Dependence on halo mass of the non-linear (
Figure 30. The distribution of one-dimensional peculiar velo
Figure 31. The ratio of mean streaming velocity of dark matter p
Figure 32. Three dimensional power spectrum of the momentum de
Figure 33. Power spectrum of the momentum: k 2 ∆pp(k). Filled circles show the sum of the power spectrum of the divergence (open circles) and the curl (crosses) of the momentum fields in the simulations. Dashed curves show the linear theory prediction, and solid curves show our nonlinear theory prediction for the total power. The solid curves are obtained by summing the dot-dashed curves, which represent the contributions to the power from the single-halo and two-halo terms discussed in the text. Bottom panel shows the fraction of the total power contributed by the divergence (open circles)…
Figure 34. Cross-spectrum of the momentum and the velocity. Fi
Figure 35. The ratio of power in redshift space compared to real
Figure 36. Weak lensing convergence power spectrum under the h
Figure 37. Weak lensing convergence (a) bispectrum and (b) tri
Figure 38. The power spectra of lensing convergence (solid) an
Figure 39. Moments of the convergence field as a function of top-
Figure 40. The bispectrum configuration dependence R l3 l1l2 as a function of l1 and l2 with l3 = 1000. Due to triangular conditions associated with l’s, only the upper triangular region in l1-l2 space contribute to the bispectrum. 8.5 Bispectrum Using the spherical harmonic moments of convergence defined in equation (174), the angular bispectrum of the convergence is defined following [54,265] as hκl1m1 κl2m2 κl3m3 i =   l1 l2 l3 m1 m2 m3   B κ l1l2l3 . (193) Here, the quantity in parentheses is the Wigner 3j symbol. Its orthonormality relation implies B κ l1l2l3 = X m1m
Figure 41. The skewness, S3(σ), as a function of angular scale. The filled symbols indicate the mean and variance computed from a set of κ planes generated in parti￾cle-mesh (PM) simulations by [289]. Under the halo model, shown here is the skew￾ness with varying maximum mass used in the calculation (solid lines ranging from 1014 to 1016 M⊙). For comparison, we also show skewness values as predicted by hyper-extended perturbation theory (HEPT) and second-order perturbation theory (PT). Figure is reproduced based on [289] and [55]. scales of few arcseconds, where it is expected to be valid.…
Figure 42. The fractional errors in the measurements of the con
Figure 43. (a) The correlation coefficient, Cˆ ij as a function of the multipole li with lj as shown in the figure. We show the correlations calculated with the full halo model and also with only the single halo term for lj = 77072. In (b), we show the non-Gaussian correlation coefficient CˆNG ij , which only involves the trispectrum (see, equation 213). The transition to full correlation is due to the domination of the single halo contribution. To better understand this behavior it is useful to isolate the purely non￾Gaussian correlation coefficient CˆNG ij = Tij q TiiTij . (213) As show
Figure 44. The ratio of non-Gaussian to Gaussian contribution
Figure 45. figure 45. Here, for simplicity, we take the same description
Figure 46. The window functions involved with the projection o
Figure 47. The expected foreground galaxy-background quasar
Figure 48. Frequency dependence of the SZ effect at a multipole o
Figure 49. Recovery of the SZ signal with Planck multifrequenc
Figure 50. The dark matter (NFW) profile and the ones predicted b
Figure 51. The variation in the density weighted temperature o
Figure 52. The (a) pressure and (b) pressure-dark matter cross
Figure 53. The baryon density (left) and temperature weighted
Figure 54. The mass dependence on the dark matter power spectru
Figure 55. The angular power spectra of SZ thermal and kinetic e
Figure 56. The SZ power spectrum based on numerical simulation
Figure 57. Line of sight projected maps of the thermal (left) an
Figure 58. Distribution of peak heights or number counts of the
Figure 59. The temperature fluctuation power (∆T 2 l = l(l + 1)/(2π)ClT 2 CMB) for a variety of methods to calculate the kinetic SZ effect. Here, we show the contribution for a reionization redshift of ∼ 8 and an optical depth to reionization of 0.05. The contributions are calculated under the assumption that the baryon field traces the non-linear dark matter (Pg(k) = Pδ(k) with Pδ(k) predicted by the halo model), the linear density field (Pg(k) = P lin(k)), and the halo model for gas, with total and the 2-halo contributions shown separately. For the most part, the kinetic SZ effect can be d
Figure 60. Contribution to temperature fluctuations through h
Figure 61. Temperature fluctuations due to galaxy clusters: (a
Figure 62. The angular power spectrum of the full ISW effect, inc

Figures reproduced from arXiv:astro-ph/0206508 by the authors.

References
How later papers used it

651 later papers cite it

Evidence: substantial. 629 of the citing papers share no author with it, and 163 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 897 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

Uses its method or tool 19 of 163 papers with this label · 2 by the same authors

By the same authors

How the literature uses it

289 arXiv papers cite this in 440 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 · 206 sentences without a confident label

Contests it 2 papers
Uses its method or tool 100 papers
Uses its data or values 1 paper
Compares with it 1 paper
Cites it as support 2 papers
Cites it as background 91 papers

Discussion

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