An analytic model for the spatial clustering of dark matter haloes

H. J. Mo; S. D. M. White

REVIEW

An analytic model for the spatial clustering of dark matter haloes

H. J. Mo, S. D. M. White

Astrophysics · Submitted 19 Dec 1995

Pith's verdict: The Eulerian halo bias formula is the real contribution and it holds up; the 'entirely analytic autocorrelation' line in the abstract is an empirical regularity the authors themselves flag as underived.

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arXiv IDastro-ph/9512127 (version 1)

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CitationH. J. Mo, S. D. M. White. "An analytic model for the spatial clustering of dark matter haloes." arXiv:astro-ph/9512127 (1995). https://pith.science/paper/PDCFJZXS

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Editor's note

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This is the paper that gives you b = 1 + (nu^2 - 1)/delta_c, and that formula is the reason to care. The Lagrangian piece is properly credited to Efstathiou et al. (1988) and Cole & Kaiser (1989); what Mo and White add is the spherical-collapse contraction term that turns the Lagrangian peak-background split into a Eulerian bias. That step is clean, has no fitted constants, and is tested hard: Figs. 2, 3 and 4b compare the bias function and the halo-mass cross-correlation against three large P3M runs over a wide range of R and delta, including the two-epoch case where haloes are tagged early and correlated late. The agreement is good, and the two-epoch test is the one that matters for galaxy biasing. Credit where due: this is a real result, not a repackaging of PS, and it is why the paper is cited everywhere now. One correction to the reader's report: it says the authors state no limitations. They do, and plainly. Section 2.4 says outright that the extended PS formalism 'does not lead directly to a model for xi_hh', and Section 3.4 presents xi_hh = b^2 xi_m as a simulation finding with b taken from linear theory. So abstract:s9 is a scope overreach, not a hidden assumption. The authors know Eq. 24 fails in Fig. 5 and say why (halo exclusion). The residual the stress-test flags is fair but smaller than it sounds: every test is scale-free P(k) in Einstein-de Sitter, and the advertised application is CDM with a curved spectrum. Eq. 20 makes b a function of nu_1 alone, and that universality is checked across n = -1.5, 0, 1 but never at matched nu_1. Minor, but worth a sentence in any follow-up. The Fig. 4a circularity is real and the authors handle it honestly by showing the lognormal alternative and letting it fail in the nonlinear regime. Who should read it: anyone doing halo occupation, cluster clustering, or trying to extract sigma_8 and Omega from tracer bias. Send it to referees. The bias formula is load-bearing for a large literature and deserves a careful report, mainly on the domain of validity of Eq. 28.

Referee report

Significance

This is a substantial and useful contribution. The bias function of Eqs. (19)-(20) is derived, not fitted, and depends on halo mass and identification redshift only through nu_1 = delta_1/sigma_1; it is a parameter-free prediction given a power spectrum and cosmology. The tests against simulation are broad in R and delta and are carried out with two independent group finders, which is a genuine robustness check. The paper is also candid about what it cannot do: §2.4 states plainly that the formalism does not lead directly to xi_hh, and the exclusion problem is diagnosed rather than papered over. The result that xi_hh/xi_m is approximately constant and equal to the linear b^2 well into the nonlinear regime is a falsifiable regularity that has proven influential. The discussion of how nu_1 connects observed halo abundances to sigma_8 and Omega is a concrete, testable programme.

Major objections (3)

[§2.4, Eqs. (24) and (28), abstract:s9] The headline claim that the results permit 'an entirely analytic estimate of the autocorrelation function of dark haloes' (abstract:s9) rests on an empirical proportionality, not a derivation. The formalism's own prediction for the average autocorrelation, Eq. (24), is shown in Fig. 5 to fail on small scales and, for n = 0, on all scales; the replacement, Eq. (28), is read off the simulations, as §2.4 states explicitly ('our model ... does not lead directly to a model for xi_hh'). What is genuinely analytic is b (Eq. 20) and xi_m (via the Jain et al. fitting formula); the constancy of the ratio xi_hh/xi_m is an empirical regularity validated in Fig. 6. The authors should qualify abstract:s9 and §3.4 accordingly, and should state the range of R, halo mass, and xi_m over which the proportionality was verified, since the analytic estimate inherits exactly that domain.
[§3.3, Fig. 4a, abstract:s4] In Fig. 4a the solid 'model' curves for the Eulerian halo-mass cross-correlation are computed with the mass overdensity PDF taken directly from the same simulations they are compared against. That comparison is therefore not an independent prediction, and the only genuinely independent analytic alternative shown (the lognormal PDF, dashed curves) degrades in the nonlinear regime. This bears on the abstract's claim of 'excellent agreement between simulation results and analytic predictions' (abstract:s4). The claim is carried by the independent tests in Figs. 2, 3, and 4b, which do not use the simulation PDF; the authors should say so explicitly and distinguish in the text which comparisons are independent and which are not.
[§3, §4, Eqs. (18) and (20)] Every test in the paper uses a scale-free power law P(k) ∝ k^n in an Einstein-de Sitter cosmology, whereas the applications advertised in §4 (CDM, measuring Omega) require a curved spectrum and, for the Omega measurement, non-EdS cosmologies. Eq. (18) is an EdS-specific interpolation, and the claim implicit in Eq. (20) that b depends only on nu_1, independent of spectral shape, is tested only for pure power laws. The three indices n = -1.5, -1, 0 do probe shape to some degree, but none has a turnover. The authors should either add a CDM-like test or state explicitly that the model's validity is established only for scale-free spectra in EdS, and flag this as an assumption in the §4 applications.

Minor objections (6)

[Eq. (18)] The interpolation coefficients relating delta_0 to delta are given without derivation or a stated accuracy. Please state the maximum fractional error of Eq. (18) over the range of delta actually used in the figures, so that readers can judge whether it contributes to the discrepancies in Figs. 2 and 3.
[§2.4] The paper uses three related but distinct quantities: the average autocorrelation xi_bar_hh (Eq. 23), the conventional count variance sigma_hh^2 (Eq. 25), and the standard two-point function xi_hh (Eq. 26). A short table summarizing their definitions and the relations among them would make §2.4 considerably easier to follow.
[§3.2, §3.3, Figs. 3 and 4b] In the two-epoch tests the position of a halo at the later epoch is taken to be that of the particle closest to its centre at a_1. How sensitive are the results in Figs. 3 and 4b to this choice of tracer particle, particularly for haloes that merge between a_1 and a_2? A brief statement of the sensitivity would strengthen these figures.
[Fig. 1] The authors note that the M > 256 samples contain few haloes and that the disagreement for the most massive bin is likely a statistics problem. Please state the number of haloes per mass bin per panel so the reader can judge which discrepancies are meaningful.
[Fig. 4a] The dashed lognormal curves are described as significantly worse in the nonlinear regime. Since the lognormal is offered only for illustration, the caption should say so, and the text should avoid implying that it is a competitive model where xi_bar_m exceeds unity.
[Typesetting] The manuscript text supplied to me is an OCR rendering with corrupted symbols (for example the minus sign and several equation glyphs). The authors should verify the final typeset forms of Eqs. (18) and (22) in particular, since both are reproduced from earlier work and small transcription errors would be easy to miss.

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The paper

The paper claims

Halo clustering can be computed from mass clustering times a single bias factor, and that factor comes from the Press-Schechter formalism plus spherical collapse.

The reading

The paper asks a narrow but consequential question: if galaxies live inside dark matter haloes, how does the spatial distribution of those haloes differ from the distribution of the mass? Mo and White answer with a model built from two ingredients that were already on the shelf — the extended Press-Schechter description of how much mass sits in collapsed objects on a given scale, and the spherical collapse of a density perturbation — and they get a bias function $b(M,R,z) = 1 + (1/6_c)(6_1^2-1)/6_1$ that says how much overdense in haloes a region of mass overdensity $6$ should be. The value of the result is that it turns halo clustering into arithmetic on the mass power spectrum: no simulation needed to ask where haloes of a given mass sit. They then check the model against large N-body simulations and find the bias relation and the halo–mass cross-correlation reproduced over a wide range of scales, halo masses, and identification epochs. The one place the formalism does not deliver is the halo autocorrelation, because haloes cannot overlap and the count variance inside a sphere is sub-Poisson; they show empirically that the standard two-point autocorrelation is nevertheless proportional to the mass autocorrelation, with the large-scale bias factor as the constant of proportionality, even where both are far above unity.

What carries the argument

The bias function $b(M,z_1) = 1 + (6_1^2-1)/6_1$, with $6_1 = 6_1/7_1$ the peak height of the halo's smoothing scale. Its two additive pieces have distinct origins: the $1$ comes from the spherical contraction (or expansion) of the region between Lagrangian and Eulerian space, and the $(6_1^2-1)/6_1$ term is the peak-background split bias imprinted on the initial Gaussian field, the same combination that appears in earlier high-peak arguments. The machinery that produces it is the conditional Press-Schechter mass fraction $f(7_1,6_1|7_0,6_0)$ of Bond et al., which gives the mean number of $M_1$ haloes inside a sphere of radius $R_0$ and overdensity $6_0$, plus the spherical-collapse relations $R/R_0 = (3/10)(1-6_0)^{-1}$ and the interpolating formula $6_0(6)$ used to invert them. The paper also defines an averaged cross-correlation $7_{hm}$ over spheres of fixed radius and overdensity, which depends only on the mean halo count and so sidesteps the scatter problem that blocks the autocorrelation.

Extended reading notes

Core claim

The central claim is that the bias of dark haloes relative to the mass is computable analytically and is accurate. Halos of mass $M$ identified at redshift $z_1$ are defined, following Press-Schechter, as regions whose linearly extrapolated overdensity exceeds $6_1 = (1+z_1)6_c$; the fraction of the mass in a larger sphere of linear overdensity $6_0$ that sits in such haloes follows from the conditional probability of Bond et al. (1991). Combining that with a spherical-collapse map from Lagrangian $(R_0,6_0)$ to Eulerian $(R,6)$ gives a mean halo overdensity $6_h = b(M,z_1)6$ with $b = 1 + (6_1^2-1)/6_1$, where $6_1 = 6_1/7_1$ is the peak height. Halos with $6_1=1$, i.e. mass $M_*$, are predicted unbiased; more massive haloes are positively biased and rarer, and low-mass haloes are mildly antibiased. The authors verify this bias function, the Lagrangian cross-correlation, and the Eulerian halo–mass cross-correlation against P$^3$M simulations with $N=10^6$ particles and several power-law spectra, finding agreement well into the nonlinear regime. They are explicit that the average autocorrelation cannot be obtained this way, because halo exclusion makes count variances substantially sub-Poisson; instead they measure the standard two-point halo autocorrelation and find it proportional to the mass autocorrelation with the large-scale bias factor squared, which is what licenses an entirely analytic estimate of halo clustering.

Load-bearing premise

The claim that halo clustering can be estimated entirely analytically rests on the empirical finding that the ratio of halo to mass autocorrelation equals the large-scale bias factor squared even on nonlinear scales; the paper itself notes that the formalism does not directly produce a model for the halo autocorrelation, so this proportionality is read off the simulations rather than derived.

Editorial extensions

If this is right

Halo clustering of any mass at any identification redshift can be predicted from the linear power spectrum alone, replacing a simulation with a formula for the bias factor.
The same bias factor applied to a known nonlinear mass correlation function (e.g. the fitting formula of Jain, Mo & White) yields the halo autocorrelation analytically, including on scales where the mass correlation is well above unity.
Because bias rises with identification redshift at fixed mass, galaxies formed at the centres of early low-mass haloes can be as strongly clustered today as much more massive haloes formed later, so a luminosity dependence of clustering is not a required feature of hierarchical models.
Expanding $6_h(6)$ to second order gives a calculable curvature $B_2$ in the bias relation, which feeds through to a predicted reduction in the skewness of halo counts relative to mass counts, potentially explaining the lower skewness measured for late-type galaxies.
If the abundance of a class of haloes is known, the peak height $6_1$ can be inferred, the mass correlation function recovered from the observed halo correlation function, and $7_8$ extracted as a function of the assumed density parameter $\Omega_0$ — which is how the method is proposed as a measurement of $\Omega$.

Reading between the lines

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

The proportionality $7_{hh} = b^2 7_m$ is the paper's most reusable export and its least derived one: it would be worth checking whether the ratio drifts with halo mass at fixed scale in simulations with larger boxes and better mass resolution than the ones available here.
If $7_{hh}/7_m$ really is scale-independent down to the exclusion scale, then the shape of the galaxy correlation function carries almost no information about halo occupation — only its amplitude does — which would sharpen the degeneracy between $7_8$ and $\Omega_0$ in exactly the direction the authors want to exploit.
The sub-Poisson count variance is treated as an obstacle; it could instead be treated as a signal, since the size of the deficit below Poisson measures the volume filling fraction of the excluded haloes and hence constrains their density profile.
The double-epoch construction — haloes identified at $a_1$, positions followed to $a_2$ — is a template for any abundance-matching scheme in which a galaxy keeps the identity of the halo it formed in even after that halo has merged.

Original abstract

We develop a simple analytic model for the gravitational clustering of dark matter haloes to understand how their spatial distribution is biased relative to that of the mass. The statistical distribution of dark haloes within the initial density field (assumed Gaussian) is determined by an extension of the Press-Schechter formalism. Modifications of this distribution caused by gravitationally induced motions are treated using a spherical collapse approximation. We test this model against results from a variety of N-body simulations, and find that it gives an accurate description of a bias function. This bias function is sufficient to calculate the cross-correlation between dark haloes and mass, and again we find excellent agreement between simulation results and analytic predictions. Because haloes are spatially exclusive, the variance in the count of objects within spheres of fixed radius and overdensity is significantly smaller than the Poisson value. This seriously complicates any analytic calculation of the autocorrelation function of dark halos. Our simulation results show that this autocorrelation function is proportional to that of the mass over a wide range in $R$, even including scales where both functions are significantly greater than unity. The constant of proportionality is very close to that predicted on large scales by the analytic model. This result permits an entirely analytic estimate of the autocorrelation function of dark haloes. We use our model to study how the distribution of galaxies may be biased with respect to that of the mass. In conjunction with other data these techniques should make it possible to measure the amplitude of cosmic mass fluctuations and the density of the Universe.

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.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We develop a simple analytic model for the gravitational clustering of dark matter haloes to understand how their spatial distribution is biased relative to that of the mass. The statistical distribution of dark haloes within the initial density field (assumed Gaussian) is determined by an extension of the Press-Schechter formalism. Modifications of this distribution caused by gravitationally induced motions are treated using a spherical collapse approximation. We test this model against results from a variety of N-body simulations, and find that it gives an accurate description of a bias function. This bias function is sufficient to calculate the cross-correlation between dark haloes and mass, and again we find excellent agreement between simulation results and analytic predictions. Because haloes are spatially exclusive, the variance in the count of objects within spheres of fixed radius and overdensity is significantly smaller than the Poisson value. This seriously complicates any analytic calculation of the autocorrelation function of dark halos. Our simulation results show that this autocorrelation function is proportional to that of the mass over a wide range in $R$, even including scales where both functions are significantly greater than unity. The constant of proportionality is very close to that predicted on large scales by the analytic model. This result permits an entirely analytic estimate of the autocorrelation function of dark haloes. We use our model to study how the distribution of galaxies may be biased with respect to that of the mass. In conjunction with other data these techniques should make it possible to measure the amplitude of cosmic mass fluctuations and the density of the Universe.

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 develop a simple analytic model for the gravitational clustering of dark matter haloes to understand how their spatial distribution is biased relative to that of the mass.	contribution 0.99	0.95	0.07	asserted 0.91
s1	The statistical distribution of dark haloes within the initial density field (assumed Gaussian) is determined by an extension of the Press-Schechter formalism.	method 0.95	0.66	0.03	asserted 1.00
s2	Modifications of this distribution caused by gravitationally induced motions are treated using a spherical collapse approximation.	method 0.99	0.77	0.03	asserted 0.98
s3	We test this model against results from a variety of N-body simulations, and find that it gives an accurate description of a bias function.	result 0.86	0.96	0.25	asserted 1.00
s4	This bias function is sufficient to calculate the cross-correlation between dark haloes and mass, and again we find excellent agreement between simulation results and analytic predictions.	result 1.00	0.96	0.15	asserted 1.00
s5	Because haloes are spatially exclusive, the variance in the count of objects within spheres of fixed radius and overdensity is significantly smaller than the Poisson value.	result 0.71	0.58	0.06	asserted 1.00
s6	This seriously complicates any analytic calculation of the autocorrelation function of dark halos.	background 0.44	0.23	0.03	asserted 0.98
s7	Our simulation results show that this autocorrelation function is proportional to that of the mass over a wide range in $R$, even including scales where both functions are significantly greater than unity.	result 1.00	0.94	0.25	asserted 1.00
s8	The constant of proportionality is very close to that predicted on large scales by the analytic model.	result 0.98	0.88	0.13	asserted 1.00
s9	This result permits an entirely analytic estimate of the autocorrelation function of dark haloes.	interpretation 0.81	0.81	0.27	asserted 0.72
s10	We use our model to study how the distribution of galaxies may be biased with respect to that of the mass.	contribution 0.75	0.90	0.06	hedged possibility 0.77
s11	In conjunction with other data these techniques should make it possible to measure the amplitude of cosmic mass fluctuations and the density of the Universe.	future work 0.68	0.09	0.09	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from simulation or computation 0.86.

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

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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 average cross-correlation function between haloes and mass in Lagrangian space, normalized by the linear average mass correlation function. Results from simulations are shown for haloes with mass M 32 (triangles), 64 (three-pointed stars), 128 (squares), and 256 (crosses). The solid curves are the model predictions. The statistics for the M > 256 samples in the simulations are poor, because they contain only small number of haloes. The power index n and the expansion factor a are shown in each panel

**Figure 4.** shows the ratio of the average cross-correlation between haloes and mass to the average autocorrelation of the mass. The heavy ticks on the horizontal axis show the values of R where m = 1. Open symbols give results for haloes in various mass ranges as identi ed by the FOF group nder. Filled dots in Figure 4a give corresponding results for the SO group nder. We see clearly that the two group nders give similar results for the average cross-correlation function hm. For the case n = 1:5, the SO haloes have slightly higher hm than the corresponding FOF haloes because the SO

Showing 2 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9512127 by the authors.

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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...
Spherical collapse in Galileon gravity: fifth force solutions, halo mass function and halo bias
astro-ph.CO 2013-08 accept novelty 6.0 of 10

The Quartic Galileon model gives roughly 40% weaker gravity inside dense halos and a large excess of high-mass halos relative to Lambda-CDM, while the Quintic variant admits no physical fifth-force solution at high density.
Halo Stochasticity from Exclusion and non-linear Clustering
astro-ph.CO 2013-05 conditional novelty 6.0 of 10

Halo shot noise departs from the Poisson value 1/n because exclusion suppresses it and non-linear clustering enhances it, with the same low-wavenumber amplitude before and after gravitational collapse.
Gravity and Large-Scale Non-local Bias
astro-ph.CO 2012-01 accept novelty 6.0 of 10

Local bias imposed at formation inevitably becomes non-local under gravity, producing Galileon-shaped tidal terms whose neglect biases the linear bias recovered from the galaxy bispectrum.
Large-scale clustering of galaxies in general relativity
astro-ph.CO 2011-07 accept novelty 6.0 of 10

The observed galaxy power spectrum on horizon scales depends on the gauge used to define linear bias; synchronous-comoving gauge is the physically correct choice, and the resulting relativistic correction mimics a loc...
Optimal Constraints on Local Primordial Non-Gaussianity from the Two-Point Statistics of Large-Scale Structure
astro-ph.CO 2011-04 conditional novelty 6.0 of 10

Optimal mass weighting plus a multi-tracer Fisher analysis of halo power spectra can tighten local fNL constraints by up to an order of magnitude over a single uniformly weighted halo sample, reaching sigma_fNL around...
Primordial non-Gaussianity in the Bispectrum of the Halo Density Field
astro-ph.CO 2010-11 conditional novelty 6.0 of 10

A full tree-level calculation of the halo bispectrum including non-linear gravity, multivariate bias and local primordial non-Gaussianity predicts that squeezed-triangle bispectra beat the power spectrum by roughly a ...
Halo Clustering with Non-Local Non-Gaussianity
astro-ph.CO 2010-08 accept novelty 6.0 of 10

Generalizing the peak-background split to arbitrary primordial bispectra shows that equilateral and self-ordering-scalar-field non-Gaussianity produce almost no scale-dependent halo bias, while folded and orthogonal m...
Galaxy bias of radio AGN and star-forming galaxies from MIGHTEE and CMB lensing
astro-ph.CO 2026-10 accept novelty 5.0 of 10

The galaxy bias of radio AGNs and star-forming galaxies is simultaneously measured from MIGHTEE and ACT CMB lensing, finding AGNs are more clustered than SFGs with bias values 2.60/D(z) and 0.52/D(z) respectively in t...
The impact of primordial magnetic fields on the formation of galaxies
astro-ph.GA 2026-09 conditional novelty 5.0 of 10

PMF-enhanced initial conditions boost high-redshift structure formation, but stellar and AGN feedback erase most late-time differences, with the peak scale of the field governing the outcome.
Assembly Bias in eRASS1 X-ray Galaxy Clusters: Accounting for Non-uniform Survey Exposure
astro-ph.CO 2026-09 accept novelty 5.0 of 10

A relative clustering analysis of eROSITA eRASS1 clusters yields no evidence for assembly bias, with Δb = 0.8 ± 1.1, consistent with FLAMINGO simulations.
COSMOS-Web: does halo mass alone shape the clustering of star-forming and quiescent galaxies?
astro-ph.GA 2026-06 unverdicted novelty 5.0 of 10

Quiescent galaxies cluster more strongly than star-forming ones by 0.5-1 dex after halo-mass matching, with one-halo conformity up to z~2 that disappears at higher redshifts.
Cosmology-dependent covariance in galaxy cluster number counts: consequences for parameter inference
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

Fixing the covariance at an incorrect cosmology in cluster count analyses leaves Ω_c, σ_8, and w estimates unbiased but distorts their uncertainties, driven by S_8 amplitude effects; a single update at the recovered b...
Columba: isolated dwarf galaxy populations in diverse cosmological environments simulated with a cold interstellar medium
astro-ph.GA 2026-06 unverdicted novelty 5.0 of 10

New hydrodynamical simulations show that dwarf galaxy stellar mass-halo mass relations and star formation histories are more influenced by host halo concentration than by the 5 cMpc scale environment.
Cosmic web stripping and starvation of low-mass filament galaxies in TNG50
astro-ph.GA 2026-05 unverdicted novelty 5.0 of 10

Low-mass filament galaxies in TNG50 exhibit smaller asymmetric cold gas discs due to cosmic web tidal fields causing altered accretion or starvation and late-time stripping, while integrated stellar and halo propertie...
Little ado about everything II: an `emergent' dark energy from structure formation to rule cosmic tensions
astro-ph.CO 2025-02 conditional novelty 5.0 of 10

An ensemble-averaged stochastic cosmology with a 'little ado' noise term produces an emergent dark energy that fits late-time data and claims to relieve the H0 and f sigma8 tensions.
Analytical modeling of the one-dimensional power spectrum of 21-cm forest based on a halo model method
astro-ph.CO 2024-11 conditional novelty 5.0 of 10

A halo-model formula for the 1D power spectrum of the 21-cm forest is presented and shown to match small-scale simulations built from the same gas and temperature assumptions.
Planck 2018 results. III. High Frequency Instrument data processing and frequency maps
astro-ph.CO 2018-07 conditional novelty 5.0 of 10

Improved mapmaking that solves per-detector calibration and bandpass parameters inside the destriper cuts intensity-to-polarization leakage enough to enable a significant HFI measurement of the reionization optical de...
Large-Scale Galaxy Bias
astro-ph.CO 2016-11 accept novelty 5.0 of 10

The paper reviews and extends the perturbative theory of galaxy bias, showing that quasi-linear galaxy clustering is captured by a finite set of bias parameters built from local gravitational observables.
Position-dependent power spectrum of the large-scale structure: a novel method to measure the squeezed-limit bispectrum
astro-ph.CO 2014-03 accept novelty 5.0 of 10

Correlating the local power spectrum with the local mean density measures the squeezed-limit bispectrum, confirmed in 160 N-body simulations.
Connection between Newtonian simulations and general relativity
astro-ph.CO 2011-01 accept novelty 5.0 of 10

Newtonian cosmological simulations compute the correct gravitational potential and particle trajectories even on super-horizon scales; only a constant initial displacement must be added to convert simulation coordinat...
Constraining Running Non-Gaussianity
astro-ph.CO 2009-06 conditional novelty 5.0 of 10

A Fisher matrix forecast shows that Planck could measure the running of primordial non-Gaussianity to Δn_NG ≈ 0.1 (local shape) or 0.3 (equilateral), with large-scale structure surveys providing comparable or better c...
Tracing Large-scale Structure with the MeerKLASS On-the-Fly Survey: Angular Clustering of Radio Sources at 816 MHz
astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

First measurement of angular clustering w(theta) for radio sources at 816 MHz over 800 deg2 yields positive signal and effective bias 1.53-2.0 depending on N(z) prior.
Risk-based framework to determine climate-informed design storms for road drainage infrastructure
physics.geo-ph 2025-08 unverdicted novelty 4.0 of 10

A risk-index method combining climate projections, physiographic hazard, and socioeconomic vulnerability scales future design storms for Ontario road drainage.
From quantum fluctuations to galaxy power spectrum multipoles
astro-ph.CO 2026-09 conditional novelty 3.0 of 10

A self-contained lecture-note derivation of galaxy power-spectrum multipoles from primordial curvature perturbations, with a closed-form FFTLog master-integral basis and illustrative validation against AbacusPNG halo ...
Large-scale structures of the Universe: physics, phenomenology, statistics
astro-ph.CO 2026-06 unverdicted

Lecture series on the physics, phenomenology, and statistics of large-scale cosmic structure evolution and non-Gaussian predictions.

How later papers used it

243 later papers cite it

Evidence: substantial. 232 of the citing papers share no author with it, and 40 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 324 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

17 later papers by other authors are listed below: 1 built on it, 14 used its method or tool, 1 used its data or values and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 paper

Generalized this paper's standard formula for the time evolution of linear bias to the case where merging is important.

Gravity and Large-Scale Non-local Bias (2012) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 15 of 28 papers with this label · 1 by the same authors

According to the theory of density perturbation, we use the approximation for spherical collapse from the equation 18 in [ref] to the linear matter density $δ_L$.

Spectroscopic Confirmation of two Extremely Massive Protoclusters BOSS1244 and BOSS1542 at $z=2.24$ (2021) · Overdensity and Present-day Masses Estimate · 1.00
In order to do that we first map each density element at $z_ini$ to a corresponding linear density contrast using the following parametric formulae from [ref] : 1+δ_NL =92 (θ- sin θ)^2(1-cos θ)^3 , δ_L=3 ×6^2/320 (θ-sin θ)^2/3 for $δ_NL > 0$,

Simulating the epoch of Helium Reionization in photon-conserving semi-numerical code SCRIPT (2024) · Subgrid Implementation of thermal history · 1.00
Took the conditional mass function to be provided by the Extended Press-Schechter formalism of this paper.

Probing power spectrum enhancement at small scales with the SKA (2026) · 0.98 · paraphrased: the licence does not allow quoting
Converted a real over-density to a linear over-density via this paper.

Lyman-alpha Emitters in Ionized Bubbles: Constraining the Environment and Ionized Fraction (2022) · 0.98 · paraphrased: the licence does not allow quoting
Used a spherical collapse model from this paper to relate a matter overdensity to a linear matter enhancement.

Probing the existence of a rich galaxy overdensity at z=5.2 (2021) · 0.96 · paraphrased: the licence does not allow quoting
Used equation 18 of this paper to convert between delta over sigma and the nonlinear density.

Small-scale clumping of dark matter and the mean free path of ionizing photons at $z=6$ (2022) · 0.93 · paraphrased: the licence does not allow quoting
Used the halo bias of this paper in writing the one-halo and two-halo components of the 21-cm power spectrum.

Prospects for measuring neutrino mass with 21-cm forest (2025) · 0.93 · paraphrased: the licence does not allow quoting
To model this bias, the peak-background split formalism [ref] is usually adopted to calculate an approximate linear halo bias, particularly in the 2-halo term of Eq. [ref], which describes the influence of the large-scale power spectrum on halo formation.

Accelerating cosmological inference of interacting dark energy with neural emulators (2025) · Halo bias · 0.93
Used the linear halo bias factor of this paper.

A Designer's Guide to Lunar Far-Side Interferometer Array: Power Spectrum Measurement and Cosmological Constraints from the Dark Ages (2026) · 0.91 · paraphrased: the licence does not allow quoting
Employed the halo bias of this paper in its two-halo term.

Cosmological free-free emission from dark matter halos in the $\Lambda$CDM model (2022) · 0.90 · paraphrased: the licence does not allow quoting
Used the halo bias of this paper, according to the Press-Schechter formalism, in its ionization fraction fluctuation model.

Cross-correlation between 21-cm radiation and CMB B modes from the cosmic birefringence in the presence of a light scalar field (2019) · 0.89 · paraphrased: the licence does not allow quoting
In particular, it predicts the probability that matter in a spherical region of mass, $M_0$, at redshift, $z_0$, and linear overdensity, $δ_0$, is contained within dark matter haloes of mass in the interval $(M_1, M_1 + dM_1)$ at redshift $z_1$: [equation] where, for a virialized structure, $δ_1=δ_c/D(z)$, and $δ_0$ is given by [The accuracy of this formula in underdense regions is examined in Appendix [ref].]: [equation] as an approximate solution of Eqs. (16-17) of [ref] .

The abundance of dark matter haloes down to Earth mass (2023) · Theoretical halo mass functions · 0.81
According to the spherical collapse model, the halo collapses when the linear density contrast $δ(z,M)$ exceeds the critical value $δ_c$, so the collapse redshift is expressed as [ref]

Anisotropies in Cosmological 21 cm Background by Oscillons/I-balls of Ultra-light Axion-like Particle (2021) · Minihalo Mass Function · 0.88
Under the peak-background split ansatz including this paper, related the large-scale halo bias to the Sheth-Tormen halo mass function.

The mass-concentration relationship of virialized halos and its impact on cosmological observables (2011) · 0.80 · paraphrased: the licence does not allow quoting

By the same authors

Followed this paper to define halo bias as the ratio between the halo-matter cross-correlation function and the matter-matter autocorrelation function.

Relating the structure of dark matter halos to their assembly and environment (2020) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Uses its data or values 1 paper

Used the fitting function in this paper to convert a density contrast of 0.8 to a linearly extrapolated density contrast of -2.81.

Voids as a Precision Probe of Dark Energy (2010) · 0.88 · paraphrased: the licence does not allow quoting

Compares with it 1 of 10 papers with this label

Showed the predictions of this paper for comparison.

First results from the IllustrisTNG simulations: matter and galaxy clustering (2017) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

108 arXiv papers cite this in 150 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 · 94 sentences without a confident label

Builds on it 1 paper

Along the way, we generalized the standard formula of the time evolution of linear bias [ref] to the case where merging is important (see Eq. [ref]).

Gravity and Large-Scale Non-local Bias · Conclusions · 1.00

Uses its method or tool 17 papers

We derive the void bias in PNG using the peak-background split formalism [ref] .

Constraint of Void Bias on Primordial non-Gaussianity · Theory · 0.94
The collapse fraction can then be calculated with the extended Press-Schechter method [ref] .

Reionization in the Warm Dark Matter Model · Method of Calculation · 1.00
Following [ref] , an application of the peak-background split approximation enables one to calculate $b(M)$ from a given mass function.

Nonlinear clustering in models with primordial non-Gaussianity: the halo model approach · Halo bias · 0.94

Uses its data or values 1 paper

In order to estimate the number of voids of a particular size at a particular redshift, we use the Press-Schechter formalism to determine the number density of voids in a redshift bin centered at $z,$ with Eulerian comoving radius between $R_E$ and $R_E + d R_E.$ Simulations indicate that the number density of voids peaks at a density contrast of $δ≈-0.85$ [ref] , we shall consider all the voids to have a density contrast of 0.8, which can be seen to correspond to a linearly extrapolated density contrast of -2.81 using the fitting function in [ref] .

Voids as a Precision Probe of Dark Energy · Estimate of Voids to be found from a survey · 0.88

Compares with it 5 papers

In particular, we compare against the fitting formulae from: * the predictions of spherical collapse [ref] : [equation] * the fit from [ref] (hereafter SMT01), motivated by the ellipsoidal collapse model and calibrated using simulations: [equation] with $a=0.707,b=0.5,c=0.6$. * the fit in [ref] (hereafter T10, see also Ref. [ref] ), also calibrated against simulations, with a more flexible form: [equation] where the parameters ($A,B,C,a,b,c$) are dependent on the virial overdensity, $Δ$, with respect to the mean density of the Universe, as detailed in Table 2 of T10.

An Improved Fit for Linear Halo Bias at High Redshift · Comparison to existing fits · 1.00
In other words, references [ref] and [ref] derive an effect on the halo/galaxy bias that, for the local model, can be shown to be equivalent, at large scales, to the one described by Eq. ([ref]), and they actually coincide if one considers as well the high-peak limit $ν/σ_R≫1$ since, in this case [b_h,1-1]δ_c≃b_2σ_R^2≃ν^2=δ_c^2/σ_R^2, once one assumes an expression such as the one derived by [ref] for the galaxy quadratic bias parameter.

Constraining Running Non-Gaussianity · Halo bias and primordial non-Gaussianity · 0.92
This differs from the more general Press-Schechter expression, $b_0,L=(ν^2-1)/δ_c$ [ref] , only by the term $-1/δ_c$, which is negligible for the massive halos considered here.

Tracing Primordial Gravitational Waves via non-Gaussian Signatures of Halo Bias · Halo Bias Correction · 0.98

Cites it as background 30 papers

The theoretical framework underlying such clustering analysis, based on the Press-Schechter theory and cosmological $N$-body simulations, have shown that galaxies are biased tracers of the dark matter distribution, with their properties determined by the halos in which they form [ref] .

ODIN: Clustering Properties of Ly$\alpha$ Blobs at $z$ $\sim$ 2.4 and 3.1 · Introduction · 0.93
In this perspective, the two main approaches are the excursion-set formalism [ref] and the theory of Lagrangian density peaks [ref] , which have been widely explored to provide theoretical models for the dark matter (DM) halo mass function (HMF), bias, halo correlation functions [ref] , and cosmic void size functions (VSF) [ref] , hereafter , [ref] .

The universal multiplicity function: counting halos and voids · Introduction · 0.94
The excursion set formalism predicts the abundance of objects crossing a critical density threshold as a consequence of hierarchical structure formation in an initially linear Gaussian field [ref] .

VERSUS: An excursion-set-inspired void-finder for the Stage-IV era · Voids in the matter field · 0.95

Discussion

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