Large scale bias and the peak background split

Ravi K. Sheth (MPA) & Giuseppe Tormen (Padova)

REVIEW

Large scale bias and the peak background split

Ravi K. Sheth (MPA) & Giuseppe Tormen (Padova)

Astrophysics · Submitted 11 Jan 1999 · 19 references

Pith's verdict: The claim that the unconditional mass function alone fixes large-scale halo bias is correct and useful; the derivation is heuristic exactly where it matters, and the authors say so themselves.

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arXiv IDastro-ph/9901122 (version 2)

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CitationRavi K. Sheth (MPA) & Giuseppe Tormen (Padova). "Large scale bias and the peak background split." arXiv:astro-ph/9901122 (1999). https://pith.science/paper/QJVFPBXJ

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Editor's note

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The thing to take away: Sheth and Tormen show you don't need merger histories to get the large-scale bias — the shape of the unconditional mass function is enough. That's the conceptual step, and it's why this paper is everywhere now. The concrete deliverables are eq. (10), a two-parameter tweak to Press–Schechter (a = 0.707, p = 0.3) that fits the GIF mass functions across SCDM, OCDM and ΛCDM and across redshift, and eq. (12), the bias that follows from it. Both are simple enough to actually use, and both hold up against the simulations. What's done well: the peak-background-split argument is short and clean, the comparison covers three cosmologies and several output times, and they test three different mass functions rather than only their own. They are honest about the residual scale dependence in b(k) and about how they extracted the small-k amplitude. The Zeldovich comparison is a good sanity check even though it doesn't beat their fit. Soft spots, in order. First, a and p are fitted to the same GIF mass functions that later get compared to GIF bias. The bias is a different observable, so Fig. 3 is a genuine out-of-sample test of eq. (12) — but it is not an independent test of the mass function fit, and the paper doesn't pretend otherwise. Second, and this is the one I'd press on: eq. (11) is derived by sending ν → ν10, which is exact for a constant barrier and a sharp-k filter. Their fitted mass function is not constant-barrier, and they say so at the end of §4 — 'the barrier shape must depend on mass.' For a moving barrier the linear bias picks up a ∂B/∂δ0 term that the constant-barrier split misses, so eq. (11) isn't literally the derivative of eq. (10). They acknowledge the inconsistency and defer the moving-barrier calculation to work in progress. That's a gap in rigor, not in the empirical claim, and the stress-test note has this right. Who it's for: anyone doing halo-model or cluster-abundance work, and anyone who wants to see how far a one-line peak-background-split argument can be pushed. Worth a referee's time.

Referee report

Major objections (3)

[§2, eq. (11)] The step from eq. (10) to eq. (11) assumes that the conditional mass function is obtained from the unconditional one by the replacement ν → ν10 with the same functional form, which is exact only for a constant barrier and a sharp-k filter. The fitted mass function of eq. (10) is not of that form, and the authors state at the end of §4 that for the barrier-crossing model to yield the GIF mass function 'the barrier shape must depend on mass.' For a moving barrier the linear Lagrangian bias acquires an additional term from the response of the barrier to the background shift, which the constant-barrier peak-background split omits. The manuscript should therefore say explicitly, in §2 and not only in §4, whether eq. (11) is the exact derivative of eq. (10) or an ansatz that reproduces its qualitative behaviour, and should indicate how large the omitted moving-barrier term is expected to be. As written, the phrase 'physically motivated formula' in §4 overstates what has been derived.
[§3, Fig. 3] The parameters a = 0.707 and p = 0.3 entering eq. (12) are fitted to the same GIF simulations whose bias is then used to test the formula in Fig. 3. The comparison is not fully independent: it tests whether the peak-background-split map from mass-function shape to bias holds within one simulation set, not whether the mass function and bias are jointly predicted. This should be stated plainly where Fig. 3 is introduced. A quantitative comparison against an independent simulation set, or at minimum a demonstration that the residuals in Fig. 3 are uncorrelated with the residuals of the Fig. 2 mass-function fit, would materially strengthen the central claim of abstract:s2.
[§4, final paragraph] The manuscript states that the authors 'have solved for this barrier shape, and are in the process of developing and testing the detailed predictions of such a moving barrier model further.' That work is not presented here, yet it is the ingredient that would justify the derivation underlying eq. (11). As it stands the reader cannot check whether a moving-barrier peak-background split reproduces eq. (12) or differs from it. Either the relevant result should be included, or the sentence should be recast so that it does not carry argumentative weight for the paper's central derivation.

Minor objections (8)

[§3, paragraph on Fig. 2] The text says the dot-dashed curve 'fits the data slightly better than the dashed line,' but the Fig. 2 caption lists only dotted (Press–Schechter), dot-dashed (Zeldovich) and solid (modified) curves. Presumably 'dashed' should read 'dotted'; please make the text and caption consistent.
[§2, eq. (11)] The numerator of the bracketed expression reads 'aν1 − 1 + 2p1 + (aν1)^p'; the '2p1' appears to be a typographical error for '2p'. Please check, since the same combination appears correctly as 2p/δ1 in eq. (12).
[§1] The citation 'Catelan et al. 1997' does not match the reference list entry 'Catelan P., Lucchin F., Matarrese S., Porciani C., 1998'. Please reconcile the year.
[References] Efstathiou et al. (1988) and Mo, Jing & White (1997) appear in the reference list but are not cited in the text; the Haehnelt, Natarajan & Rees entry is missing its year. Please check the reference list against the text.
[§3, Fig. 1 caption] The bin definition ('the lower limit of the mth bin is at 10 × 2^m particles per halo, m > 0') should state the range of m explicitly, since the text also says the lowest mass halo considered has twenty particles; as written the indexing takes a moment to reconstruct.
[§3, eq. (10)] The normalisation A is defined only implicitly by requiring that the integral of f(ν) over all ν be unity. Please quote its numerical value for a = 0.707 and p = 0.3, and report the goodness of fit and, if available, the uncertainty on a and p, so that readers can judge how tightly the mass function is constrained.
[Abstract and §2] The claim that 'knowledge of the unconditional mass function alone' suffices is slightly stronger than what eq. (12) requires: the formula is written in terms of ν1 = (δc/σ)², so the mapping from mass to σ(m) and the value of δc(z) are also needed. It would be more precise to say that the shape of νf(ν) is the only non-standard input.
[§1] The word 'reaonably' in the third paragraph should read 'reasonably'.

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The paper

The paper claims

Halo bias follows from the mass function alone

The reading

The paper asks whether the large-scale clustering of dark matter haloes can be predicted from one input: the abundance of haloes as a function of mass. Using the peak-background split, the authors argue that a long-wavelength density shift changes the collapse threshold, and that this change can be read off from the unconditional mass function without tracking merger histories. Plugging in the mass function measured in the GIF simulations yields a single bias formula, Eq. (12), that depends on halo mass and redshift through $��=(��_c/��)^2$. Comparisons with SCDM, OCDM and $��$CDM simulations show the formula tracks the measured bias for low-mass and high-mass haloes, including the case where haloes form at one redshift and are observed later. If right, it removes merger history as a required input for large-scale halo clustering.

What carries the argument

The peak-background split is the mechanism: separate the density field into a long-wavelength background and small-scale peaks, then note that a background overdensity shifts the effective collapse threshold. In this paper it lets the conditional mass function be obtained by replacing $��$ with $��_{10}=(��_1-��_0)^2/(��_1^2-��_0^2)$ in the unconditional mass-function form. The Lagrangian bias is the linear response of that conditional abundance to $��_0$, and the Eulerian bias is one plus the Lagrangian bias. The shape of the unconditional mass function therefore carries all the mass dependence of the large-scale bias.

Extended reading notes

Core claim

The central claim is that on large scales the bias factor of dark matter haloes is fixed by the shape of the unconditional halo mass function. The authors derive this by treating a long-wavelength density perturbation as a shift in the collapse threshold, so the conditional mass function is the unconditional one evaluated at a shifted $��$; differentiating that response gives the Lagrangian bias, and adding one converts it to Eulerian bias. For the GIF mass function, $�� f(��)=A(1+1/��'^p)(��'/2)^{1/2}e^{-��'/2}/��$ with $��'=a��$, $a=0.707$ and $p=0.3$, this gives $b_{��m Eul}=1+(a��_1-1)/��_1+(2p/��_1)/(1+(a��_1)^p)$. The paper reports that this formula matches the measured large-scale bias in SCDM, OCDM and $��$CDM simulations better than the Press-Schechter-based formula, and that low-mass haloes are less anti-biased and high-mass haloes slightly less biased than Press-Schechter predicts.

Load-bearing premise

The load-bearing premise is that the peak-background split can be applied with a single, mass-independent collapse barrier, so that the conditional mass function is the unconditional one with $��$ shifted to $��_{10}$; the authors acknowledge at the end of Section 4 that a constant barrier cannot produce the GIF mass function and that the barrier shape must depend on mass, so the bias formula is derived under an assumption they know is not exactly consistent with the mass function they insert.

Editorial extensions

If this is right

Galaxy formation models can assign large-scale clustering to haloes using only the halo mass function, without requiring merger trees or detailed formation histories.
Low-mass haloes hosting galaxies are predicted to be more clustered, or less anti-biased, than Press-Schechter-based bias formulae imply, which changes galaxy clustering predictions at high redshift.
High-mass haloes are predicted to be slightly less biased than Press-Schechter-based formulae imply, which affects estimates for cluster-scale and high-redshift quasar host haloes.
The same formula applies when haloes form at $z_{\rm form}$ and are observed at lower $z_{\rm obs}$, so evolved halo populations can be modeled with the same mass-function input.
Because different mass-function shapes produce different bias relations, improving the fit to the halo mass function directly improves predictions for large-scale halo clustering.

Reading between the lines

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

Any proposed mass function can be converted into a bias prediction and checked against large-scale clustering, so bias measurements become a consistency test for mass-function models.
If the collapse barrier is mass-dependent, as the authors suggest, then a moving-barrier calculation could sharpen the high-mass end of the bias prediction beyond the constant-barrier approximation used here.
The logic can be inverted: measured large-scale bias as a function of mass might constrain the mass function in regimes where direct abundance measurements are noisy, especially at high redshift.
Because the formula depends on $��$ alone, it predicts a self-similar scaling of bias with redshift and cosmology; deviations from that scaling would point to additional baryonic or assembly effects.

Original abstract

Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,871 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.30 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.92 · 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.

Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.

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	Dark matter haloes are biased tracers of the underlying dark matter distribution.	background 1.00	0.06	0.04	asserted 1.00
s1	We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales.	method 0.66	0.93	0.06	asserted 0.95
s2	Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor.	result 0.93	0.92	0.77	asserted 1.00
s3	Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias.	method 0.99	0.93	0.05	asserted 0.99
s4	Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones.	result 1.00	0.92	0.34	asserted 1.00
s5	In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.	result 0.96	0.93	0.35	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from simulation or computation 0.92.

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 square root b(k) of the ratio of the power spectra of dark matter haloes to that of the dark matter, computed at the time the haloes were identified: z = 0, z = 2 and z = 4 for the top, middle and bottom rows, respectively. The different line types in each panel correspond to haloes of different mass ranges (mass bins increase in size by factors of two). The more massive haloes at a given redshift typically have larger values of b(k). averaged halo and mass correlation functions, obtained using a counts-in-cells procedure (Mo & White 1996; Sheth & Lemson 1999). Others have used…

**Figure 2.** The unconditional mass functions from five different output times (filled triangles, open triangles, open squares, filled circles, open circles show results for z = 0, z = 0.5, z = 1, z = 2 and z = 4) in the GIF simulations plotted as a function of the scaled variable ν. Dotted curve shows the Press–Schechter prediction, dot-dashed curve shows the mass function associated with the Zeldovich approximation, and solid curve shows our modified fitting function. puted using our peak background split model depends on the unconditional mass function. Solid and dotted curves show equation…

**Figure 3.** The large scale Eulerian bias relation at zobs = zform between haloes which are identified at zform, and the mass. Solid curves show the relation we predict using the GIF mass function, dotted curves show the relations which follow from the Press–Schechter mass function, and dot-dashed curves show the relations associated with the Zeldovich mass function. time of each model, we will simply use this one function with the given constants a and p. This fortuitous rescaling of the mass function is useful because it means that we need compute the peakbackground split limit associated…

**Figure 5.** The square root of the ratio of the halo and matter power spectra for haloes that formed at zform = 4 (top panels) and zform = 2 (bottom panels) but were observed at zobs = 0. data from the same output times as before. They were obtained by transforming the large scale Eulerian bias relation as indicated by the labels on the axes. The curves show the theoretical predictions for the three different mass functions we have been considering. There are two points to be made about this plot. The first is that our peak background split model (solid curves) provides a reasonable…

**Figure 6.** The large scale Eulerian bias relation at zobs = 0 for haloes that formed at zform. Solid curves show the relation we predict using the GIF mass function, dotted and dot-dashed curves show the relations which follow from the Press–Schechter and Zeldovich mass functions, respectively. therefore, that our formula for the bias relation is reasonably accurate in this two epoch context as well. 4 DISCUSSION Collapsed haloes are biased tracers of the dark matter distribution. This bias depends on halo mass. In general, to quantify this bias requires a detailed knowledge of the merger…

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9901122 by the authors.

References

Works cited (19)

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Vlasov Perturbation Theory with a dispersion-scale input predicts the two-loop matter power spectrum in LambdaCDM at percent-level accuracy without EFT-style free parameters.
Joint 21-cm and CMB Forecasts for Constraining Self-Interacting Massive Neutrinos
astro-ph.CO 2025-04 conditional novelty 6.0 of 10

21-cm power spectrum forecasts show HERA and CMB-S4 combined can constrain the neutrino self-coupling G_eff to about 10 percent across strong, moderate, and mild interaction models, with futuristic lunar arrays reachi...
The Irrelevance of Primordial Black Hole Clustering in the LVK mass range
astro-ph.CO 2025-02 accept novelty 6.0 of 10

Initial spatial clustering of primordial black holes is irrelevant for binary mergers in the LVK mass range because FIRAS spectral-distortion constraints limit the clustering scale to below the merger-relevant separat...
The ACCEL2 Project: Precision Measurements of EFT Parameters and BAO Peak Shifts for the Lyman-$\alpha$ Forest
astro-ph.CO 2024-12 conditional novelty 6.0 of 10

A one-loop EFT model matches ACCEL2 Lyman-alpha forest simulations well and predicts BAO scale shifts of roughly -0.2 to -0.3 percent at z=2.
Accelerating HI density predictions during the Epoch of Reionization using a GPR-based emulator on N-body simulations
astro-ph.CO 2024-12 conditional novelty 6.0 of 10

A Gaussian-process emulator that samples collapse fractions from simulation-calibrated conditional distributions reproduces reionization-era HI power spectra to about 10% on large scales while using far less memory th...
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.
Simulations of Axion Minihalos
astro-ph.CO 2021-01 conditional novelty 6.0 of 10

N-body simulations of white-noise axion perturbations yield a calibrated minihalo mass function whose extrapolated present-day abundance sits above forecast pulsar-timing and microlensing detection thresholds.
Probing Small-Scale Power Spectra with Pulsar Timing Arrays
astro-ph.CO 2020-12 conditional novelty 6.0 of 10

A full time-domain Monte Carlo of pulsar timing residuals from dark matter subhalos finds standard cold dark matter undetectable after Milky Way tidal stripping, while early matter domination below 1 GeV reheating and...
Primordial Non-Gaussianity from Biased Tracers: Likelihood Analysis of Real-Space Power Spectrum and Bispectrum
astro-ph.CO 2020-10 conditional novelty 6.0 of 10

Joint power-spectrum plus bispectrum analysis of simulated halos tightens local f_NL constraints by about a factor of five to seven, but only if all stochastic amplitudes and non-Gaussian bias parameters are left free.
Galaxy bias and primordial non-Gaussianity: insights from galaxy formation simulations with IllustrisTNG
astro-ph.CO 2020-06 conditional novelty 6.0 of 10

For stellar-mass-selected galaxies, the standard universality relation b_phi = 2 delta_c (b_1 - 1) understates the PNG bias; the IllustrisTNG separate-universe simulations prefer b_phi = 2 delta_c (b_1 - p) with p bet...
Cubic Halo Bias in Eulerian and Lagrangian Space
astro-ph.CO 2018-02 conditional novelty 6.0 of 10

Cross-correlating cubic bias operators with simulated halo fields yields significant detections of local and non-local cubic halo bias, partly at odds with co-evolution predictions from a local Lagrangian bias model.
Structure Formation and Microlensing with Axion Miniclusters
astro-ph.CO 2017-07 conditional novelty 6.0 of 10

Microlensing data from Subaru HSC bound the fraction of axion dark matter collapsed into miniclusters to below roughly 8 percent for the QCD axion.
Linking High-Energy Cosmic Particles by Black-Hole Jets Embedded in Large-Scale Structures
astro-ph.HE 2017-03 conditional novelty 6.0 of 10

Black-hole jets in galaxy clusters can simultaneously account for ultrahigh-energy cosmic rays, IceCube's PeV neutrinos, and the non-blazar Fermi gamma-ray background.
The Gaussian streaming model and Lagrangian effective field theory
astro-ph.CO 2016-09 conditional novelty 6.0 of 10

The authors derive new expressions for the real-space correlation function, mean pairwise velocity, and pairwise velocity dispersion of biased tracers using Lagrangian EFT with a generalized bias expansion, and compar...
keV Sterile Neutrino Dark Matter from Singlet Scalar Decays: The Most General Case
hep-ph 2016-09 conditional novelty 6.0 of 10

A full distribution-function Boltzmann computation of sterile neutrino dark matter from scalar decays, combined with a half-mode power-spectrum analysis, shows that free-streaming estimates misclassify viable non-ther...
Precision measurement of the local bias of dark matter halos
astro-ph.CO 2015-11 accept novelty 6.0 of 10

Separate-universe N-body simulations give percent-level measurements of the linear, quadratic and cubic local halo bias, cross-checked against halo power spectrum and bispectrum fits.
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...
On the reach of perturbative descriptions for dark matter displacement fields
astro-ph.CO 2015-05 accept novelty 6.0 of 10

A residual, non-perturbative stochastic displacement in dark matter, uncorrelated with Lagrangian perturbation theory terms, limits perturbative predictions to about 1% on the nonlinear power spectrum at k = 0.2 h/Mpc...
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.
On Squeezed Limits in Single-Field Inflation - Part I
hep-th 2013-03 accept novelty 6.0 of 10

Excited states in single-field inflation can violate the squeezed-limit consistency condition only for wavenumber ratios below about 10^-2, making the violation unobservable in current and planned surveys.
Peak-Background Split, Renormalization, and Galaxy Clustering
astro-ph.CO 2012-12 accept novelty 6.0 of 10

The paper proves to all orders that the coarse-graining-dependent terms in the local bias expansion of galaxy clustering resum exactly into peak-background-split bias parameters defined as derivatives of the mean trac...
Evidence for Quadratic Tidal Tensor Bias from the Halo Bispectrum
astro-ph.CO 2012-01 conditional novelty 6.0 of 10

The matter-matter-halo bispectrum in N-body simulations requires a non-zero tidal tensor bias coefficient that grows with halo mass, so local density bias alone cannot model halo clustering.
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 ...
Effects of Scale-Dependent Non-Gaussianity on Cosmological Structures
astro-ph 2007-11 conditional novelty 6.0 of 10

Scale-dependent primordial non-Gaussianity can evade CMB bounds while changing cluster counts by 10 to 40 percent, and forthcoming cluster surveys can constrain its running parameter kappa.
Gravitational lensing effects on the gamma-ray burst Hubble diagram
astro-ph 2006-04 conditional novelty 6.0 of 10

Gravitational lensing magnification bias shifts the best-fit dark energy parameters from simulated Swift GRBs, but the true cosmology stays within the 68% confidence region.
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.
Spherical Collapse and Halo Formation in a Cosmology with Decaying Dark Matter and a Semi-Cosmographic Dark Energy
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

In a decaying dark matter plus semi-cosmographic dark energy cosmology constrained by DESI BAO and ShapeFit, the spherical collapse threshold stays close to the ΛCDM value while massive halo abundance shows pronounced...
Too Heavy to Hide: Gamma-Ray Constraints on Annihilating Dark Matter beyond Unitarity
hep-ph 2026-06 unverdicted novelty 5.0 of 10

Gamma-ray upper limits from five high-energy observatories constrain the annihilation cross sections of composite dark matter in the mass range 10^5--10^12 GeV.
Supermassive black hole seeds from direct collapse of CDM-curvature peaks
gr-qc 2026-05 unverdicted novelty 5.0 of 10

Broad compensated primordial CDM curvature peaks collapse directly into 10^3 to 10^6 solar mass black hole seeds at redshifts greater than 5, as modeled with LTB and Szekeres solutions.
A Variable-Slope Smooth-$k$ Filter for Modeling Halo Abundances with Damped and Oscillatory Power Spectra
astro-ph.CO 2026-02 conditional novelty 5.0 of 10

Introduces a variable-slope smooth-k filter within Press-Schechter theory that uses separate slope parameters for small and intermediate mass regimes, claiming a single parameter set matches N-body halo mass functions...

See all 92 Pith citations

How later papers used it

615 later papers cite it

Evidence: substantial. 608 of the citing papers share no author with it, and 302 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 894 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

21 later papers by other authors are listed below: 19 used its method or tool, 1 used its data or values and 1 compared its own results with it.

Uses its method or tool 19 of 259 papers with this label

By convolving the total power spectrum with a top-hat filter, we apply the PS formalism [ref] to compute the HMF, $dn/dM_h$, incorporating corrections for ellipsoidal collapse [ref] .

How do Primordial Black Holes change the Halo Mass Function and Structure? (2024) · PBH Halo Mass Function · 1.00
In the following, we assume a parametrisation of the halo mass function as given by the Sheth-Tormen formula [ref] : [equation] where $ρ_m$ is the mean matter density, $δ_c$ is the linearly extrapolated spherical collapse threshold which we compute using the formula by [ref] , and [equation] is the variance of the linear density field smoothed on a spherical volume of radius $R$ enclosing the mass $M=4/3πρ_m R^3$ with [equation] and $P(k,z)$ being the linear matter power spectrum at a given redshift $z$, which we compute assuming the linear transfer function from [ref] .

Cosmological Constraints from Galaxy Cluster Sparsity, Cluster Gas Mass Fraction and Baryon Acoustic Oscillations Data (2021) · Halo Sparsity · 1.00
Adopted this paper's prescription of the halo mass function for its constraint pipeline.

A general framework to test gravity using galaxy clusters V: A self-consistent pipeline for unbiased constraints of $f(R)$ gravity (2021) · 1.00 · paraphrased: the licence does not allow quoting
Used the Press-Schechter halo mass function to calculate the collapse fraction and its derivative, normalizing them with this paper's Sheth-Tormen correction.

Reconstruction of Cosmological Initial Density Field with Observations from the Epoch of Reionization (2023) · 1.00 · paraphrased: the licence does not allow quoting
Computed the effective bias of the FoF halos using this paper's Sheth and Tormen model.

The Void Size Function in Dynamical Dark Energy Cosmologies (2019) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's Sheth-Tormen model of ellipsoidal collapse to compute the halo mass function.

High-redshift, small-scale tests of ultralight axion dark matter using Hubble and Webb galaxy UV luminosities (2024) · 1.00 · paraphrased: the licence does not allow quoting
Calculated both the Tinker halo mass function and this paper's Sheth-Tormen halo mass function for each set of cosmological parameters.

Cold dark energy constraints from the abundance of galaxy clusters (2017) · 1.00 · paraphrased: the licence does not allow quoting
Initial conditions for peculiar velocity and density fields are set at $z=300$ with a Gaussian random field in Lagrangian space, before being evolved with the Zel'dovich approximation [ref] to match the mean collapse fraction for the conditional Sheth-Tormen halo mass function [ref] .

Probing ultra-light axions with the 21-cm Signal during Cosmic Dawn (2021) · The 21-cm hydrogen line · 1.00
Used this paper's bias formalism in a web tool to obtain fractional errors on its normalisation.

Galaxy luminosity functions at redshifts 0.6 to 1.2 in the Chandra Deep Field South (2022) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's Sheth-Tormen form for the first-crossing distribution.

Warm dark matter constraints from the JWST (2023) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the halo mass function from this paper's Sheth and Tormen formalism.

Probing power spectrum enhancement at small scales with the SKA (2026) · 1.00 · paraphrased: the licence does not allow quoting
Constructed a halo mass function following this paper.

Explaining the LIGO black hole mass function with field binaries: Revisiting Stellar Evolution at low Metallicity or Invoking Growth via gas accretion? (2021) · 1.00 · paraphrased: the licence does not allow quoting
Obtained the halo mass function and bias using the semi-analytic excursion set formalism of this paper.

Ultra-light axions and the kinetic Sunyaev-Zel'dovich Effect (2021) · 1.00 · paraphrased: the licence does not allow quoting
Assumed this paper's formulation to describe the CDM field halo mass function.

Joint constraints on thermal relic dark matter from strong gravitational lensing, the Lyman-$\alpha$ forest, and Milky Way satellites (2020) · 1.00 · paraphrased: the licence does not allow quoting
Used the Press-Schechter approach as modified by this paper's Sheth and Tormen prescription to compute the mass function.

Galaxy cluster number count data constraints on cosmological parameters (2011) · 0.99 · paraphrased: the licence does not allow quoting
In what follows, I use the [ref] dark matter halo mass function $dn(M,z)/dM$ — the number of dark matter halos of mass $M$ per unit mass per unit comoving volume at redshift $z$ — to compute the comoving number density of halos above a given halo mass threshold, [equation] and the comoving mass density in halos more massive than $$, [equation] These translate directly to upper limits on the statistics of galaxies through the straightforward assumption that the largest stellar content a halo can have given its cosmic allotment of baryons is $=$.

Stress Testing $\Lambda$CDM with High-redshift Galaxy Candidates (2022) · Assumptions · 0.91
Made use of this paper's Sheth and Tormen halo mass distribution function and linear bias in all its halo model calculations.

Multi-tracer intensity mapping: Cross-correlations, Line noise & Decorrelation (2021) · 0.99 · paraphrased: the licence does not allow quoting
Following [ref] , we can express the LF as [equation] where $d n / d M_h$ is the halo mass function [ref] and the observable $𝒪$ is taken to be either the 1500Å UV continuum magnitude, $M_UV$, or the H$α$ luminosity, $L_Hα$, in this paper.

Constraining bursty star formation histories with galaxy UV and H$\alpha$ luminosity functions and clustering (2024) · Luminosity function · 0.98
Presented results for three analytic forms of the CDM halo mass function, including this paper's Sheth-Tormen form.

Coupled Quintessence and the Halo Mass Function (2011) · 0.97 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 18 papers with this label

Used this paper's dark matter halo mass function at z about 6 to evaluate the duty cycle.

Subaru High-$z$ Exploration of Low-Luminosity Quasars (SHELLQs). XVIII. The Dark Matter Halo Mass of Quasars at $z\sim6$ (2023) · 0.84 · paraphrased: the licence does not allow quoting

Compares with it 1 of 29 papers with this label

Tested its own mass function in GR by comparing it to this paper's theoretical predictions.

N-body Simulations for f(R) Gravity using a Self-adaptive Particle-Mesh Code (2010) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

304 arXiv papers cite this in 446 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 · 228 sentences without a confident label

Contests it 3 papers

The conditional PS [ref] and ST [ref] mass functions can only be used to get the field at a resolution of $Δx ≥1$, and any finer resolution causes the values to turn out negative.

Accelerating HI density predictions during the Epoch of Reionization using a GPR-based emulator on N-body simulations · Variation · contest 0.71
We found that the Gaussian mass function of [ref] was a poor description of the simulation data.

Nonlinear clustering in models with primordial non-Gaussianity: the halo model approach · Conclusions · contest 0.84
While semi-analytical prescriptions, such as the conditional Press-Schechter [ref] and Sheth-Tormen [ref] mass functions, can be used to generate the field, they are known to be approximations of the more complex physics of halo formation and are quite inaccurate compared to N-body simulations at the redshifts relevant to reionization [ref] .

Conditioning halos on the tidal environment for fast and accurate HI power spectra during reionization · Introduction · contest 0.81

Builds on it 1 paper

[ref] further refined the mass function by calibrating the function $f(σ)$ motivated by [ref] against N-body simulation suites with a box size spanning from $∼100 ()^3$ to $∼1000 ()^3$ populated with $512^3$ to $1024^3$ particles, resulting in a fitting function with an accuracy of $∼5 %$.

Cosmology with Galaxy Clusters · Theoretical Background · 0.80

Uses its method or tool 134 papers

Mass function.– To compute the mass function, we use the Press-Schechter (PS) approach [ref] , as modified by Sheth and Tormen (ST) [ref] .

Galaxy cluster number count data constraints on cosmological parameters · IIa. Theory · 0.99
The total number of halos within that radius is N = ∫M_v ∫_0^z_s z_l ∫_0^ r_max r_l 2π(1+z_l)^2 r_l/H(z_l) n(z_l)/M_v , where $n(z_l)/M_v$ is the HMF which we compute using the Sheth-Tormen approximation [ref] with the CDM power spectrum [ref] .

The dark timbre of gravitational waves · Total lens effect · 0.98
The UVLF is computed by convolving the Sheth–Tormen halo mass function [ref] with the UV magnitude mapping using the Kennicutt relation [ref] and Gaussian scatter $$ [ref] (Supplemental Material B).

Breaking the UV Luminosity Function Degeneracy:Self-Interacting Dark Matter Constraints from Reionization Topology · Model and Data · 0.99

Uses its data or values 11 papers

We use $A=0.322$, $a=0.707$ and $p=0.3$ [ref] , for all the quintessence models considered.

Coupled Quintessence and the Halo Mass Function · Mass functions · 0.89
Below in our calculation, we would use the halo mass function from [ref] , specifically their Eq. (B3), and the halo bias from [ref] .

Cross-correlating the Thermal Sunyaev-Zel'dovich Effect and the Distribution of Galaxy Clusters · The SZ Effect and the SZ Power Spectrum · 0.89
Next we calculate the relative probability distribution of LMBHs as a function of logarithmic halo mass , [equation] Here the halo mass distribution $n/log[M_h]$ is taken as the Sheth-Tormen distribution at $z=0$ [ref] .

Dark Matter-Induced Low-Mass Gap Black Hole Echoing LVK Observations · Lower Mass Black hole distribution · 0.87

Compares with it 14 papers

Our fits are also consistent (to $< 1.5σ$) with the peak-background split estimates of the biases [ref] , where $b_n^L∼(b_1^L)^n$.

The One-Loop Power Spectrum of Fast Radio Burst Dispersion Measures · Validation with Hydrodynamical Simulations · 0.94
Since this procedure yields halo masses that are broadly comparable to spherical-overdensity (SO) definitions, we compare our results to the SO-calibrated fitting function of [ref] , which adopts a Sheth-Tormen-like form [ref] , [equation] where $ν≡δ_c^2/σ^2$ is the peak height and $δ_c≃1.686$ is the linear collapse threshold.

Non-linear Structure Formation in Planck+DESI Favoured Interacting Dark Energy Cosmologies · Simulations · 0.92
We test our mass function in GR firstly by comparing it to the theoretical predictions proposed by Sheth and Tormen [ref] , and by Tinker [ref] .

N-body Simulations for f(R) Gravity using a Self-adaptive Particle-Mesh Code · GR · 1.00

Cites it as background 33 papers

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 original mass function of Press & Schechter [ref] , derived with support of 1000-particle N-body simulations, has evolved into a number of forms with billion-particle simulation support, among them the Sheth-Tormen [ref] , Jenkins [ref] , Evrard [ref] , Warren [ref] and Tinker [ref] mass functions.

Sensitivity of galaxy cluster dark energy constraints to halo modeling uncertainties · Introduction · 0.99
Extensive effort has gone into modeling and calibrating this fully nonlinear observable in the standard cosmological constant plus Cold Dark Matter ($Λ$CDM) paradigm (e.g. [ref] ), and some work in this direction has been carried out for alternative dark energy models and gravity theories (e.g. [ref] ).

Cluster abundance in chameleon $f(R)$ gravity I: toward an accurate halo mass function prediction · Introduction · 0.90

Discussion

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