An excursion set model of hierarchical clustering: Ellipsoidal collapse and the moving barrier

Giuseppe Tormen (Padova); Ravi K. Sheth (Fermilab)

REVIEW

An excursion set model of hierarchical clustering: Ellipsoidal collapse and the moving barrier

Ravi K. Sheth (Fermilab), Giuseppe Tormen (Padova)

Astrophysics · Submitted 7 May 2001 · 26 references

Pith's verdict: The analytic first-crossing formula is the real contribution and it holds up; the 'wide variety of barriers' line is two shapes and an ad hoc truncation, which the authors half-admit.

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CitationGiuseppe Tormen (Padova), Ravi K. Sheth (Fermilab). "An excursion set model of hierarchical clustering: Ellipsoidal collapse and the moving barrier." arXiv:astro-ph/0105113 (2001). https://pith.science/paper/Y3HNQ6RI

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Worth your time, mainly for equation (4). The setup is familiar — excursion set, moving barrier from Sheth–Mo–Tormen — but what's new here is a usable closed-ish form for the first-crossing distribution of a barrier that isn't constant or linear: a Taylor series in B(S) truncated at five terms, dropped into the standard Gaussian expression. Equation (7) does the same for the two-barrier conditional problem. Those two formulae are why this paper gets cited, and they deserve it. The Monte Carlo check in Fig. 1 is a genuine independent test of the approximation, not a fit to the GIF simulations, and the conditional mass functions in Figs. 3–8 are out-of-sample once a is fixed from the unconditional function. Appendix B is a nice bonus: the demonstration that δ and r² are independent, with r² a χ²₅ variate, is clean algebra and justifies the one-dimensional reduction properly rather than hand-waving it. Credit for that. It also explains why the six-dimensional walk gives essentially the same answer, which is a real result, not a consistency check dressed up as one. On the reader's circularity worry: I think it's overweighted. The a≈0.7 calibration is acknowledged, and the comparison that matters is against the Monte Carlo, which owes nothing to the simulations. Where the stress-test note lands is the 'wide variety of moving barriers' claim in the abstract. Two shapes are tested, one rising and one falling, and the n=5 truncation has no convergence criterion. Fig. 14 is the tell: five terms and ten terms give visibly different fits to Monaco's barrier and the paper doesn't say which is right or how to pick n in advance. The authors do say they've only applied it to two shapes, so this is a phrasing problem more than a hidden one, but the abstract oversells. Related: the accuracy claim is stated to improve as the starting distance from the barrier grows, and for the ellipsoidal case that distance is only √a≈0.84 — the marginal regime. The non-unit normalization (the ~10% un-crossed fraction) is asserted, not checked against the simulations. The small-z conditional mass function failure in Section 2.5 is real and honestly reported; the scale-correlation explanation is plausible but untested. None of this sinks the unconditional mass function result. Send it out. Anyone building semi-analytic halo models or merger trees will use equations (4) and (7), and the referee should push on the truncation and the normalization, not on the physics.

Referee report

Significance

The paper supplies a closed-form, easily evaluated approximation to first-crossing distributions for a class of moving barriers, which is genuinely useful because no exact solution exists for the ellipsoidal barrier. The validation strategy is sound where it is independent: the Fig. 1 histogram is a direct Monte Carlo of the same barrier, so the agreement with the solid curve is a real test of Eq. (4) rather than a fit to the simulations, and the conditional mass functions in Figs. 3–8 are largely out-of-sample once a is fixed. The paper is also candid about failure: §2.5 and Figs. 9–10 report that the model has the wrong shape at small lookback times, and §3 reports that the six-dimensional walk does not repair it. I checked the algebra of Appendix B and Eq. (B6) is correct: with δ = −y1 and r² = (y2²+…+y6²)/5, the cross terms in d11d22 + d11d33 + d22d33 cancel exactly. The negative result in §3 is a real contribution and is, if anything, undersold.

Major objections (5)

[§2.1, Eq. (4); Fig. 14; abstract:s7] Eq. (4) is a Taylor series truncated at n = 5 with no stated convergence criterion, and the abstract's claim that the formulae are 'accurate for a wide variety of moving barriers' rests on exactly two tested shapes (the rising ellipsoidal barrier and Monaco's falling barrier). Fig. 14 makes the truncation dependence concrete: for Monaco's barrier the five-term and ten-term curves are visibly different, and the text does not say which is correct or how n should be chosen a priori. The paper does acknowledge that only two shapes were applied, so the concern is not concealment but the missing selection rule. Please add either a convergence diagnostic (e.g. the size of the n = 6 term relative to the sum, evaluated across the range of S that contributes) or an explicit statement of the domain of validity of the n = 5 truncation, and state plainly that 'wide variety' is an extrapolation from two cases.
[§2.1, Fig. 1; Appendix A] Eq. (4) is not normalized to unity, and the deficit is given a physical interpretation — the fraction of walks that never cross is identified with the ~10% of mass not in bound haloes. This is a falsifiable prediction that the paper does not test. Fig. 1 shows the solid curve lying below the dashed curve, but the size of the deficit is never quoted, and §2.4 uses the same machinery for n(m|δ) without checking whether the predicted unbound fraction is consistent with the simulations. Since the authors themselves note that the simulation unbound fraction is uncertain because of resolution, a direct comparison may not be decisive, but the predicted deficit should at least be stated numerically for the ellipsoidal barrier and compared with the simulation value, with the caveat noted.
[§2.1 and §4, Fig. 13] The parameter a is fixed at 0.707 by normalization to the simulations, and §4/Fig. 13 then states that changing a to 0.75 'appears to be all that is necessary to reduce the discrepancy' with Jenkins et al. substantially. All the conditional and local-density comparisons in Figs. 3–10 nonetheless use a = 0.707. Because a enters the barrier height and hence every crossing distribution, the reader cannot tell how much of the agreement or disagreement in Figs. 3–10 is sensitive to this choice. Please quantify the effect of a = 0.75 on at least one conditional mass function panel, or state explicitly that the qualitative conclusions are unchanged and by how much the curves shift.
[§3 and §4] The conclusion that the small-lookback discrepancy is caused by the neglect of correlations between scales is reached by elimination, not by direct test. The six-dimensional walk of §3 still takes independent steps in σ², so it shares the neglect of scale correlations with the one-dimensional walk; its failure is therefore consistent with both candidate explanations (over-simplified one-dimensional parametrization and neglected correlations), and rules out the first only if the six-dimensional walk is accepted as a faithful representation of ellipsoidal collapse. The inference is reasonable but should be presented as an elimination argument with its two premises stated, and the alternative — a direct test using correlated or constrained walks — should be named as the decisive check that this paper does not perform.
[§2.4, Eq. (8)] Eq. (8), the fitting formula relating the local nonlinear density δ to an effective δ0, is load-bearing for the entire local-density analysis of §2.4 and Figs. 5–8, yet its accuracy is asserted ('we have checked that the following simple modification … is reasonably accurate for all cosmologies of interest') without a figure or a quantitative error estimate. Please show the comparison behind this statement — the range of δ over which the fit holds and the fractional accuracy — since the visual agreement in Figs. 5–8 cannot separate error in Eq. (8) from error in the conditional mass function.

Minor objections (7)

[§2, notation] The symbol ν is defined twice with different meanings: ν ≡ [δsc(z)/σ(m)]² in Eq. (1) and ν ≡ S*/S after Eq. (3). Similarly S is used for σ² throughout. Please add a short notation paragraph or a table reconciling these, since the reader must track the conversions across Eqs. (1)–(7).
[§2.1, Eq. (5)] The coefficient 0.094 in the second line of Eq. (5) appears without derivation. State that it is a numerical evaluation of the truncated series at the ellipsoidal barrier parameters, so the reader does not look for an analytic origin.
[§2.2] The algorithm aν = |x|^3.6/(1+|x|^1.6) is claimed accurate to within a percent over 0.01 ≤ ν ≤ 100, but no figure or table supports the claim. A small inset showing the ratio of generated to target cumulative distributions would make this useful rather than asserted.
[Appendix B] The statement that the independence of δ and r² 'does not seem to have been noticed before' is a priority claim that should be softened. The independence follows immediately from Eq. (B1), in which δ = −y1 while r² involves only y2…y6, and this structure is implicit in the Bardeen et al. (1986) decomposition the authors cite. I have verified Eq. (B6) and it is correct; the issue is only the novelty framing.
[References] Several works cited in the text are missing from the reference list: Bardeen, Bond, Kaiser & Szalay (1986) and Catelan & Theuns (1998) in Appendix B, and Whitmore (1978) in Appendix A. In addition, the Sheth & Pitman entry lacks a year, and 'AA, 337, 96' for Del Popolo & Gambera should read A&A.
[Figs. 2 and 13] Fig. 2's claim that 'for S/S* < 0.5 the height of the barrier is approximately constant' should be quantified, since the subsequent argument about the effective redshift difference rests on it. In Fig. 13 the caption should identify which of the several dashed curves is the new a = 0.75 variant, as the text introduces it only in passing.
[Typographical] There are several typographical errors: 'graviational' (Introduction), 'appraoch' (§4), 'happes' (Appendix A), 'idenﬁnfying' (§2.5), 'an small z' (§2.5), and 'publically' (Acknowledgements).

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The paper

The paper claims

The paper argues that dark matter halo abundances and progenitor counts are better described by random walks crossing a barrier that rises with the smoothing scale than by the classical flat barrier of spherical collapse, and gives a closed

The reading

The excursion set approach turns the counting of collapsed dark matter haloes into a random-walk problem: a walk starts at zero overdensity and a halo forms where the walk first crosses a barrier. For most of the model's history that barrier was taken to be flat, which is what spherical collapse implies. This paper asks what happens when the barrier rises with the variance, the shape ellipsoidal collapse implies, on the reasoning that a low-mass patch must hold itself together against stronger tides than a massive one, and it supplies a formula that works across a whole family of such rising barriers rather than for one hand-tuned shape. If the claim holds, one short expression yields the abundance of haloes, the abundance of their progenitors inside a given parent, and the way halo counts shift between underdense and overdense regions, without running a simulation for each case.

What carries the argument

The load-bearing object is the moving barrier $B(S,z)=sqrt{a},delta_{sc}(z)[1+beta(anu)^{-alpha}]$ with $nu equiv [delta_{sc}(z)/sigma(m)]^2$, together with the ansatz that the first-crossing distribution is the constant-barrier Gaussian form multiplied by a truncated Taylor series $T(S)=sum_{n=0}^{5}(-S)^n/n!,partial^n B/partial S^n$ of the barrier. The same substitution rule, $B(s)to B(s)-B(S)$ and $Sto s-S$, carries the formula over to the two-barrier problem that defines the conditional mass function, which is what makes progenitor counts and local-density mass functions accessible without a fresh numerical fit for every parent mass. Two auxiliary devices do real work: a sampling trick that turns a Gaussian variate $x$ into a mass via $anu=|x|^{3.6}/(1+|x|^{1.6})$, accurate to about a percent, and a six-dimensional random walk built from the deformation tensor, which is used to check that replacing the scatter in the shear field by its mean is a fair reduction.

Extended reading notes

Core claim

The central claim is that the first-crossing distribution of a Brownian walk absorbed on a moving barrier $B(S)$ is well approximated by $f(S)=|T(S)|e^{-B(S)^2/2S}/(Ssqrt{2pi S})$, where $T(S)$ is the sum of the first five terms of the Taylor expansion of $B(S)$ in $S$; the expression is exact for constant and linear barriers, where the series terminates, and remains accurate for the ellipsoidal barrier $B=sqrt{a},delta_{sc}(z)[1+beta(anu)^{-alpha}]$ with $beta approx 0.485$, $alpha approx 0.615$ and $a approx 0.7$. Feeding it the ellipsoidal barrier reproduces both the unconditional mass function and the conditional (progenitor) mass function measured in the GIF $N$-body simulations better than the constant-barrier spherical-collapse prediction, and the same machinery, with local density mapped to an effective redshift, gives the mass function inside dense and underdense cells. The authors are explicit that the improvement holds at large lookback times: at small lookback times, and for progenitors more than half the parent mass, the moving-barrier conditional mass function has the wrong shape, and they trace that to the excursion set neglect of correlations between smoothing scales rather than to the barrier shape itself.

Load-bearing premise

The premise everything rests on is that the collapse condition can be written as a barrier depending only on the smoothing scale, with the scatter a real patch feels from its surrounding shear field averaged away, and that the barrier's overall amplitude $a approx 0.7$ is legitimately fixed against the same simulations the resulting formula is then tested on.

Original abstract

The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the nonlinear processes of collapse and virialization are modelled. We present simple analytic approximations which allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high density regions are collapsed haloes viewed at small lookback times. Although we have only applied our analytic formulae to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.

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The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the nonlinear processes of collapse and virialization are modelled. We present simple analytic approximations which allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high density regions are collapsed haloes viewed at small lookback times. Although we have only applied our analytic formulae to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.

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	The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately.	background 1.00	0.10	0.04	asserted 0.98
s1	The predictions of this approach depend on how the nonlinear processes of collapse and virialization are modelled.	background 0.99	0.06	0.02	asserted 1.00
s2	We present simple analytic approximations which allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse.	contribution 0.88	0.95	0.06	asserted 0.97
s3	In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today.	contribution 0.98	0.93	0.06	asserted 0.99
s4	We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times.	result 1.00	0.94	0.78	asserted 1.00
s5	Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function which quantifies the tendency for massive haloes to populate denser regions than less massive haloes.	result 0.38	0.85	0.07	asserted 0.54
s6	This happens because low density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high density regions are collapsed haloes viewed at small lookback times.	interpretation 1.00	0.29	0.04	asserted 0.92
s7	Although we have only applied our analytic formulae to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers.	result 0.82	0.94	0.10	asserted 1.00
s8	We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.	future work 1.00	0.49	0.04	hedged possibility 0.97

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.55.

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references and citing sentences
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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.** First crossing distributions and the universal unconditional halo mass function. Histogram shows the distribution obtained by simulating random walks which are absorbed on the ellipsoidal collapse moving barrier; solid curve shows our analytic approximation to this distribution (equation 4). Dashed curve shows the distribution which fits the halo mass function in cosmological simulations (equation 2), and dotted curve shows the distribution associated with spherical collapse. and linear (Sheth 1998 and Appendix A of this paper) barriers. These solutions suggest the following…

**Figure 2.** Examples of random walks used to construct the conditional mass functions associated with the ellipsoidal collapse moving barriers at z = 0 (dotted curve) and z = 2 (solid curve). Bond et al. (1991) and Lacey & Cole (1993) argued that conditional mass functions could be estimated by considering the successive crossings of boundaries associated with different redshifts. The first crossing of two constant barriers of different heights has an analytic solution, so they were able to provide analytic estimates for the conditional mass function associated with the spherical collapse…

**Figure 3.** Excursion set conditional mass functions at z, for parent haloes identified at the present time. The two sets of curves in each panel are for parent haloes in the mass range 1 ≤ M/M∗ ≤ 2 (upper curves) and 16 ≤ M/M∗ ≤ 32 (lower curves); the upper curves have been shifted upwards by a factor of ten for clarity. Symbols with error bars show the distributions measured in the ΛCDM simulations, histograms show the result of generating the first crossing distribution by simulating an ensemble of 104 random walks, smooth solid curves show the analytic approximation discussed in the text,…

**Figure 4.** Conditional mass functions in the SCDM simulations (symbols with error bars). The two sets of curves in each panel show parent haloes with masses in the range 1 ≤ M/M∗ ≤ 2 (upper curves) and 16 ≤ M/M∗ ≤ 32 (lower curves); the upper curves have been shifted upwards by a factor of ten for clarity. Dotted curves show the spherical collapse prediction (Lacey & Cole 1993), dashed curves show the distribution one gets by rescaling the unconditional mass function (equation 2), and solid curves show our analytic approximation to the random walk with moving-barrier simulations (equation 7).…

**Figure 5.** Mass functions as a function of local density in the SCDM simulations (symbols with error bars), plotted as a function of relative mass, so as to resemble the conditional mass functions presented earlier. Dotted curves show the spherical collapse prediction, dashed curves show the distribution one gets by rescaling the unconditional mass function (equation 2), and solid curves show our analytic approximation to the random walk with moving-barrier simulations (equation 7). The curves have been offset upwards by a factor of ten and a hundred, in the case of the middle and topmost…

**Figure 9.** Rescaled conditional mass functions in the SCDM model. Panels show the different redshift bins we studied earlier. Symbols in each panel show the different mass ranges we considered. Dotted line shows the constant barrier prediction (in these variables it is the same as the unconditional mass function), dashed line shows the result of rescaling the actual unconditional mass function, and solid curves show the result of rescaling the moving barrier predictions. the rescaled conditional mass function constructed using a friends-of-friends algorithm look very similar to the spherical…

**Figure 10.** Same as the previous figure, but for ΛCDM. the mass which is got by summing up the masses of all the progenitor subclumps (which may depend on redshift). The figures above are for the case in which a subclump is any clump which donates at least one particle to the final object, and the parent mass is defined as the mass at the final time (so it is independent of redshift). Although the actual conditional mass functions depend slightly on which combination of the above choices we make, the generic results shown above are independent of this choice. Before moving on, we think it…

**Figure 11.** Distribution of overdensities and scales at which the six-dimensional random walks crossed the ellipsoidal collapse barrier. Solid curve shows the approximation used by Sheth, Mo & Tormen (2001) in their, considerably simpler, one-dimensional random walks. patch may differ from patch to patch. Appendix B provides a simple prescription for choosing a set of patches which have the correct ensemble averaged properties—in essence, this requires studying the first crossing distribution of sixdimensional random walks. Because a six-dimensional walk is computationally expensive, rather…

**Figure 13.** Comparison of the various mass functions described in this paper, with the one which fits the Hubble volume simulations presented in Jenkins et al (2001). Upper curves show the residuals between our analytic formulae and the Jenkins et al. fitting formula

**Figure 14.** The unconditional mass function associated with a model of collapse which was presented in Monaco (1997a,b). Whereas the height of the barrier studied in our paper increases with distance, the one proposed by Monaco decreases with distance. Despite the dramatic differences between the two barriers, inserting this shape into our equation (4) provides a good fit (solid curve) to the exact result (histogram). their shape is not so different from ours, whereas Monaco’s really is quite different. The fact that this barrier decreases with S means that all walks are guaranteed to cross…

Showing 10 of the paper's 14 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/0105113 by the authors.

References

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The IllustrisTNG simulation yields calibrated halo-model ingredients for cosmic neutral hydrogen, including a halo HI mass function, density profiles, velocity dispersions, bias, and shot-noise, from redshift 0 to 5.
Constraining Dark Matter-Baryon Scattering with Linear Cosmology
astro-ph.CO 2013-11 accept novelty 6.0 of 10

Combining Planck CMB and SDSS Lyman-alpha data, elastic dark matter-baryon scattering is bounded to be far too weak for a proton in a Milky Way-like halo to scatter off dark matter within the age of the Universe.
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.
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...
Configuration Requirements for 21-cm Forest Background Quasar Searches with the Moon-based Interferometer
astro-ph.CO 2025-04 conditional novelty 5.0 of 10

A Moon-based radio array with roughly SKA-like collecting area could detect enough high-redshift radio-loud quasars at z≈10 to enable 21-cm forest studies, while smaller arrays reach only z≈5-6.
Modelling the matter bispectrum at small scales in modified gravity
astro-ph.CO 2019-09 conditional novelty 5.0 of 10

No existing matter-bispectrum model is accurate enough for Stage IV lensing surveys across the tested modified gravity theories, and a halo-model-corrected fitting formula is the most accurate option tested.
Constraining Non-Cold Dark Matter Models with the Global 21-cm Signal
astro-ph.CO 2018-04 conditional novelty 5.0 of 10

The redshift at which the first stars lit up the 21-cm line, if EDGES measured it correctly, excludes thermal warm dark matter below 6.1 keV, ultra-light axions below 8e-21 eV, and hot fractions above 17 percent.
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.
Limits on Dark Matter Annihilation Signals from the Fermi LAT 4-year Measurement of the Isotropic Gamma-Ray Background
astro-ph.CO 2015-01 accept novelty 5.0 of 10

No dark matter annihilation signal appears in the Fermi LAT isotropic gamma-ray background, and the resulting cross-section limits reach tens of TeV masses, the strongest Fermi LAT bounds reported at those energies.
Prospects for constraining the shape of non-Gaussianity with the scale-dependent bias
astro-ph.CO 2012-04 conditional novelty 5.0 of 10

A Dark Energy Task Force stage IV-like galaxy survey could constrain the squeezed-limit slope of the primordial bispectrum, distinguishing local-like from orthogonal-like shapes when the non-Gaussian amplitude is of o...
Sensitivity Projections for Dark Matter Searches with the Fermi Large Area Telescope
astro-ph.HE 2016-05 accept novelty 4.0 of 10

With 15 years of Fermi LAT data and roughly 60 dwarf spheroidal galaxies, indirect dark matter searches would reach the thermal relic annihilation cross section for WIMP masses above 400 GeV in the b-bbar channel.
The nature of the Diffuse Gamma-Ray Background
astro-ph.CO 2015-02 accept novelty 3.0 of 10

The review concludes that known astrophysical populations can plausibly account for most of the Fermi LAT diffuse gamma-ray background, leaving limited but not excluded room for a dark matter contribution.

How later papers used it

233 later papers cite it

Evidence: substantial. 225 of the citing papers share no author with it, and 112 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 354 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

25 later papers by other authors are listed below: 25 used its method or tool.

Uses its method or tool 25 of 89 papers with this label

Used a halo mass function conditioned on pixel overdensity following this paper to statistically sample unresolved low-mass halos.

Simulating the Large-Scale Structure of HI Intensity Maps (2015) · 1.00 · paraphrased: the licence does not allow quoting
Computed the comoving halo mass function as a function of redshift and halo mass using the excursion set approach from another work accounting for ellipsoidal collapse from this paper.

Can primordial black holes explain the overabundance of bright super-early galaxies? (2025) · 1.00 · paraphrased: the licence does not allow quoting
Followed this paper for the halo mass function and another work for the halo bias in its halo model approach.

Beyond $\Lambda$CDM with HI intensity mapping: robustness of cosmological constraints in the presence of astrophysics (2019) · 1.00 · paraphrased: the licence does not allow quoting
Computed the halo mass function using the Sheth-Tormen multiplicity function.

Soliton Merger Rates and Enhanced Axion Dark Matter Decay (2023) · 1.00 · paraphrased: the licence does not allow quoting
Utilized a sub-grid prescription based on this paper's conditional ellipsoidal mass function to compute the halo field.

A fast method of reionization parameter space exploration using GPR trained SCRIPT (2023) · 1.00 · paraphrased: the licence does not allow quoting
Populated each grid cell with halos using a conditional ellipsoidal collapse model from this paper.

SCRIPT in the Cosmic Dawn: Distinguishing Redshift-Evolving Galaxy Populations with 21-cm Fluctuations (2026) · 1.00 · paraphrased: the licence does not allow quoting
Applied this paper's conditional mass function to populate grid cells with halos above a minimum mass.

Unbiased constraints on reionization model parameters in presence of the foreground wedge (2018) · 1.00 · paraphrased: the licence does not allow quoting
Computed the collapsed mass fraction in haloes using a subgrid prescription based on this paper's conditional ellipsoidal collapse model.

Probing the Physics of Reionization Using kSZ Power Spectrum from Current and Upcoming CMB Surveys (2023) · 1.00 · paraphrased: the licence does not allow quoting
Computed the collapsed mass fraction in haloes using a subgrid prescription based on this paper's conditional ellipsoidal collapse model.

Disentangling patchy reionization signatures from primordial gravitational waves using CMB $E$-mode and $B$-mode polarization (2023) · 1.00 · paraphrased: the licence does not allow quoting
We now elaborate on the computation of each of the components that goes into Eq.[ref]: (i) In our calculations, we use the Sheth-Tormen HMF [ref]

Constraints on the fuzzy dark matter mass window from high-redshift observables (2024) · UV luminosity functions · 1.00
We obtain the HMF by using the classical [ref] formalism, which uses an ellipsoidal model for perturbation collapse[Throughout the paper, we assume a flat Universe with the following cosmological parameters: $Ω_M = 0.3075$, $Ω_Λ = 1- Ω_M$, and $Ω_b = 0.0486$, $h=0.6774$, $σ_8=0.826$, where $Ω_M$, $Ω_Λ$, and $Ω_b$ are the total matter, vacuum, and baryon densities, in units of the critical density; $h$ is the Hubble constant in units of $100$, and $σ_8$ is the late-time fluctuation amplitude parameter [ref] .].

On the stunning abundance of super-early, massive galaxies revealed by JWST (2022) · Building the luminosity function · 0.99
A key connection between astrophysics and PMFs is encoded in the halo mass function $dn_h/dlnM$, which we compute from the matter power spectrum [ref] within the excursion-set formalism formalism [ref] with ellipsoidal collapse [ref] .

JWST Constraints on Primordial Magnetic Fields (2026) · Introduction · 0.99
Started from this paper's Sheth-Tormen halo mass function and assumed each halo to contain a gas mass linked to halo mass through the cosmological baryon-to-dark matter ratio.

Are the Newly-Discovered $z \sim 13$ Drop-out Sources Starburst Galaxies or Quasars? (2022) · 0.99 · paraphrased: the licence does not allow quoting
Used this paper's Sheth-Tormen mass function, as in another work but with a fit calibrated to high-redshift N-body simulations from another work.

Insights from HST into Ultra-Massive Galaxies and Early-Universe Cosmology (2023) · 0.92 · paraphrased: the licence does not allow quoting
Adjusted this paper's mass function for finite volume effects by correcting the mass variance and shifting the collapse threshold by the subregion overdensity.

The halo mass function and inner structure of ETHOS haloes at high redshift (2021) · 0.92 · paraphrased: the licence does not allow quoting
Generated dark matter halo mass functions as a function of redshift following this paper, then assigned stars to each halo.

Feedback-regulated star formation and escape of LyC photons from mini-haloes during reionisation (2016) · 0.81 · paraphrased: the licence does not allow quoting
Adopted this paper's Sheth-Tormen halo mass function model.

The Star Formation Efficiency During Reionization as Inferred from the Hubble Frontier Fields (2023) · 0.98 · paraphrased: the licence does not allow quoting
Adapted the collapse formalism developed by Sheth and Tormen, giving a halo mass function with specified parameters.

First Constraints on Small-Scale Non-Gaussianity from UV Galaxy Luminosity Functions (2020) · 0.99 · paraphrased: the licence does not allow quoting
Used this paper's Sheth-Tormen form of the comoving number density of dark matter haloes per unit logarithmic halo mass to estimate halo masses from Lyman-alpha luminosities.

Intergalactic Lyman-$\alpha$ haloes before reionization are detectable with JWST (2024) · 0.99 · paraphrased: the licence does not allow quoting
Used the Sheth-Tormen mass function, which extends the Press-Schechter formalism to ellipsoidal collapse as in this paper.

Milky Way satellite velocities reveal the Dark Matter power spectrum at small scales (2023) · 0.96 · paraphrased: the licence does not allow quoting
Used this paper's Sheth-Tormen approximation for the halo mass function following another work for all dark matter models.

Can EDGES observation favour any dark matter model? (2019) · 0.93 · paraphrased: the licence does not allow quoting
Made use of the Sheth-Tormen mass function given by this paper.

New Roads to the Small-Scale Universe: Measurements of the Clustering of Matter with the High-Redshift UV Galaxy Luminosity Function (2021) · 0.99 · paraphrased: the licence does not allow quoting
Used the Sheth-Tormen mass function given by this paper in its fiducial model.

GALLUMI: A Galaxy Luminosity Function Pipeline for Cosmology and Astrophysics (2021) · 0.97 · paraphrased: the licence does not allow quoting
Used a semi-analytical prescription based on sampling a conditional mass function given by this paper's fitting form.

Photon number conservation and the large-scale 21 cm power spectrum in semi-numerical models of reionization (2018) · 0.93 · paraphrased: the licence does not allow quoting
Regarding the halo mass function, we assume the [ref] prescription (Eq. [ref]) with modified parameter $q=0.85$.

A halo model approach for the 21-cm power spectrum at cosmic dawn (2020) · Results · 0.87

How the literature uses it

109 arXiv papers cite this in 172 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 · 92 sentences without a confident label

Uses its method or tool 45 papers

Confronting JWST Massive Galaxies. — By assuming that each dark-matter halo at high $z$ contains a single central galaxy, the expected number of galaxies with stellar mass larger than some observational threshold $M_⋆^obs$ can be obtained as [ref] : [equation] where Vol $=Ω(r_max^3 - r_min^3)/3$ is the comoving survey volume, determined by the surveyed redshift range (from $r(z_min)$ to $r(z_max)$) and the survey angular coverage $Ω$, and $dn/dM_h$ is the halo mass function[We will use the Sheth-Tormen [ref] mass function as in Ref. [ref] , but with a fit calibrated to high-redshift N-body simulations from Ref. [ref] .].

Insights from HST into Ultra-Massive Galaxies and Early-Universe Cosmology · Introduction · 0.92
To this purpose we use the Sheth & Tormen mass function [ref] dndM = -√(2ãπ)A ł[1+ł(ã δ_c^2D^2σ_M^2)̊^-p]̊ ρ_mM^2δ_cDσ_M × dlnσ_Mdln M expł(-ã δ_c^22D^2σ_M^2)̊ , where $ã=0.707$ [We changed the commonly adopted notation to avoid confusion with the scale factor $a$.], $p=0.3$, $A=0.2162$, $δ_c$ is the linear critical density contrast derived in Section [ref], $D=δ^L_m/δ^L_m(a=1)$ is the linear growth factor, and $σ_M$ is the variance of the linear matter power spectrum defined as [ref] σ_M^2 = 12π^2∫^∞_0 dk k^2W^2(kR) P^L(k) , where the window function is defined as W(kR) = 3sin(kR)-kRcos(kR)(kR)^3 , being $R$ the comoving radius enclosing the mass $M=4π3ρ_mR^3$.

Growth of non-linear structures and spherical collapse in the Galileon Ghost Condensate model · Mass function · 0.98
Since our simulations do not resolve all the low mass halos that are expected to host HI, we have used the coarse density field in each pixel to statistically sample these halos using a halo mass function conditioned on the overdensity of the pixel following [ref] .

Simulating the Large-Scale Structure of HI Intensity Maps · Conclusions · 1.00

Uses its data or values 2 papers

The bias terms are given as (for the Sheth-Tormen and Tinker mass functions respectively) [ref] [equation] where $a_2=-17/21$ and we list other constants below [equation] We re-emphasize that all these constants are fit to Newtonian, $Λ$CDM simulations.

Modelling the matter bispectrum at small scales in modified gravity · Details of the halo model spectra · 0.80
The corresponding halo masses and number densities ($M_h, dn/dlnM_h$) are approximately ($7 ×10^10 , 7 ×10^-6 Mpc^-3$) at $z = 12$, ($2 ×10^11 , 4 ×10^-6 Mpc^-3$) at $z = 10$, and ($2 ×10^12 , 9 ×10^-7 Mpc^-3$) at $z = 7$ [ref] .

Grain-size evolution and rapid dust growth in high-redshift galaxies · Model parameters and fiducial setup · 0.85

Compares with it 9 papers

Our main arguments for the validity of our approach are the good agreement of the resulting unconditional halo mass function (see Figure [ref]) and large scale bias (see Figures [ref] and [ref]) with the analytical results from [ref] .

Simulating the Large-Scale Structure of HI Intensity Maps · HI intensity maps · 0.97
In Fig. [ref] we compare our HMFs calculated from $N$-body simulations with HMFs from the Sheth-Tormen (ST) semi-analytic formulae [ref] .[See [ref] for other semi-analytic fits to the mass function, which we found to work less well for models with massive neutrinos.] The ST fit is based on the fact that, as first pointed out by Press and Schechter [ref] , the HMF can be written as [equation] with $ν≡[δ_sc(z)/σ(M)]^2$, where $δ_sc(z)=1.686$ is the overdensity required for spherical collapse at $z$, and $ρ̅= Ω_m ρ_c$. $dn(M,z)$ is the number density of halos in the mass interval $M$ to $M + dM$.

Neutrinos in Non-linear Structure Formation - The Effect on Halo Properties · The halo mass function · 0.99
Note that this expression matches the first two terms of [ref] which are equivalent to [ref] for $D_B=0$.

Spherical collapse and halo mass function in f(R) theories · Halo mass function prediction for uncorrelated random walk and generic barrier · 1.00

Cites it as support 1 paper

However, [ref] showed that $q_λ$ and $δ_m$ encode complementary physical information and follow different distributions.

Astrophysical systematics in Kinematic Lensing: quantifying an Intrinsic Alignment analog · Characterizing the environment · 0.99

Cites it as background 14 papers

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
A large amount of research has been devoted to identifying mass functions that accurately describe the simulations and data of galactic halos, see, e.g., [ref] .

Toward More Realistic Mass Functions For Ultradense Dark Matter Halos · Introduction · 0.98
Over the years, several layers of complexity have been added to the initial model, with the inclusion of the peak constraint [ref] , halos preferentially form at peaks in the initial field, the excursion set constraint to solve the cloud in cloud problem [ref] , and more realistic collapse models, halos are likely to form in a ellipsoidal fashion [ref] .

Weighing neutrinos with the halo environment · Analytical considerations on environmental bias · 0.84

Discussion

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