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Large-scale non-Gaussian mass function and halo bias: tests on N-body simulations

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M. Grossi (MPA), L. Verde (ICREA&ICE), C. Carbone (ICE), K. Dolag (MPA), E. Branchini (Roma 3), F. Iannuzzi (Bologna), S. Matarrese (Padova), L. Moscardini (Bologna)

Cosmology and Nongalactic Astrophysics · Submitted 12 Feb 2009 · 55 references

Pith's verdict: A solid, well-scoped calibration of the non-Gaussian mass function and halo bias, whose one real weakness is that q = 0.75 is imported rather than fitted and the agreement is judged without error bars.

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CitationC. Carbone (ICE), E. Branchini (Roma 3), F. Iannuzzi (Bologna), K. Dolag (MPA), L. Moscardini (Bologna), L. Verde (ICREA&ICE), M. Grossi (MPA), S. Matarrese (Padova). "Large-scale non-Gaussian mass function and halo bias: tests on N-body simulations." arXiv:0902.2013 (2009). https://pith.science/paper/YRQXHGJB
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The paper

The paper claims

The analytic formulas that predict how many dark-matter halos form, and how strongly they cluster, under non-Gaussian initial conditions match large N-body simulations once a single collapse-threshold rescaling of about 0.75 is applied.

The reading

Primordial non-Gaussianity — a departure from the simplest Gaussian random field of density fluctuations — is one of the few handles we have on how inflation actually happened, and the most promising way to measure it is through the abundance and clustering of massive dark-matter halos. The analytic machinery for that prediction is old and rests on the Press-Schechter approximation: spherical collapse, sharp k-space filtering, and a Gaussian probability distribution. This paper asks whether those formulas survive when the initial conditions are non-Gaussian, by running a suite of large N-body simulations with $f_{NL} = 0, -200, -100, 100, 200$ in a $(1200h^{-1}{Mpc})^3$ box and comparing the halo counts and the scale-dependent bias directly against the analytic predictions. The answer is that both the mass-function corrections of Matarrese et al. (2000) and LoVerde et al. (2008) and the bias correction of Matarrese & Verde (2008) fit the simulations well, provided the density threshold for collapse is rescaled — by $1/2$ in the exponent for the mass function, i.e. $q^{3/2}$ overall, and by $q$ for the bias — with $q$ about 0.75. If that holds generally, then the analytic expressions, not simulation-calibrated fitting formulas, can be trusted for cosmologies and $f_{NL}$ values nobody has simulated, which is what survey forecasts need.

What carries the argument

The load-bearing object is a single number, $q = 0.75$, transplanted from the Sheth-Tormen ellipsoidal-collapse fit to Gaussian simulations into the non-Gaussian correction. In the mass function it enters as $q^{3/2}$ in the exponential of the ratio $R_{NG}$; in the bias it enters as a multiplicative factor $q$ on the fractional correction $2 f_{NL} M(k)$. The justification is structural: the non-Gaussian correction in the Matarrese & Verde (2008) derivation carries a factor of $^3/^3_M$ with $= /$, and that factor contains one Lagrangian Gaussian bias to the second power plus an extra $/^2$; if the Gaussian bias is itself corrected by $q$ for ellipsoidal collapse, the non-Gaussian correction inherits $q$ too. The paper is careful to distinguish its $q$ (from the bias) from the alternative of replacing $c$ with the full Sheth-Tormen critical density, which would give $q$ rather than $q$ — Afshordi & Tolley (2008) took that route, and the simulations here prefer the former.

Extended reading notes

Core claim

The paper's central claim is that the Press-Schechter-based description of non-Gaussian halo statistics is correct, but only after a correction inherited from ellipsoidal collapse. Specifically, the non-Gaussian mass function should be written as the simulated Gaussian mass function multiplied by a ratio $R_{NG}$, computed from either the Matarrese et al. (2000) saddle-point expression or the LoVerde et al. (2008) Edgeworth expression, with the collapse threshold replaced by $q$-corrected values: effectively $n(M,z,f_{NL}) = n^{sim}_G(M,z) R_{NG}(M,z,f_{NL})$ with $q = 0.75$ in the threshold. For the halo bias, the fractional non-Gaussian correction $2 f_{NL} q M(k)$ — with the standard formula multiplied by the same $q$ — reproduces the simulation's scale, mass, and redshift dependence on scales $k < 0.03h/{Mpc}$. The authors verify that the Gaussian mass function itself is well fit by Sheth & Tormen (1999) across $z = 0$ to $3.23$, that their non-Gaussian mass-function ratios agree with two independent simulation suites (Pillepich et al. 2008; Desjacques et al. 2008) once the differing $f_{NL}$ conventions are reconciled, and that the mass dependence of the bias correction is negligible for $M > 10^{13} M_/h$. They also flag a bookkeeping hazard: papers that use the CMB convention for $f_{NL}$ or fail to extrapolate the skewness to $z=0$ carry a factor of about 1.3, and the $q$ correction contributes roughly $0.75^{3/2} = 0.65$ for the mass function and 0.75 for the bias — a coincidence that makes the bias forecasts accidentally right while the mass-function comparisons look discrepant.

Load-bearing premise

The whole calibration rests on the assumption that a single number, $q = 0.75$, taken from a fit to Gaussian ellipsoidal-collapse simulations, can be carried unchanged into the non-Gaussian correction at every mass, redshift, and $f_{NL}$ value — an ansatz the paper states as such and never derives independently for non-Gaussian fields.

Editorial extensions

If this is right

  • Forecasts for primordial non-Gaussianity from cluster and galaxy surveys can use the analytic expressions (Eqs. 5, 6, 7, 9) rather than fitting formulas, because they extrapolate better outside the simulated range of mass, redshift, and $f_{NL}$.
  • Existing constraints from scale-dependent halo bias should be rescaled by roughly $q = 0.75$; combined with the CMB-to-LSS convention factor of 1.3, this largely cancels, so published bias-based limits are close to correct by accident.
  • Mass-function-based constraints need a larger correction, roughly $0.75^{3/2} \times 1.3 \approx 0.8$, which may explain reported disagreements between simulations and analytic mass-function predictions.
  • The paper tabulates $S_3^{(1)}$, $\sigma_M$, and $\alpha_M$ for a WMAP5 cosmology so the formulas can be applied without recomputing the numerical integrals.

Reading between the lines

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

  • If $q$ is really a proxy for the difference between spherical and ellipsoidal collapse, then it should depend on the shape of the primordial bispectrum, not just its amplitude; a simulation with a different triangle configuration (equilateral, orthogonal) would test whether 0.75 is universal or specific to the local template.
  • The paper's own Fig. 10 hints that the effective $q$ drifts with the sign and size of $f_{NL}$; a dedicated high-signal simulation at $|f_{NL}| \gtrsim 500$ could turn that hint into a measured function $q(f_{NL})$ and would tell forecasters how much of their error budget is currently hidden in a constant.
  • Because the correction enters the mass function as $q^{3/2}$ and the bias as $q$, joint fits to abundance and clustering from the same survey could in principle break the degeneracy between $q$ and $f_{NL}$ internally, rather than assuming $q$ from Gaussian-calibrated fits.
Original abstract

The description of the abundance and clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of order unity on the non-Gaussianity parameter f_{NL}. We present tests on N-body simulations of analytical formulae describing the halo abundance and clustering for non-Gaussian initial conditions. We calibrate the analytic non-Gaussian mass function of Matarrese et al.(2000) and LoVerde et al.(2008) and the analytic description of clustering of halos for non-Gaussian initial conditions on N-body simulations. We find excellent agreement between the simulations and the analytic predictions if we make the corrections delta_c --> delta_c X sqrt{q} and delta_c --> \delta_c X q where q ~ 0.75, in the density threshold for gravitational collapse and in the non-Gaussian fractional correction to the halo bias, respectively. We discuss the implications of this correction on present and forecasted primordial non-Gaussianity constraints. We confirm that the non-Gaussian halo bias offers a robust and highly competitive test of primordial non-Gaussianity.

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, 5,124 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.15 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The description of the abundance and clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of order unity on the non-Gaussianity parameter f_{NL}. We present tests on N-body simulations of analytical formulae describing the halo abundance and clustering for non-Gaussian initial conditions. We calibrate the analytic non-Gaussian mass function of Matarrese et al.(2000) and LoVerde et al.(2008) and the analytic description of clustering of halos for non-Gaussian initial conditions on N-body simulations. We find excellent agreement between the simulations and the analytic predictions if we make the corrections delta_c --> delta_c X sqrt{q} and delta_c --> \delta_c X q where q ~ 0.75, in the density threshold for gravitational collapse and in the non-Gaussian fractional correction to the halo bias, respectively. We discuss the implications of this correction on present and forecasted primordial non-Gaussianity constraints. We confirm that the non-Gaussian halo bias offers a robust and highly competitive test of primordial non-Gaussianity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The description of the abundance and clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of order unity on the non-Gaussianity parameter f_{NL}. background 1.00 0.04 0.03 hedged possibility 0.78
s1 We present tests on N-body simulations of analytical formulae describing the halo abundance and clustering for non-Gaussian initial conditions. contribution 0.94 0.94 0.04 asserted 0.90
s2 We calibrate the analytic non-Gaussian mass function of Matarrese et al.(2000) and LoVerde et al.(2008) and the analytic description of clustering of halos for non-Gaussian initial conditions on N-body simulations. contribution 0.56 0.92 0.04 asserted 0.96
s3 We find excellent agreement between the simulations and the analytic predictions if we make the corrections delta_c --> delta_c X sqrt{q} and delta_c --> \delta_c X q where q ~ 0.75, in the density threshold for gravitational collapse and in the non-Gaussian fractional correction to the halo bias, respectively. result 1.00 0.96 0.61 asserted 1.00
s4 We discuss the implications of this correction on present and forecasted primordial non-Gaussianity constraints. interpretation 0.98 0.51 0.06 unclear 0.83
s5 We confirm that the non-Gaussian halo bias offers a robust and highly competitive test of primordial non-Gaussianity. interpretation 0.62 0.74 0.64 asserted 1.00

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

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. Skewness S3 of the smoothed initial density field for fNL = 100, 200, 500, 1000. Symbols show the numerical results of the initial conditions code (averaged over 5 realizations) and are plotted against the analytical predictions for smoothing radii rs = 4, 6, 8, 10M pc/h of a spherical top-hat filter. 2 4 6 8 10 bHM 2 4 6 8 10 bHH fNL=-200 fNL=200 fNL=-100 fNL=100 fNL=0 0.01 < k < 0.03
Figure 2. The bias of the halo power spectrum bhh compared to the bias of the cross (halo-matter) power spectrum bhm. As expected, when the number density of halos is high there is good agreement between the two quantities. At low halo number densi￾ties the two quantities are affected differently by shot noise, with bhm being the least affected. Note that Afshordi & Tolley (2008) arrived to a similar yet not identical expression when considering ellipsoidal col￾lapse, i.e. they suggest that δc should be substituted by the critical density of Sheth et al. (2001), which in our limit would…
Figure 3. Multiplicity mass function for the Gaussian simu￾lation computed using a Friends-of-Friends halo finder. Points denote the simulations results at different redshift: z = 0, 0.44, 1.02, 1.53, 2.26 and 3.23 (top to bottom). Solid (green) lines are the Sheth & Tormen (1999) formula, dashed (red) lines are the Warren et al. (2006) one and dotted (blue) are the Jenkins et al. (2001). the subscript s denotes quantities measured from the simu￾lation. In principle, the quantity one is interested in would be the bias of the halo power spectrum bhh = p Phh/Pmm, but bhm is a less noisy…
Figure 5. Comparison between the halo mass function recovered in our simulations with the work of Desjacques et al. (2008) and Pillepich et al. (2008). In the left panel we show the ratio between the non-Gaussian and Gaussian simulation at redshift 1 for our simulations with fNL = ±100, three points we read out from
Figure 6. Ratio of the non-Gaussian (fNL = ±100) to Gaussian mass function for different redshift snapshots: top left z = 0.61; top right z = 1.02; bottom left z = 1.53; bottom right z = 1.86. The dashed line is the mass function of Matarrese, Verde & Jimenez (2001) and the dot-dashed lines are that of LoVerde et al. (2008), both including the q-correction. (M >∼ 1013M⊙), it is reliable to assume that the correction from the “non-spherical collapse” can be encapsulated in the factor q in front of δc(zo)/(σ 2 MD(zo)). 6 NON-GAUSSIAN HALO BIAS Eq. (8) shows that the redshift and scale…
Figure 7. Ratio of the non-Gaussian (fNL = ±200) to Gaussian mass function for different redshift snapshots: top left z = 0.61; top right z = 1.02; bottom left z = 1.53; bottom right z = 1.86. The dashed line is the mass function of Matarrese, Verde & Jimenez (2001) and the dot-dashed lines are that of LoVerde et al. (2008), both including our q-correction. taking halos in six different mass bins
Figure 8. Black solid line: the large-scale Gaussian halo Eulerian bias bG of Eq. (11). Blue dotted line: the Gaussian halo bias bG 0 ≡ 1 + δc(zo)/(σ 2 M D(zo)) as defined in Efsthatiou et al. (1988) and Kaiser (1984). Green dotted-dashed line: 1+q(bG 0 −1). Red dashed line: the contribution bG 1 ≡ 1+[qδc(zf )/σ2 M −1/δc(zf )]/D(zo) to the total bias of Eq. (11). c 0000 RAS, MNRAS 000, 000–000
Figure 9. Weak mass dependence of ∆b/b at scales k < 0.03 h/Mpc. Different lines correspond to different redshift snapshots between z = 0 and z = 1.5. The overall normalization is arbitrary. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 z -0.4 -0.2 0.0 0.2 0.4 (bNG-bG)/(bG-1) fNL=100 fNL=-100 fNL=200 fNL=-200 q=0.75
Figure 10. The redshift dependence of the non-Gaussian correc￾tion to the halo bias: points are the values measured from the simulations, lines are the theoretical predictions, Eq. (9). Only k < 0.026 h/M pc were used. 0.001 0.010 k 10-5 10-4 10-3 10-2 10-1 α (k) fNL=-100 fNL=100
Figure 11. Scale dependence of Eq. (9). The thin lines corre￾spond to different redshifts for halos with mass above 1013M⊙/h and the thick black line is their average. The dotted line is the theory prediction with q = 0.75. At k > 0.03h/Mpc the effect of non-Gaussianity is very small and the measurement become extremely noisy. ment become extremely noisy. These results are in qualita￾tive agreement with the findings of Pillepich et al. (2008). We conclude that Eq. (9), with q ∼ 0.75, provides a good fit to non-Gaussian simulations. 7 COMPARISON WITH PREVIOUS WORK After the discussion of §2,…
Figure 12. Left panel: comparison between simulations points and fits from this work and the polynomial fit of Pillepich et al. (2008). Right panel: Our correction to the Matarrese, Verde & Jimenez (2001) and LoVerde et al (2008) non-Gaussian mass function fits, the polynomial fit of Pillepich, Porciani & Zahn (2009) and points from Desjaques et al (2009) (black squares) and Pillepich, Porciani & Zahn (2009) (green triangles). See text for more details. constraints on fNL from the literature rescaled to f CMB NL and corrected for our factor q. This confirms that constraints on fNL achievable…

Showing 11 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:0902.2013 by the authors.

References
How later papers used it

56 later papers cite it

Evidence: some. 44 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 9 of the 102 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 paper · 1 by the same authors

By the same authors

  • Followed this paper in limiting all fits to modes with k ≤ 0.03 h/Mpc.

    Non-Gaussian halo assembly bias (2010) (same authors) · 0.92 · paraphrased: the licence does not allow quoting

How the literature uses it

43 arXiv papers cite this in 83 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 · 64 sentences without a confident label

Uses its method or tool 1 paper
  • Following Ref. [ref] , we limit all fits to modes with $k ≤0.03 h/$Mpc, which after discarding modes with wavenumber $2π/L_box$ and $√(2) ×2π/L_box$, amounts to 366 modes.

    Non-Gaussian halo assembly bias · Fitting non-Gaussian bias · 0.92

Uses its data or values 1 paper
  • We use the Grossi et al. (2009) [ref] set of 5 simulations ($f_NL^LSS = 0, ±100, ±200$, or equivalently, $f_NL^CMB = 0, ±75, ±151$), where all simulations use the same initial condition field $ϕ$ in Equation [ref] to suppress cosmic variance.

    Non-Gaussian halo assembly bias · Simulation Results · 0.95

Compares with it 4 papers
Cites it as support 5 papers
Cites it as background 7 papers
  • Now, let us consider the case of non-Gaussian initial conditions that is often characterized by the primordial non-Gaussianity parameter $f_NL$ as $ψ(x)=ϕ(x) + f_NL(ϕ^2(x)- ⟨ϕ^2(x)⟩)$ where $ϕ$ is a Gaussian random field and $ψ(x)$ is the linearly extrapolated gravitational potential at $z=0$ [ref] .

    The Mass Function of Void Groups as a Probe of the Primordial non-Gaussianity · AN ANALYTIC MODEL · 0.92

  • The fact that extended PS theory gives a qualitatively correct answer but fails at the quantitative level has led many authors either to resort to fits to the $N$-body simulations, see e.g. [ref] , or to look for improvements of the spherical collapse model.

    The Halo Mass Function from Excursion Set Theory. II. The Diffusing Barrier · The ellipsoidal collapse barrier and the diffusing barrier · 0.80

  • The halo mass function has long been known to be sensitive to the presence of primordial non-Gaussianity [ref] and these effects have been replicated within N-body simulations [ref] .

    Exact Extreme Value Statistics and the Halo Mass Function · Extreme Value Statistics of the Halo Mass Function · 0.89

What it cites · 12 sentences without a confident label

Uses data or values from 1 paper
Compares with 3 papers
Cites as background 2 papers

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