Void probability function in the Quijote simulations

Snehasish Bhattacharjee

REVIEW

Void probability function in the Quijote simulations

Snehasish Bhattacharjee

Cosmology and Nongalactic Astrophysics · Submitted 16 Sep 2026 · Published version · 58 references

Pith's verdict: Systematic VPF comparison across 13 cosmologies is well executed, but the central quantitative ranking is based on one simulation per cosmology and the paper's own appendix shows that's not stable.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 17, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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T3 Formal statement The claims written out in a formal math language a computer can read.
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arXiv ID2609.18221 (version 1)

Pith IDPWKBICNR, Pith's permanent record of this paper and its review

CitationSnehasish Bhattacharjee. "Void probability function in the Quijote simulations." arXiv:2609.18221 (2026). https://pith.science/paper/PWKBICNR

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Claim paper

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Editor's note

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The paper does something genuinely useful: it runs the same VPF estimator over 13 cosmological models in the Quijote suite and produces a clean set of tables and figures showing how the VPF responds to neutrino mass, f(R) gravity, PNG, and vanilla ΛCDM parameters. The method is standard count-in-cell, the error propagation is honest, and the appendix on cosmic variance is the right thing to include. The qualitative results — halo VPF follows GH at z=0, galaxy VPF follows NB, redshift space shifts toward NB, morphology dependence — confirm prior literature, which is fine; the intended novelty is the cross-cosmology ranking.

Here is the soft spot, and it is load-bearing for the paper's claim: all the rankings in Tables 3–8 come from one realization per cosmology. The author shows in Appendix .2 that the chosen ΛCDM realization has RMS deviation 0.069±0.034, while the mean over 10 independent ΛCDM realizations is 0.031±0.017. That single realization is a 2.2σ outlier. If similar variance affects the other cosmologies — and there is no reason it wouldn't — then the reported ranking (ΛCDM smallest deviation, Mnu_ppp largest at ~7σ) could partly be cosmic variance. The abstract and conclusions present the ranking as a robust finding, but the statistical basis is insufficient. This is not a deep flaw in the method; it is a sample-size problem that is straightforward to fix by repeating the analysis on the full ensemble of realizations.

The paper is a serious piece of work, well-written, and the code should be shared. But the headline quantitative claims need rechecking before they can be taken at face value. For a reading group, I'd show it as a good example of how systematic comparisons can be undermined by cosmic variance if you do not use the available realizations. I would accept a peer-review invitation, but I'd request the author redo the analysis over all realizations and fold cosmic variance into the quoted errors.

Referee report

Major objections (2)

[§3, Table 3, Appendix .2] The central ranking of cosmological models by deviation from the GH model (e.g., Mnu_ppp at ~7σ, ΛCDM at ~2σ) is derived from one realization per cosmology. Appendix .2 shows that the chosen ΛCDM realization has RMS deviation 0.069, while the mean of 10 independent ΛCDM realizations is 0.031—the single realization is about 2.2σ high. This demonstrates that cosmic variance is comparable to or larger than the differences between some models. Without using the full ensemble of realizations for every cosmology (or at least a representative subset), the ranking is not robust. The authors should either: (a) repeat the analysis using all available realizations for each cosmology and re-derive the tables with cosmic-variance error bars, or (b) if those data are not available, substantially weaken the quantitative claims (abstract, Section 6) to state that the ranking is tentative and subject to cosmic-variance scatter. At minimum, the error bars in Tables 3–8 must include a cosmic-variance component (as in Appendix .2) rather than only counting-statistics uncertainties.
[Tables 3–8, §.1–.2] The quoted uncertainties in all tables are derived solely from the binomial counting error in P0 (Eq. 3.4–3.5). Appendix .2 shows that cosmic variance from using a single simulation box is of order 0.01–0.02 in RMS deviation, comparable to the values in Table 3 (e.g., ΛCDM: 0.069±0.034 from counting stats, but mean over 10 boxes is 0.031±0.017). This means the precision in the tables is artificially high. The main comparison across cosmologies should incorporate cosmic-variance errors. Without that, the significance of differences between models (e.g., the claimed 7σ deviation for Mnu_ppp) is strongly overestimated.

Minor objections (6)

[Table 3] For cosmology fR_pppp, the RMS deviation at z=0 (0.0268±0.0041) and z=0.5 (0.0268±0.0041) are identical; please check if this is a duplication error or correct.
[Figure captions] In Figures 1, 2, 3, 4 the bottom panels are labeled 'GH' when the model used is the GH model, but in Figure 5 the lower panels are labeled 'NB'. This is correct but could be made more explicit in the caption: state which model the residual refers to.
[Section 4.2.1] The text refers to 'Figure 3' for subsampling; the three panels correspond to 25%, 50%, 75% removal. Please also refer to the specific panel in the text for clarity.
[General] The notation for the void statistic: in Eq. 2.6 it is defined as χ = -ln(P0)/N, but in figures it is labeled 'log(P0)/N'. Since log is typically base 10 in plots but ln in equations, make the base explicit to avoid confusion.
[Section 2] The hierarchical scaling relation Eq. 2.3 is taken from prior literature; consider citing the original derivation (Bernardeau & Schaeffer 1999 or Fry 1986) as done, but also note that the specific form S_n = n^{n-2} is one choice among many.
[Section 6] The conclusions mention that the galaxy catalogs are HOD-based; this is a strength but also a limitation (baryonic effects ignored). The paper acknowledges this, which is good. However, the concluding sentence could note that observational applications will require careful treatment of survey geometry and completeness.

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The paper

The paper claims

The void probability function responds systematically to cosmology: ΛCDM shows the smallest deviation from the geometric hierarchical model while massive neutrinos produce the largest departure.

The reading

Using the Quijote N-body simulations, this paper tests whether the void probability function (VPF) — the probability of finding no tracers in a volume — can distinguish the standard ΛCDM model from extensions such as massive neutrinos, modified gravity, and primordial non-Gaussianity. At z=0, dark-matter haloes' VPFs follow the geometric hierarchical (GH) model, but the statistical significance of the deviation from that model depends strongly on cosmology: ΛCDM has the smallest deviation (~2σ) while a model with 0.4 eV neutrinos has the largest (~7σ). At higher redshift and in redshift space the VPF shifts toward the negative binomial (NB) model, and for galaxies ellipticals and spirals follow the GH and NB models respectively. The paper concludes that the VPF is a computationally cheap probe that can help break degeneracies in upcoming large galaxy surveys.

What carries the argument

The central mathematical id entity is the hierarchical scaling relation for the void statistic χ = −ln P_0 / N = function of Nξ, where N is the mean count in a spherical cell and ξ is the volume-averaged two-point correlation function. The geometric hierarchical (GH) model assumes χ = 1/(1 + ξ N /2), derived from a particular choice of hierarchical amplitudes S_n = $n^{{n-2}}$. The negative binomial (NB) model assumes χ = ln(1 + Nξ)/(Nξ). These two analytic forms are the templates against which the simulated VPF are compared.

Extended reading notes

Core claim

The central claims are: (1) the halo VPF scales as predicted by the GH model at z=0, but the amplitude of the deviation (measured as the RMS residual from the GH prediction) varies systematically across cosmologies, with the standard ΛCDM showing the smallest deviation (significance ~2σ) and a massive-neutrino cosmology (M_nu = 0.4 eV) showing the largest (~7σ); (2) at higher redshifts (z ≳ 0.5) and in redshift space, the VPF departs from the GH model and approaches the NB model, with primordial non-Gaussianity producing the largest shifts; (3) the VPF of galaxies depends on morphology — ellipticals follow GH, spirals follow NB — and their combined sample fits neither model well.

Load-bearing premise

The analysis uses one random simulation realization per cosmology, but the scatter among realizations for the same cosmology (tested only for ΛCDM) is large enough that the measured RMS deviations of individual models could reorder if all realizations were used.

Editorial extensions

If this is right

If correct, the VPF can break degeneracies among cosmological parameters because different extensions produce characteristic patterns of deviation from the GH and NB models.
The strong signal from massive neutrinos (0.4 eV) implies that void statistics may be especially effective for constraining the total neutrino mass.
The shift from GH to NB in redshift space confirms that redshift-space distortions must be modelled carefully when using VPF to constrain cosmology with spectroscopic surveys.
The dependence on halo mass — with low-mass haloes most deviant in ΛCDM but intermediate-mass haloes most deviant in modified gravity — suggests that combining VPF from different mass bins can discriminate among gravity models.
For galaxy surveys, the morphological distinction (ellipticals vs spirals) implies that galaxy-type selection can optimise the VPF signal for particular cosmological parameters.

Reading between the lines

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

The paper uses only a single Quijote realization per cosmology; applying the same analysis to all available realizations (typically 500–15000 per cosmology) would provide robust error bars and confirm the ranking of models reported here.
The VPF's sensitivity to massive neutrinos arises from the suppression of small-scale structure; this suggests a cross-check with other small-scale probes (e.g., halo mass function, power-spectrum ratio) could isolate the neutrino signature.
The result that the combined galaxy VPF fits neither model well implies that simple two-population models ( just ellip and spiral) may be insufficient; a more nuanced HOD with environmental dependence may be needed.
Budgeting cosmic variance (the single-realization 2.2σ offset found for ΛCDM) suggests that future work should state error budgets for all models from the ensemble, not just ΛCDM.

Original abstract

The void probability function (VPF) is the likelihood of finding no tracers (galaxies or dark-matter haloes) within a given volume and is sensitive to the large-scale matter distribution. We investigate the VPF of dark matter haloes and galaxies using the \textsc{Quijote} N-body simulations across a range of cosmological models, including $\Lambda$CDM and several extensions. We examine its dependence on cosmology, redshift, halo mass, morphology, and sample dilution, and compare the results with theoretical predictions from the geometric hierarchical (GH) and negative binomial (NB) models. We find that the halo VPF is sensitive to cosmology and halo mass and follows the GH model at $z=0$, with $\Lambda$CDM showing the smallest deviation and the massive-neutrino cosmology the largest. At higher redshift and stronger sample dilution, the VPF departs from the GH model and approaches the NB model. The dependence on halo mass varies with cosmology, with low-mass haloes showing the largest deviations in $\Lambda$CDM, while intermediate- and high-mass haloes show stronger deviations in the modified-gravity scenario. In redshift space, the VPF shifts toward the NB model, with primordial non-Gaussianity producing the largest deviations. For galaxies, the VPF depends on morphology and cosmology, with ellipticals following the GH model and spirals following the NB model. We also find that cosmic variance introduces non-negligible scatter in the VPF measurements. Overall, these results suggest that the VPF could be useful as a computationally efficient probe to distinguish extensions of the standard $\Lambda$CDM model in upcoming large-volume galaxy surveys.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,749 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.82 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The void probability function (VPF) is the likelihood of finding no tracers (galaxies or dark-matter haloes) within a given volume and is sensitive to the large-scale matter distribution. We investigate the VPF of dark matter haloes and galaxies using the \textsc{Quijote} N-body simulations across a range of cosmological models, including $\Lambda$CDM and several extensions. We examine its dependence on cosmology, redshift, halo mass, morphology, and sample dilution, and compare the results with theoretical predictions from the geometric hierarchical (GH) and negative binomial (NB) models. We find that the halo VPF is sensitive to cosmology and halo mass and follows the GH model at $z=0$, with $\Lambda$CDM showing the smallest deviation and the massive-neutrino cosmology the largest. At higher redshift and stronger sample dilution, the VPF departs from the GH model and approaches the NB model. The dependence on halo mass varies with cosmology, with low-mass haloes showing the largest deviations in $\Lambda$CDM, while intermediate- and high-mass haloes show stronger deviations in the modified-gravity scenario. In redshift space, the VPF shifts toward the NB model, with primordial non-Gaussianity producing the largest deviations. For galaxies, the VPF depends on morphology and cosmology, with ellipticals following the GH model and spirals following the NB model. We also find that cosmic variance introduces non-negligible scatter in the VPF measurements. Overall, these results suggest that the VPF could be useful as a computationally efficient probe to distinguish extensions of the standard $\Lambda$CDM model in upcoming large-volume galaxy surveys.

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 void probability function (VPF) is the likelihood of finding no tracers (galaxies or dark-matter haloes) within a given volume and is sensitive to the large-scale matter distribution.	background 1.00	0.04	0.02	asserted 1.00
s1	We investigate the VPF of dark matter haloes and galaxies using the \textsc{Quijote} N-body simulations across a range of cosmological models, including $\Lambda$CDM and several extensions.	contribution 0.70	0.94	0.03	asserted 0.96
s2	We examine its dependence on cosmology, redshift, halo mass, morphology, and sample dilution, and compare the results with theoretical predictions from the geometric hierarchical (GH) and negative binomial (NB) models.	method 0.63	0.91	0.03	asserted 0.84
s3	We find that the halo VPF is sensitive to cosmology and halo mass and follows the GH model at $z=0$, with $\Lambda$CDM showing the smallest deviation and the massive-neutrino cosmology the largest.	result 1.00	0.97	0.19	asserted 1.00
s4	At higher redshift and stronger sample dilution, the VPF departs from the GH model and approaches the NB model.	result 1.00	0.93	0.11	asserted 1.00
s5	The dependence on halo mass varies with cosmology, with low-mass haloes showing the largest deviations in $\Lambda$CDM, while intermediate- and high-mass haloes show stronger deviations in the modified-gravity scenario.	result 1.00	0.96	0.06	asserted 1.00
s6	In redshift space, the VPF shifts toward the NB model, with primordial non-Gaussianity producing the largest deviations.	result 1.00	0.92	0.06	asserted 1.00
s7	For galaxies, the VPF depends on morphology and cosmology, with ellipticals following the GH model and spirals following the NB model.	result 1.00	0.93	0.07	asserted 1.00
s8	We also find that cosmic variance introduces non-negligible scatter in the VPF measurements.	result 0.91	0.94	0.04	asserted 1.00
s9	Overall, these results suggest that the VPF could be useful as a computationally efficient probe to distinguish extensions of the standard $\Lambda$CDM model in upcoming large-volume galaxy surveys.	interpretation 0.99	0.50	0.87	hedged possibility 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 v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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