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Cosmological Constraints using the Void Size Function Data from BOSS DR16

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Voids in BOSS DR16 constrain dark energy to $w=-1.263^{+0.329}_{-0.396}$ and matter density to $\Omega_{\rm m}=0.293^{+0.060}_{-0.053}$.

desk verdict A competent, honest DR16 VSF extension with constraints consistent with DR12; the caveats are the empirical model-to-void mapping and scale-cut systematics that the paper itself reports. read the letter →

arxiv 2501.07817 v2 pith:XGDVIQDL submitted 2025-01-14 astro-ph.CO

Yingxiao Song , Yan Gong , Xingchen Zhou , Haitao Miao , Kwan Chuen Chan , Xuelei Chen This is my paper · ORCID
classification astro-ph.CO MSC 85A4083F05 PACS 98.80.Es
keywords cosmicvoidsvoidsizefunctiondarkenergyequationofstatematterdensitysigma8BOSSDR16excursion-settheoryredshiftspacedistortions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper measures the void size function—the number density of cosmic voids as a function of radius—from 1.9 million BOSS DR16 galaxies in two redshift bins ($z=0.2$–$0.5$ and $0.5$–$0.8$), and fits it with an excursion-set model. The headline result is a dark-energy equation of state $w=-1.263^{+0.329}_{-0.396}$, matter density $\Omega_{\rm m}=0.293^{+0.060}_{-0.053}$, and fluctuation amplitude $\sigma_8=0.897^{+0.159}_{-0.192}$ in a $w$CDM model. A sympathetic reader should care because void counts probe large-scale structure in underdense regions, giving degeneracy directions that are nearly orthogonal to those from galaxy clustering and therefore useful for joint cosmological constraints.

What carries the argument

The load-bearing object is the $V\,\mathrm{d}n$ void size function, $\mathrm{d}n_v/\mathrm{d}\ln R_v = (3/4\pi R_v^3)\,F(\nu,\delta_v,\delta_c)\,\mathrm{d}\nu/\mathrm{d}\ln R_L$. The first-crossing factor $F$ is the Sheth–van de Weygaert distribution for random walks that hit the void barrier $\delta_v$ before the collapse barrier $\delta_c=1.686$, with significance $\nu=|\delta_v|/\sigma_M(z)$. Lagrangian and Eulerian void sizes are linked by the spherical mapping $R_L\simeq R_v(1-\delta_v/c_v)^{c_v/3}$ with $c_v=1.594$, and observed radii are corrected by $R^{\rm obs}_v = q_{\rm RSD}\,q_{\rm AP}\,R_v$, with one free RSD parameter $B$ per redshift bin. These pieces convert counts of watershed voids into a cosmological likelihood.

What would settle it

Run the identical void finder and MCMC fitter on mock galaxy catalogs with known $w$, $\Omega_{\rm m}$, and $\sigma_8$ and with BOSS DR16-like geometry and density; if the recovered values depart from the truth by more than the reported 68% intervals, the empirical model-to-void mapping is biased. A simpler data-side test is to map how the best-fit parameters shift as the ellipticity threshold moves in steps of 0.01 around 0.15.

Watch

Extended reading notes

Core claim

The paper reports that the abundance of nearly spherical voids identified by a Voronoi-tessellation watershed algorithm in BOSS DR16 can, by itself, produce competitive cosmological constraints. Selecting voids with ellipticity $\epsilon_v<0.15$ and excluding small voids below roughly $2.5\times$ the mean galaxy separation, it derives void size functions in two redshift bins and jointly fits $w$, $\Omega_{\rm m}$, and $\sigma_8$ together with a void linear underdensity threshold $\delta_v$ and an RSD nuisance parameter $B$ in each bin. The fitted values are consistent with earlier BOSS DR12-based void results and the paper argues the constraints are insensitive to the width of the Planck-based priors on $h$, $\Omega_{\rm b}$, and $n_s$. It also finds that the best-fit $\delta_v$ increases toward zero at higher redshift, which it interprets as more linear void evolution and a closer match between Eulerian and Lagrangian void radii.

Load-bearing premise

The load-bearing premise is that the empirical connection between the excursion-set void model and the watershed voids—with $\delta_v$ and $B$ left free and the ellipticity cut fixed at 0.15—is accurate enough that any mismatch does not bias the recovered cosmological parameters.

Editorial extensions

If this is right

  • From one spectroscopic survey, void size functions alone constrain $w$ to about $\pm0.3$–$0.4$ and $\Omega_{\rm m}$ and $\sigma_8$ to about $\pm20$ percent.
  • Combining the VSF with galaxy clustering can break the $\Omega_{\rm m}$–$\sigma_8$ degeneracy, because the two probes' contours are nearly orthogonal.
  • The same fitting pipeline can be applied to upcoming spectroscopic surveys, where void counts are expected to increase by one to two orders of magnitude.
  • The fitted void barrier $\delta_v$ rises from $-0.162^{+0.022}_{-0.024}$ to $-0.123^{+0.014}_{-0.015}$ between the low- and high-redshift bins, a trend the paper attributes to more linear evolution and galaxy bias.

Reading between the lines

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

  • A natural extension is to combine the VSF with weak lensing or cluster counts to shrink the broad $w$–$\sigma_8$ plane, which this paper leaves open.
  • Because the ellipticity threshold is empirical and the paper itself notes that larger voids are rounder, a size-dependent ellipticity prior could reduce the largest systematic of the method.
  • The reported $w<-1$ could be checked with end-to-end mock catalogs of known cosmology: if the pipeline recovers wrong $w$ at the level of the 68% error bars, the model–data mapping, not the Universe, is the source.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper measures the void size function (VSF) from BOSS DR16 galaxies in two redshift bins (0.2<z<0.5 and 0.5<z<0.8) using VIDE watershed voids selected by ellipticity <0.15 and a minimum radius of about 2.5 times the mean galaxy separation. It fits the Jennings et al. (2013) volume-conserving Vdn excursion-set model, with free void linear underdensity thresholds delta_v and RSD parameters B in each bin, together with wCDM parameters w, Omega_m, and A_s, using MCMC with a jackknife covariance and 10-sigma Planck Gaussian priors on A_s, h, Omega_b, and n_s. The headline results are w = -1.263^{+0.329}_{-0.396}, Omega_m = 0.293^{+0.060}_{-0.053}, and sigma8 = 0.897^{+0.159}_{-0.192}. The paper argues that the VSF provides a complementary probe to galaxy clustering, and it explicitly acknowledges that the relation between the theoretical model and the observed watershed void catalog is empirical and that no theoretical framework fixes the optimal ellipticity threshold.

Significance. If the model-to-data mapping were validated, this paper would be a useful demonstration that VSF measurements from spectroscopic surveys can constrain w and Omega_m with uncertainties competitive with some clustering analyses and with a complementary degeneracy direction. The analysis uses standard public tools (VIDE, CAMB, emcee), a reasonable jackknife covariance, and transparent prior choices, and the paper explicitly acknowledges several limitations. I do not share the characterization of the joint fitting of delta_v and B as circular in the statistical sense; these are nuisance parameters estimated from the data. The decisive weakness is that the mapping between the spherical excursion-set model and VIDE watershed voids is empirical and is unvalidated at the number density and selection of the BOSS DR16 catalog, and the paper's own robustness tests show >1-sigma shifts in key parameters when the selection is changed. In addition, the sigma8 result is largely prior-driven. If the authors address these systematics, the paper could be a solid contribution; in its current form, the headline 1-sigma intervals understate the model uncertainty.

major comments (3)
  1. [Section 3, Eqs. (5)-(9); Section 4, delta_v comparison] The load-bearing model-to-data connection is explicitly empirical. Section 3 states that the relation between the Jennings et al. (2013) excursion-set model and the watershed-void abundance is empirical, and that no theoretical framework fixes the optimal ellipticity threshold. Since delta_v and B are fitted jointly with w, Omega_m, and A_s from the same VSF data, any bias in this mapping propagates directly into the cosmological parameters. The paper's own Section 4 notes that the DR16 delta_v best fits differ from the mock-calibrated values because of the lower galaxy number density and larger void size, so the mock validation in Song et al. (2024a) does not transfer to this catalog. The quoted 1-sigma intervals therefore omit the dominant systematic in the model-to-data connection; please add mock-based validation at the BOSS DR16 density and selection, or include an explicit systematic term and soften the central claim.
  2. [Section 3, radius- and ellipticity-robustness paragraphs] The robustness tests reported in Section 3 show that lowering the minimum radius cut from ~2.5xMGS to ~2.3x or ~2.0xMGS shifts Omega_m, w, and delta_v by more than 1 sigma, and that ellipticity cuts outside 0.14-0.16 produce >1-sigma deviations in Omega_m. These cuts define the very data vector used for the headline constraints, so the shifts are selection systematics rather than small perturbations. They are not propagated into the statistical errors in Table 2 or the abstract. The assertion that the current choice 'can provide reliable constraint results' is not supported by these tests; at minimum, the final intervals should include the selection systematic, or the analysis should be restricted to a selection with demonstrated stability.
  3. [Section 4, Table 2] The sigma8 constraint is not an independent VSF measurement. sigma8 is derived from A_s, and A_s is sampled with a 10-sigma Planck Gaussian prior N(2.105, 0.3) (Table 2). The A_s posterior, 2.181^{+0.293}_{-0.297}, is essentially the prior, so the sigma8 posterior is dominated by the Planck prior volume rather than by the void abundance data. The abstract and summary present sigma8 = 0.897^{+0.159}_{-0.192} as a headline VSF result; this should be qualified as a prior-influenced derived parameter, with w, Omega_m, and the void parameters highlighted as the directly constrained quantities.
minor comments (5)
  1. [Section 2.1, Table 1] The text refers to the 'MSG value'; this should be 'MGS' for mean galaxy separation.
  2. [Eq. (7)] The infinite series and the closed-form exponential expression in Eq. (7) are not exactly equal; please state which expression is used in the fits.
  3. [Section 3, Fig. 3] The minimum radius cuts are described as ~2.5xMGS, but for the first redshift bin 40 h^-1 Mpc divided by MGS=15 h^-1 Mpc is 2.67, not 2.5; please reconcile the wording with the numbers.
  4. [Abstract; Section 3] The abstract emphasizes that voids are identified 'without assuming any void shape', while the VSF model assumes spherical voids and the analysis keeps only voids with ellipticity<0.15; this wording should be reconciled.
  5. [Section 4] The MCMC section reports burn-in and thinning but no convergence diagnostic; reporting a Gelman-Rubin statistic would strengthen the reproducibility of the chains.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the cosmological constraints are parameter estimates from the same VSF data, with the model-to-data mapping explicitly acknowledged as empirical; the cited mock calibrations are external support, not a circular load-bearing chain.

full rationale

The paper's derivation chain is not circular. The measured VSF is compared to the Jennings et al. (2013) Vdn excursion-set model via a chi-square likelihood, and the cosmological parameters w, Omega_m, and sigma8 are inferred jointly with the nuisance parameters delta_v^i and B^i. No quantity is called a prediction and then also used as an input: delta_v and B are fitted nuisance parameters, not independent predictions, and sigma8 is obtained by converting the sampled A_s, which carries a deliberately broad 10-sigma Planck prior. The paper explicitly states that "the relation between the theoretical model based on the excursion-set model of Jennings et al. (2013) and the abundance derived from the observed watershed void catalog is empirical," so it does not present the model-data mapping as a first-principles derivation. The self-citations to Song et al. (2024a) are used to cite mock-calibration tests of the method and to interpret delta_v trends; these are externally generated simulation checks, not a uniqueness theorem or an equation that reduces to itself. The robustness tests showing more than 1-sigma shifts when the ellipticity or radius cuts are changed are a systematic-uncertainty concern, not a circularity: they indicate the quoted 1-sigma intervals may understate model dependence, but they do not make any prediction equivalent to a fitted input. Accordingly, no specific circular step can be quoted.

Assumptions & free parameters 13 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the excursion-set VSF model, a spherical collapse mapping, and an empirical identification between watershed voids and model voids; none of these are derived in this paper. Ten model parameters are fitted, and the selection thresholds (ellipticity and minimum radius) are chosen by hand. The four void/RSD nuisance parameters are the least theoretically constrained.

free parameters (13)
  • w = -1.263^{+0.329}_{-0.396}
    Dark energy equation of state, target of the constraint, prior U(-2,0).
  • Omega_m = 0.293^{+0.060}_{-0.053}
    Total matter density, target of the constraint, prior U(0.1,0.5).
  • A_s = 2.181^{+0.293}_{-0.297} (x10^{-9})
    Primordial amplitude, prior N(2.105,0.3)x10^{-9}; sigma8 is derived from it.
  • h = 0.711^{+0.033}_{-0.032}
    Reduced Hubble constant, 10-sigma Gaussian prior from Planck 2018.
  • Omega_b = 0.049 +/- 0.003
    Baryon density, 10-sigma Gaussian prior from Planck 2018.
  • n_s = 0.971^{+0.037}_{-0.038}
    Spectral index, 10-sigma Gaussian prior from Planck 2018.
  • delta_v1 = -0.162^{+0.022}_{-0.024}
    Void linear underdensity threshold in the 0.2-0.5 redshift bin, prior U(-2,0); absorbs model-data mismatch.
  • delta_v2 = -0.123^{+0.014}_{-0.015}
    Void linear underdensity threshold in the 0.5-0.8 redshift bin, prior U(-2,0); absorbs model-data mismatch.
  • B1 = 0.777^{+0.160}_{-0.267}
    RSD parameter in the 0.2-0.5 bin, prior U(0,1); combines beta and delta_Rv.
  • B2 = 0.289^{+0.335}_{-0.209}
    RSD parameter in the 0.5-0.8 bin, prior U(0,1); combines beta and delta_Rv.
  • Minimum void radius cut, bin 1 = 40 h^-1 Mpc (2.5 x MGS)
    Hand-chosen scale cut excluding small voids; constraints shift more than 1 sigma for cuts of 2.0-2.3 x MGS.
  • Minimum void radius cut, bin 2 = 45 h^-1 Mpc (2.5 x MGS)
    Hand-chosen scale cut excluding small voids; constraints shift more than 1 sigma for cuts of 2.0-2.3 x MGS.
  • Ellipticity cut = epsilon_v < 0.15
    Hand-chosen from peak of ellipticity distribution; deviations larger than 1 sigma in Omega_m for thresholds outside 0.14-0.16.
assumptions (6)
  • domain assumption Excursion-set VSF model (SvdW/Jennings) describes void abundance
    Invoked in Eq. 5; the model is assumed valid for the measured watershed voids, which the paper itself flags as empirical.
  • domain assumption Spherical collapse mapping with c_v = 1.594 (Eq. 9)
    Taken from Bernardeau 1994; maps Lagrangian to Eulerian void radii assuming spherical symmetry.
  • standard math delta_c = 1.686 is the collapse barrier
    Standard spherical collapse constant used in Eq. 7.
  • domain assumption RSD correction q_RSD = 1 - (1/3) delta_Rv beta Delta(R_v)
    Model from Correa et al. 2021; B = beta delta_Rv is fitted per bin.
  • domain assumption Jackknife with 200 sky-area subsamples estimates the VSF covariance
    Assumes jackknife converges to the true covariance for this survey geometry, per Favole et al. 2021.
  • domain assumption Fiducial cosmology (Planck 2018) for distance conversion
    Redshifts converted to comoving coordinates with Omega_m = 0.3111, h = 0.6766; affects void volumes and radii.

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Cite this review

Pith. "Pith review of Cosmological Constraints using the Void Size Function Data from BOSS DR16." pith.science (2026). https://pith.science/paper/XGDVIQDL

@misc{pith2026250107817,
  author       = {Pith},
  title        = {Pith review of: Cosmological Constraints using the Void Size Function Data from BOSS DR16},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGDVIQDL}},
  note         = {Machine review of arXiv:2501.07817}
}
abstract

We measure the void size function (VSF) from the Baryon Oscillation Spectroscopic Survey (BOSS DR16) and perform the cosmological constraints. The BOSS DR16 galaxy sample is selected in the redshift range from $z = 0.2$ to 0.8, considering the selection criteria based on galaxy number density. We identify non-spherical voids from this galaxy catalog using the Voronoi tessellation and watershed algorithm without assuming any void shape. We select the void samples based on the void ellipticity, and derive the VSFs in two redshift bins, i.e. $z=0.2-0.5$ and $0.5-0.8$. The VSF model we use is based on the excursion-set theory, including the void linear underdensity threshold $\delta_{\rm v}$ and the redshift space distortion (RSD) parameter $B$. The Markov Chain Monte Carlo (MCMC) method is applied to perform the joint constraints on the cosmological and void parameters. We find that the VSF measurement from BOSS DR16 gives $w = -1.263_{-0.396}^{+0.329}$, $\Omega_{\rm m} = 0.293_{-0.053}^{+0.060}$, and $\sigma_8 = 0.897_{-0.192}^{+0.159}$, which can be a good complementary probe to galaxy clustering measurements. Our method demonstrates the potential of using the VSF to study cosmological models, and it can provide a reference for future VSF analysis in the upcoming galaxy spectroscopic surveys.

Figures

Figures reproduced from arXiv: 2501.07817 by the authors.

Figure 1
Figure 1. The galaxy number density distribution of our catalog from BOSS DR16 in 0 < 𝑧 < 1. The redshift bins are divided by the gray vertical dashed lines. by a redshift fitting software Redrock2 . A total of 1,911,476 galax￾ies are retained in 0 < 𝑧 < 2. We assume a fiducial cosmology with Ωm = 0.3111 and ℎ = 0.6766, based on the flat ΛCDM model and the Planck 2018 results (Planck Collaboration et al. 2020), to convert gal… view at source ↗
Figure 2
Figure 2. The void ellipticity distributions of the redshift bins 𝑧 = 0.2 − 0.5 and 0.5 − 0.8, respectively. The void volume 𝑉 and effective radius are calculated using the cell properties of the tracer, i.e. galaxy in our analysis. Voronoi tessellation assigns a cell to each galaxy, and the effective radius of the non-spherical void 𝑅v is determined based on the volume of each cell 𝑉 𝑖 cell in this void, that we have 𝑅v =  … view at source ↗
Figure 3
Figure 3. The void size functions from BOSS DR16 with 𝜖v < 0.15 (red) and all voids (gray) in the two redshift bins. The blue curves are the best-fits of the theoretical VSF model obtained from the MCMC process, and the blue shaded regions indicate the uncertainty of the model calculated by using the 1𝜎 errors of Ωm and 𝑤. The gray areas represent the excluded scales based on ∼ 2.5×MGS in the fitting process. VIDE. Therefore,… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The normalized covariance matrix of the void size function by its diagonal components. The two redshift bins are separated by the vertical and horizontal white dashed lines. pling. The uncertainty of the best-fit curve derived from the 1𝜎 errors of Ωm and 𝑤 is also sho…
Figure 5
Figure 5. Figure 5: The 1D PDFs and contour maps at 68% and 95% CL for 𝑤, Ωm, and 𝜎8 (left panel), and 𝛿 𝑖 v and 𝐵 𝑖 in the two redshift bins (right panel), using the VSF data from BOSS DR16 data. other parameters, including the void linear underdensity threshold parameter 𝛿v and the RSD …

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Forward citations

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