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Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Counts-in-Cells Statistics

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Radio sources in LOFAR's sky survey are not Poisson-distributed: above 2 mJy their counts-in-cells follow a negative binomial distribution, whose cell-size scaling recovers the two-point correlation function cheaply.

desk verdict A solid counts-in-cells analysis of LoTSS-DR2 that overreaches when it attributes the full variance excess to multi-component sources, since the same data show angular clustering that is ignored in the generative model. read the letter →

arxiv 2504.20723 v1 pith:JFHFBD7C submitted 2025-04-29 astro-ph.CO

Morteza Pashapour-Ahmadabadi , Lukas Böhme , Thilo M. Siewert , Dominik J. Schwarz , Catherine L. Hale , Caroline Heneka , Prabhakar Tiwari , Jinglan Zheng This is my paper · ORCID
classification astro-ph.CO
keywords counts-in-cellsstatisticsnegativebinomialdistributionCoxprocesstwo-pointcorrelationfunctionLOFARLoTSS-DR2large-scalestructure
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 tries to establish which probability distribution governs the number of radio sources in equal-sized cells of sky, using the 4.4-million-source LOFAR Two-Metre Sky Survey Data Release 2. Above a 2 mJy flux-density threshold, the counts are shown to follow a negative binomial distribution far better than a Poisson or compound Poisson distribution, and the paper traces that shape to the multi-component nature of radio objects: cores, lobes, and resolved spirals are each counted as several catalogue entries. Viewed as a function of cell size, the same variance statistic recovers the angular two-point correlation function with an exponent $1-\gamma$ between $-1.05$ and $-0.8$, in agreement with direct pair-counting measurements. The counts-in-cells route estimates clustering in time proportional to the number of sources rather than the number of pairs, an advantage for the much larger surveys now being planned.

What carries the argument

The central object is the Cox process $N_i = \sum_{j=1}^{O_i} C_{ji}$, in which the number of physical objects $O_i$ in a cell is Poisson with intensity $\lambda$ and each object contributes $C_{ji}$ catalogue entries. Choosing $C_{ji}$ to follow a logarithmic distribution with parameter $p$ — the 'educated guess' of Section 2.2 — makes the generating function of $N_i$ collapse exactly to that of a negative binomial distribution with $r = -\lambda/\ln(1-p)$, so one parameter $p$ carries the whole overdispersion. The clustering measurement then runs through the reduced normalised variance $\Psi_2$, which for a power-law angular correlation $w(\vartheta) = A_0(\vartheta/\vartheta_0)^{1-\gamma}$ evaluates to $\Psi_2(\Theta) = A_0 C_\gamma (\Theta/\Theta_0)^{1-\gamma}$, with a numerically computed coefficient $C_\gamma$ of order unity.

What would settle it

The empirical component-count distribution from the LoTSS-DR2 value-added catalogue, which is complete above 4 mJy, can be compared directly with the logarithmic law the negative binomial model assumes: if the measured distribution of components per physical object is not logarithmic, or if its fitted parameter changes with flux density, sky position, or source morphology, then the reported preference for the negative binomial is an artifact of the assumed family rather than a property of the radio-source population. A second check uses the model's own moment predictions: the negative binomial fixes skewness as $g_1 = (2n_c - 1)/(\mu^{1/2} n_c^{1/2})$, so comparing the empirical third and fourth moments of the counts at 2 mJy with these predictions would falsify the model if they disagree within the quoted uncertainties.

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Extended reading notes

Core claim

The central claim is that the counts-in-cells distribution of LoTSS-DR2 radio sources above a 2 mJy flux-density threshold is strongly better described by a negative binomial distribution than by a Poisson or compound Poisson distribution. The argument builds on a Cox process: physical objects are distributed as a Poisson process, and each object contributes a random number of detected radio components drawn from a logarithmic distribution; composing those generating functions yields exactly the negative binomial family, with a fitted mean of 1.27 components per source at 2 mJy. The paper supports this preference with a reduced chi-square test, a Monte-Carlo-calibrated Kolmogorov-Smirnov test, and Bayes factors of 25.4, 17.2, and 15.9 at 2, 4, and 8 mJy, and it notes that even the preferred model is formally rejected by the KS test for the default mask, indicating residual effects beyond the model. The paper further claims that the scaling of the reduced normalised variance of counts with cell size follows a single power law whose exponent $1-\gamma$ lies between $-1.05$ and $-0.8$ at 2 mJy and SNR 7.5, consistent with the angular two-point correlation function measured directly with the Landy-Szalay estimator and with earlier optical, infrared, and radio surveys.

Load-bearing premise

The load-bearing premise is that every physical radio object generates a number of catalogue components drawn from one logarithmic distribution with a single parameter $p$ that is the same across the whole survey and across flux densities; the paper introduces this as 'an educated guess'.

Editorial extensions

If this is right

  • The fitted negative binomial mean of 1.27 components per source at 2 mJy is consistent with independent component-association counts (the value-added catalogue gives 1.13 above 4 mJy; a LOFAR survey cross-match gives about 1.33), so the model's parameter has a direct physical reading.
  • The variance-of-counts method estimates the two-point correlation function with linear rather than quadratic scaling in the number of sources, making it a practical clustering probe for future surveys with tens of millions of detections.
  • Higher moments of the same counts extend the machinery to higher-order correlation functions at the same linear cost, opening a route to non-Gaussianity tests on large angular scales.
  • The fitted slope steepens with flux-density threshold and flattens with higher signal-to-noise cuts, which the paper reads as a population effect: AGN dominance above 2 mJy raises clustering, while brighter, more isolated sources cluster less.

Reading between the lines

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

  • If the logarithmic component-count law is universal, the fitted parameter $p$ becomes a portable descriptor of how radio morphology maps into catalogue entries; measuring it as a function of flux density and source type across surveys would test whether the law is truly constant or itself a function of the source population.
  • Because the variance-based and Landy-Szalay exponents agree, the cheap variance route could double as a systematics check for pair-counting pipelines: a disagreement between the two at a given angular scale would point to masking or completeness problems rather than to cosmology.
  • The paper's admitted 'educated guess' on the component-count distribution is directly testable: the value-added catalogue, complete above 4 mJy, provides the empirical component-count distribution, so one can check whether it is really logarithmic or a mixture that varies with morphology — the Bayes factor in favour of the negative binomial depends on this.
  • Below roughly 0.1 degrees, multi-component associations dominate the variance, so the usable angular range of the method is set by how well the component law is modelled; pushing to sub-mJy thresholds would require modelling the star-forming galaxy population separately.
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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 / 3 minor

Summary. The paper analyzes the counts-in-cells statistics of the LoTSS-DR2 radio source catalogue (about 4.4 million sources over 5,635 square degrees) and compares three models for the counts: Poisson, compound Poisson, and negative binomial. Using a Cox-process construction, the authors argue that the negative binomial distribution is strongly preferred above a 2 mJy flux density threshold, with a fitted mean of 1.27 radio components per physical source. They also use the scaling of the normalised variance of counts with cell size (the reduced normalised variance Ψ2) to fit a power-law angular two-point correlation function, obtaining an exponent 1−γ between about −0.8 and −1.05 at 2 mJy and SNR 7.5, and they compare this with a direct Landy-Szalay estimate. The paper explicitly acknowledges that the distribution of the number of components per object is an 'educated guess' (Sect. 2.2) and that the KS test still rejects the negative binomial for the default mask (Sect. 6.1, Table 4).

Significance. If the central claims hold, the paper would (i) establish that a one-parameter logarithmic component-count distribution explains the counts-in-cells of a large radio survey, (ii) provide a physical interpretation of the overdispersion in terms of a mean component count of about 1.27, and (iii) validate a computationally cheap counts-in-cells route to the angular two-point correlation function that is linear in the number of sources. The paper's strengths include the use of a large, public dataset; clear derivations of the moments and generating functions for the three models; and cross-checks against the value-added catalogue and against a Landy-Szalay estimator on the same survey. The falsifiable predictions (e.g., the scaling exponent and the comparison with the value-added catalogue) are useful. However, the central model premise is an ad hoc parametric choice, and the statistical evidence is partly overstated given that the KS test rejects the preferred model and the power-law fits have large reduced chi-square values.

major comments (3)
  1. [Sect. 2.2, Eq. (6)] The generative model in Eq. (6) treats the number of physical objects O_i as Poisson with no spatial correlation, yet the same catalogue exhibits strong angular clustering measured in Sect. 6.2 with w(θ) fitted to a power law. Since p is estimated from the total variance (Eq. (23)), any variance contributed by object clustering is absorbed into the logarithmic component-count distribution, biasing the inferred p and the mean component count of 1.27 (Eq. (20)). The Bayes factor of 25.4 reported in Sect. 6.1 is computed as a product over roughly 84,625 cells that are spatially correlated, so the effective sample size is smaller than assumed and the 'strong evidence' is likely overstated. A concrete check would be to compare the variance implied by the fitted logarithmic distribution with the component-count variance estimated from the value-added catalogue, or to include a clustered object field (e.g., a Poisson cluster process) in the model; until such a test is performed, the physical interpretation of p as a component-count parameter is not secure.
  2. [Sect. 6.1, Table 4] The KS test in Table 4 rejects the negative binomial distribution at all flux density thresholds for mask d (dn=0.0078 vs dα=0.0030 at 2 mJy), and the text in Sect. 7 acknowledges this, yet the abstract and conclusions state there is 'strong evidence in favour of the negative binomial distribution' and the text in Sect. 6.1 describes it as an 'excellent fit'. A model that is formally rejected at 99% confidence cannot simultaneously be claimed as the strongly favoured model without a careful statement of the approximate nature of the fit; the Bayes factor comparison should be reported with an effective sample size or as conditional on the assumed model family.
  3. [Sect. 6.2, Table 5] The power-law fits to Ψ2 yield reduced chi-square values as high as 8.4 (2 mJy, Nside 16-512) and 3.4 (2 mJy, Nside 16-256), well above unity, indicating that the single power-law ansatz is formally unacceptable for these data. The abstract's claim that the scaling is 'in good agreement with a power-law model' is not supported by these statistics; the authors should either provide a justification for the extra variance (e.g., cosmic variance, mask systematics, non-linear clustering), or restrict the quoted exponent range to angular scales where the fit is acceptable, or revise the claim.
minor comments (3)
  1. [Sect. 7.1] The comparison of the fitted mean component count (1.27) with the value-added catalogue value (1.13) is made at 2 mJy, while Sect. 5 states that the value-adding process is only complete above 4 mJy; the text should explicitly state that this comparison is an extrapolation and note the associated uncertainty.
  2. [References and Fig. 4 caption] The reference list contains inconsistent entries: the in-text citation 'de Gasperin et al. 2023' in the Fig. 4 caption does not match the listed reference 'de Gasperin et al. 2021', and Shimwell et al. (2022) appears twice with slightly different formatting.
  3. [Sect. 2.1 and Sect. 6.2] The notation for the central moments m3 and m4 is introduced in Eq. (2) but used later in Eq. (31)-(32) without reminding the reader that these are sample estimates; a brief restatement would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: fits are validated against independent catalogues and estimators.

full rationale

The derivation chain is self-contained and not circular. The negative binomial model is obtained by a standard generating-function calculation (Eqs. 13-15) from the explicitly stated modeling assumption that C_ji follows a Logarithmic(p) distribution, which the authors label an 'educated guess' (Sect. 2.2); the compound-Poisson alternative is likewise a stated ansatz from Siewert et al. (2020), used for comparison rather than as proof. Parameters are fitted to the first two empirical moments (Eqs. 21-23), but the model selection and goodness-of-fit tests (chi-square, KS, Bayes factor) then compare the full predicted histogram, so the higher-order shape of the distribution is tested rather than assumed. The mean component count 1.27 is a fitted transformation of p, but it is checked against independent external catalogues (LoTSS-DR2 value-added catalogue, Boehme et al. 2023), and the correlation-function exponent is independently cross-checked with the Landy-Szalay estimator; these are external validations, not outputs of the same fit. Self-citations (Siewert et al. 2020, Boehme et al. 2023) supply data products and comparison models but are not load-bearing for the central negative-binomial preference. The paper honestly reports that the KS test still rejects the NB fit and that the component distribution is not derived from first principles; these are model-misspecification limitations, not circular steps.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results rest on four fitted parameters (p, kappa, gamma, A) and on phenomenological assumptions about the component-count distribution and the power-law form of clustering. The paper is explicit that the component distribution is an educated guess, which is the main model risk.

free parameters (4)
  • p (negative binomial) = 0.37 (2 mJy), 0.31 (4 mJy), 0.26 (8 mJy), mask d
    Fit by matching the sample mean and variance (Eq. 23); controls the component-count distribution and the clustering parameter nc=1/(1−p).
  • kappa (compound Poisson) = 0.58 (2 mJy), 0.45 (4 mJy), 0.34 (8 mJy), mask d
    Fit via Eq. 22; the alternative overdispersion model that is compared and rejected.
  • gamma (power-law exponent) = 1.88 to 2.52 depending on flux cut, SNR, and fitted range
    Fitted to the normalized variance vs cell size (Tables 5 and 6); the central clustering parameter.
  • A0 (power-law amplitude) = 0.001 to 0.003
    Fitted amplitude of the angular correlation power law; determines the strength of clustering.
assumptions (5)
  • domain assumption The underlying distribution of physical objects per cell is Poisson with a single intensity lambda (Cox process).
    Adopted in Eq. (6) to model the counts; no test of variability of lambda across the masked survey is performed.
  • ad hoc to paper The number of components per physical object follows a logarithmic distribution with parameter p, chosen without first-principles derivation.
    Sect. 2.2 calls the choice of C_ji distribution 'an educated guess'; it guarantees at least one component per object.
  • domain assumption The angular two-point correlation function is a single power law over the fitted angular scales (0.11 to 3.66 degrees).
    Used in Eq. (25) to convert the normalized variance scaling into gamma and A; fit quality is only partially satisfactory (chi-square/dof up to 8.4).
  • domain assumption The random mock catalogue correctly represents survey systematics, so subtracting its normalized variance removes systematic contamination.
    The random catalogue of Hale et al. (2024) includes completeness, noise, and beam effects; the subtraction assumes these effects are additive.
  • ad hoc to paper Counts-in-cells in disjoint cells are treated as independent samples in the chi-square and KS tests.
    Clustering induces correlations between neighbouring cells, which is not accounted for in the reported test statistics.

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Cite this review

Pith. "Pith review of Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Counts-in-Cells Statistics." pith.science (2026). https://pith.science/paper/JFHFBD7C

@misc{pith2026250420723,
  author       = {Pith},
  title        = {Pith review of: Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Counts-in-Cells Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JFHFBD7C}},
  note         = {Machine review of arXiv:2504.20723}
}
read the original abstract

We investigate the statistical distribution of source counts-in-cells in the second data release of the LOFAR Two-Metre Sky Survey (LoTSS-DR2) and we test a computationally cheap method based on the counts-in-cells to estimate the two-point correlation function. We compare three stochastic models for the counts-in-cells which result in a Poisson distribution, a compound Poisson distribution, and a negative binomial distribution. By analysing the variance of counts-in-cells for various cell sizes, we fit the reduced normalised variance to a single power-law model representing the angular two-point correlation function. Our analysis confirms that radio sources are not Poisson distributed, which is most likely due to multiple physical components of radio sources. Employing instead a Cox process, we show that there is strong evidence in favour of the negative binomial distribution above a flux density threshold of 2 mJy. Additionally, the mean number of radio components derived from the negative binomial distribution is in good agreement with corresponding estimates based on the value-added catalogue of LoTSS-DR2. The scaling of the counts-in-cells normalised variance with cell size is in good agreement with a power-law model for the angular two-point correlation. At a flux density threshold of 2 mJy and a signal-to-noise ratio of 7.5 for individual radio sources, we find that for a range of angular scales large enough to not be affected by the multi-component nature of radio sources, the value of the exponent of the power law ranges from -0.8 to -1.05. This closely aligns with findings from previous optical, infrared, and radio surveys of the large scale structure. The scaling of the counts-in-cells statistics with cell size provides a computationally efficient method to estimate the two-point correlation properties, offering a valuable tool for future large-scale structure studies.

Figures

Figures reproduced from arXiv: 2504.20723 by the authors.

Figure 1
Figure 1. Counts-in-cells of the LoTSS-DR2 radio source catalogue in Mollweide view and equatorial coordinates without a flux density cut. The counts-in-cells are based on HEALPix with a resolution of 13.74 square arcmin. the mean, are mj ≡ 1 Ncell X Ncell i=1 (ki − µ) j , (2) with the variance σ 2 ≡ m2, and the coefficients of skewness and excess kurtosis (Zwillinger & Kokoska 2000) g1 ≡ m3 m 3/2 2 , g2 − 3 ≡ m4 m2 2 − 3, (3… view at source ↗
Figure 2
Figure 2. Example of complete pointings in the HETDEX field and incom￾plete pointings at the boundary of the survey area, and pointings which are not mosaiced with other neighbouring pointings. 3.2. Random mock catalogue For the comparison of the observations with a Poissonian source distribution, we use the random mock catalogue generated by Hale et al. (2024). As outlined in Sect. 3.2 of Hale et al. (2024), the generation p… view at source ↗
Figure 3
Figure 3. For ‘mask d’ the light grey regions remain after masking, while dark grey is the survey area [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Left: Comparison of the differential source counts from mask d, mask 1017 (explained in App. A.3), and mask 50 (explained in App. A.2) with the results from the LoTSS Deep Fields (Mandal et al. 2021), and LoTSS-DR1 (Siewert 2021). The ending ‘radio’ refers to the appli…
Figure 5
Figure 5. Figure 5: Histograms of counts-in-cells of mask d for LoTSS-DR2 and the random mock catalogue at the flux density thresholds 2, 4 and 8 mJy (left, top to bottom), and the CDFs (right, top to bottom) with the best-fit Poisson and compound Poisson and negative binomial distributio…
Figure 6
Figure 6. Figure 6: Empirical variance of Ψ2 estimated analytically (blue) and from a bootstrap method (orange) with 500 realisations, each at 2 mJy flux density threshold at different values of Nside of the HEALPix scheme. Bayes factor (Jeffreys 1961; Gelman et al. 2003) BNB−CP = p(data …
Figure 7
Figure 7. Figure 7: Fitted single power law for variance variations of re-scaling HEALPix maps for Nside = 16 − 512 corresponding to 3.8 deg to 0.11 deg at different flux density thresholds (left) and for Nside = 16 − 256 corresponding to 3.8 deg to 0.23 deg (right), plotted for Nside = 1…
Figure 8
Figure 8. Figure 8: Fitted single power law for variance variations of re-scaling HEALPix maps for Nside = 16 − 512 (left) and for Nside = 16 − 256 (right) at flux density threshold 2 mJy and different SNR cuts plotted for Nside = 16 − 2048 (solid points represent the data points for whic…
Figure 9
Figure 9. Figure 9: Fitting results for power laws to different angular ranges at 2 mJy flux density threshold and SNR 7.5. (dashed lines). The dots represent the results of direct measurements of the two-point correlation function using the Landy-Szalay estimator. The inner plot shows th…

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    astro-ph.IM 2025-06 conditional novelty 6.0 of 10

    A diffusion-model pipeline generates realistic simulated LOFAR survey maps with controllable galaxy sizes, closely matching real flux and size distributions.

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