One- and Two-point Source Statistics from the LOFAR Two-metre Sky Survey First Data Release

Catherine Hale; Cyril Tasse; David J. Bacon; Dominik J. Schwarz; Glenn J. White; Huub R\"ottgering; Jose Sabater; Kenneth J. Duncan; Marian Biermann; Martin J. Hardcastle

REVIEW 3 major objections 4 minor 92 references

One- and Two-point Source Statistics from the LOFAR Two-metre Sky Survey First Data Release

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A clean 2 mJy sample of 19,719 LoTSS radio sources shows angular clustering consistent with the standard cosmological model and with percent-level statistical isotropy of the radio sky.

desk verdict A careful first clustering measurement from LoTSS-DR1; trust the 2 mJy result, but the Planck agreement remains conditional until photo-zs are validated. read the letter →

arxiv 1908.10309 v3 pith:CVSSXG2J submitted 2019-08-27 astro-ph.CO

Thilo M. Siewert , Catherine Hale , Nitesh Bhardwaj , Marian Biermann , David J. Bacon , Matt Jarvis , Huub Röttgering , Dominik J. Schwarz

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Timothy Shimwell Philip N. Best Kenneth J. Duncan Martin J. Hardcastle Jose Sabater Cyril Tasse Glenn J. White Wendy L. Williams

This is my paper · ORCID

classification astro-ph.CO

keywords angularcorrelationfunctioncounts-in-cellsradiosourcecountsLOFARsurveyphotometricredshiftslarge-scalestructurestatisticalisotropycosmologicalprinciple

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

The paper uses the first data release of the LOFAR Two-metre Sky Survey, covering 424 square degrees and 318,520 sources, to ask whether the large-scale distribution of faint radio sources matches the standard cosmological model. It establishes that the angular two-point correlation function of a clean, low-noise subsample above 2 mJy is consistent with the prediction built from the best-fit Planck 2018 cosmology, the photometric redshift distribution of the sources, and a literature bias model, with no fitted parameters. It also shows that counts-in-cells reject a Poisson distribution and are well described by a compound Poisson distribution, and that differential source counts agree with earlier low-frequency surveys and with two independent simulations of the radio sky. The paper's conclusion is that the radio sky is statistically isotropic at the percent level and that current-generation radio surveys can already serve as cosmological probes.

What carries the argument

The argument pivots on two objects. For the one-point statistics it is the counts-in-cell distribution and its moments, especially the clustering parameter $n_c=\mathrm{Var}[k]/\mathbb{E}[k]$, with a compound Poisson distribution (a Poisson number of clusters, each contributing a Poisson number of components) as the model that fits the data. For the two-point statistics it is the angular correlation function $w(\theta)$, measured through the Landy-Szalay pair-counting estimator $\hat w(\theta)=(DD-2DR+RR)/RR$, where $DD$, $DR$, and $RR$ are normalized data-data, data-random, and random-random pair counts. The random catalogue is a noise-weighted mock built from a simulated radio sky and the actual LoTSS noise maps, so survey masks and completeness variations are folded into the null expectation. The theoretical comparison carries through a redshift-space window: the measured photo-$z$ histogram, the Planck 2018 parameters, and a bias-redshift relation are fed into a Boltzmann-solver code that produces the predicted $w(\theta)$, with the finite-survey integral constraint subtracted.

What would settle it

Recompute the predicted $w(\theta)$ for the 2 mJy low-noise sample using a spectroscopic redshift distribution rather than the photometric one, and check whether the prediction still lies inside the measured 1-$\sigma$ band between 0.2 and 2 degrees; if it drifts outside, the claimed agreement depended on the assumed $N(z)$.

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Extended reading notes

Core claim

The paper's central claim is that the angular clustering of LoTSS-DR1 radio sources above a flux-density threshold of 2 mJy, restricted to cells with local rms noise below the survey median, is described by $w(\theta)=A(\theta/1\,\mathrm{deg})^{-\gamma}$ with $A=(5.1\pm0.6)\times10^{-3}$ and $\gamma=0.74\pm0.16$, and that between 0.1 deg and 6 deg this matches the theoretical $w(\theta)$ computed from the Planck 2018 best-fit cosmology, the observed photometric redshift distribution, and the bias relation $b(z)=1.6+0.85z+0.33z^2$, without adjusting any model parameter. The same analysis finds $w(\theta)<10^{-2}$ at angular scales above 1 deg and a radio sky that is statistically isotropic at the percent level. A second, independent claim is that the counts-in-cells are non-Poissonian: a compound Poisson distribution fits the data well, with a clustering parameter $n_c$ that exceeds unity at low flux densities and approaches unity above roughly 1 mJy.

Load-bearing premise

The standard-model comparison stands or falls on the assumption that the photometric redshifts for the roughly half of sources that have them faithfully represent the true redshift distribution of the 2 mJy sample, since the paper does not propagate redshift errors or account for the missing half.

Editorial extensions

If this is right

The most reliable clustering measurement is the 2 mJy, low-noise sample, with amplitude $A=(5.1\pm0.6)\times10^{-3}$ and slope $\gamma=0.74\pm0.16$ at 1 deg.
Below 2 mJy, pointing-to-pointing flux-scale errors contaminate the correlation function; above 4 mJy, shot noise dominates.
The counts-in-cell distribution is non-Poissonian and well fitted by a compound Poisson distribution, with $n_c$ above unity at sub-mJy fluxes.
The radio sky is statistically isotropic at the percent level for angular scales above 1 deg.
With the larger sky coverage and better flux calibration of future LoTSS releases, the same statistical pipeline can begin to constrain cosmological parameters such as $\sigma_8$ and the evolution of radio-source bias.

Reading between the lines

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

I infer that the agreement with the standard model is only as strong as the photo-$z$ window: since half the sources lack redshifts and the paper does not propagate photo-$z$ errors, a selection bias in which sources receive a redshift could shift the predicted $w(\theta)$ by more than the quoted uncertainties.
I read the low-redshift overprediction as evidence that mixed AGN and star-forming samples need separate bias functions; a direct extension would fit bias per redshift bin after accounting for photo-$z$ scatter.
A testable corollary is that weighting sources by photo-$z$ uncertainties, or repeating the analysis with a spectroscopic subsample as it becomes available, should keep the predicted curve inside the measured band if the claimed agreement is robust.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read this one if you care about radio continuum surveys as LSS tracers. It gives the first one- and two-point statistics from LoTSS DR1, and the central 2 mJy low-noise measurement is solid. The paper is unusually careful about systematics: completeness injections, mocks built from the local rms noise, a comparison of five estimators, a TreeCorr accuracy check, and jack-knife regional splits. They identify flux-calibration problems at 1 mJy and don't paper over them; instead they restrict their headline result to the 2 mJy mask-1 sample, where the clustering amplitude is A=(5.1±0.6)e-3 and slope gamma=0.74±0.16. That's a real milestone for LOFAR/SKA pathfinder work.

Now the soft spot, and it's the one you'd expect. The 'agreement with Planck 2018' in Sec 6.3 is parameter-free in the sense that they fix cosmology, use the measured z_best histogram as the window function, and use an off-the-shelf bias model from Nusser & Tiwari. But only about 48% of the 2 mJy sources have z_best, and the authors explicitly say they did not estimate or propagate photo-z errors. A biased or smeared N(z) changes the predicted w(theta) directly. And in Fig. 26 the NVSS-derived bias model overshoots the low-z bin (b1=1.2 fits better than 1.6), which suggests the agreement in the full 'Anyz' sample is an average over populations, not a clean validation of the standard model. The authors themselves flag all of this in Sec 6.2 and the conclusions. So the conditional reading is right: the clustering measurement is solid, the cosmological consistency claim is promising but not yet load-bearing.

Who gets value? Cosmologists who want to know what LoTSS DR1 can already say about clustering; people using radio sources for ISW, dipole, or fNL forecasts; and anyone planning to use future LoTSS data releases—the masking and quality-control recipe here is directly reusable. The paper deserves a serious referee and, with minor revision to make the photo-z caveat more prominent in the abstract and conclusions, it should be published. I would bring it to the reading group, and I'd cite the 2 mJy clustering result.

Recommended decision: accept with revisions, not a desk reject.

Referee Report

3 major / 4 minor

Summary. This paper uses the LOFAR Two-metre Sky Survey first data release (LoTSS-DR1) to measure one- and two-point statistics of 144 MHz radio sources over 424 square degrees. The authors estimate point-source completeness by injecting sources into residual maps, define a set of masks and flux-density thresholds, and find that the counts-in-cell distribution is not Poissonian and is well described by a compound Poisson distribution. The differential source counts agree with earlier low-frequency surveys and with the SKADS and T-RECS simulations. The angular two-point correlation function, estimated with the Landy-Szalay estimator and TreeCorr, is fitted by a power law with best-fit values A = (5.1 +/- 0.6) x 10^-3 and gamma = 0.74 +/- 0.16 for the 2 mJy, low-noise ('mask 1') sample. The paper then compares w(theta) in the redshift-selected 'Anyz' sample with CAMB predictions based on Planck 2018 parameters, the z_best photometric-redshift histogram, and the Nusser and Tiwari (2016) bias model, reporting agreement between 0.1 and 6 degrees and concluding that the radio sky is statistically isotropic at the percent level.

Significance. If the conclusions hold, this paper establishes LoTSS-DR1 as a usable cosmological data set and provides one of the first robust measurements of the angular two-point correlation function at 144 MHz over a few hundred square degrees. The data-quality analysis is a clear strength: completeness is assessed by injecting sources and re-running PyBDSF, the random catalogues are built from local rms noise maps, five estimators are compared in Appendix B, and TreeCorr is validated against a brute-force code in Appendix C. The authors are also commendably explicit about their limitations. However, the headline cosmological consistency claim is conditional: the theoretical prediction depends on an external bias model and on the photometric-redshift distribution of only about half of the sources, with no propagation of photo-z errors. The measurement itself is solid, but the cosmological interpretation needs to be either strengthened by sensitivity tests or clearly downgraded to a consistency check under stated assumptions.

major comments (3)

[Section 6.3, Eqs. (14) and (32), Fig. 25] The central claim that the measured w(theta) 'agrees well with the expectation of the cosmological standard model' is not yet established at the claimed precision, because the theoretical curve is computed from the z_best histogram as the source window function without propagating photometric-redshift errors, and with the external bias model b(z) = 1.6 + 0.85z + 0.33z^2 from Nusser and Tiwari (2016). The authors state in Section 6.2 that 'we did not estimate and propagate errors on the redshift estimation'; a biased or smeared N(z) shifts the predicted w(theta) directly, so the agreement shown in Fig. 25 could be fortuitous. I request a sensitivity analysis, such as convolving N(z) with the photo-z error distribution or repeating the prediction with an alternative redshift or bias model, and a correspondingly softened statement in the abstract and conclusions.
[Section 5.3, Table 5, and Section 6.2] Only 24,420 out of 50,977 sources above 2 mJy in mask z (47.9%) have a z_best value, and Section 7 explicitly states that it is currently unclear how representative these sources are of the full sample. The agreement of the clustering parameter nc between sources with and without redshifts (Fig. 18) is a one-point statistic and does not guarantee that the two-point clustering of the 'Anyz' subsample equals that of the full LoTSS-DR1 population. To support the conclusion that LoTSS-DR1 sources as a whole are consistent with the standard model, the paper should either measure w(theta) for the No-z sample or model the selection function; otherwise the consistency statement should be restricted to the redshift-selected subsample.
[Section 6.3, Fig. 26] The binned-redshift comparison shows that the adopted bias model fails in the lowest-redshift bin: the authors must reduce b1 from 1.6 to 1.2 to bring the model into agreement with the data. This demonstrates that the 'no adjusted parameters' agreement in the full Anyz sample is not a clean test of the standard model, since it can be an average over populations with different bias and redshift selection. The full-sample comparison should be presented as a consistency check under an assumed bias model, with the dependence of the conclusion on the bias assumption quantified.

minor comments (4)

[Abstract and Section 7] The phrase 'The deviation from a nmhg distribution' contains a typographical error; it should presumably read 'Poisson' or 'compound Poisson distribution'.
[Section 6.3] The text says the CAMB window function uses the observed redshift distribution 'for sources with z <= 2', but Fig. 16 shows counts up to z ~ 4; please clarify how the z > 2 tail is treated and whether omitting it could bias the prediction at small angular scales.
[Table 6] The column labelled 'z' is used for both the redshift-binned rows and the rows with 'n.a.' for full samples; the caption should state explicitly that the reported A and gamma parameters are power-law fits to the angular correlation function, not constraints on cosmological parameters.
[Fig. 5 caption] The caption refers to 'see also text in Sec. 5.3' for mask z, but mask z is first described in Section 6.2; the cross-reference should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: prediction uses external Planck parameters, external NVSS bias model, and measured N(z); no fitted parameter is relabeled as a prediction.

full rationale

The central claim—that the measured angular two-point correlation of the LoTSS-DR1 2 mJy sample is consistent with the Planck 2018 ΛCDM expectation—is a genuine parameter-free comparison. The inputs to the CAMB prediction are the Planck 2018 best-fit parameters (external), the photometric redshift histogram from the LoTSS-DR1 value-added catalogue, and the bias model b(z) = 1.6 + 0.85z + 0.33z^2 from Nusser & Tiwari (2015)/Tiwari & Nusser (2016), a fit to NVSS data external to this paper. The paper states explicitly that 'we did not adjust any model parameter' (Sec. 6.3), and the power-law amplitudes and slopes in Table 6 are descriptive fits to the data, not inputs to the prediction. Using the same catalogue to provide both N(z) and w(θ) is not tautological: it is a conditional consistency test with fixed external cosmology and bias. The paper also flags its limitations honestly, noting in Sec. 6.2 that 'We did not estimate and propagate errors on the redshift estimation' and that only half the sources have redshifts, which is a systematic-uncertainty concern, not circular reasoning. The cited companion papers with overlapping authorship (Williams et al. 2019 for the value-added catalogue, Duncan et al. 2019 for photometric redshifts, Hale et al. 2018 for the mock-generation method) are data-products and methodology references, not self-citations carrying the theoretical argument; the mock is used only to build the random catalogue for the Landy-Szalay estimator and is verified to have negligible auto-correlation. No derivation step reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction. Hence the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The analysis rests on standard cosmological inputs (Planck 2018, CAMB, Halofit), an external bias model from NVSS, and the measured photometric redshift distribution. The only fitted parameters are the descriptive power-law amplitude and slope of w(theta), which are not used to claim parameter constraints. The main caveat is that the source bias and the photometric redshift distribution are assumed rather than derived within this work.

free parameters (2)

Power-law amplitude A (mask 1, 2 mJy) = 5.1 +/- 0.6 x 10^-3
Fitted to the measured angular correlation function w(theta) = A (theta/1 deg)^(-gamma) over 0.2 to 2.0 deg.
Power-law slope gamma (mask 1, 2 mJy) = 0.74 +/- 0.16
Fitted together with A as a descriptive model of the angular correlation function.

assumptions (4)

domain assumption The theoretical angular power spectrum from CAMB sources, with Planck 2018 cosmological parameters and Halofit, is a correct prediction of the angular clustering of matter.
Invoked in Section 6.3 to generate w(theta) via Eq. (14); the authors rely on the public CAMB sources code and the Planck 2018 best-fit model.
domain assumption The bias model b(z) = 1.6 + 0.85z + 0.33z^2 (Nusser & Tiwari 2016, fitted to NVSS) approximately describes the bias of LoTSS radio sources at 144 MHz.
Used in Section 6.3 as the fiducial source bias; the paper later shows it overestimates low-redshift clustering and replaces it with piecewise constant values b1=1.2, b2=2, b3=3.
domain assumption The photometric redshifts 'z_best' in the LoTSS-DR1 value-added catalogue are accurate enough for the redshift-split correlation measurements.
Used in Sections 5.3 and 6.2 with an explicit disclaimer that redshift errors are not propagated and that misidentification can move sources between bins.
domain assumption The mock catalogue generated from SKADS flux densities and LoTSS noise maps is an unbiased random catalogue for the Landy-Szalay estimator.
Section 4: mock sources are generated assuming unresolved sources and a fixed spectral index alpha=0.7; the paper verifies that data-random and data-mock agree, which supports this.

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Cite this review

Pith. "Pith review of One- and Two-point Source Statistics from the LOFAR Two-metre Sky Survey First Data Release." pith.science (2026). https://pith.science/paper/CVSSXG2J

@misc{pith2026190810309,
  author       = {Pith},
  title        = {Pith review of: One- and Two-point Source Statistics from the LOFAR Two-metre Sky Survey First Data Release},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CVSSXG2J}},
  note         = {Machine review of arXiv:1908.10309}
}

abstract

The LOFAR Two-metre Sky Survey (LoTSS) will map the complete Northern sky and provide an excellent opportunity to study the distribution and evolution of the large-scale structure of the Universe. We study the completeness of the LoTSS first data release (DR1) and find a point-source completeness of 99 % above flux densities of 0.8 mJy and define a suite of quality cuts. We determine the count-in-cell statistics and differential source counts statistic and measure the angular two-point correlation function of the LoTSS radio sources. The counts-in-cell statistic reveals that the distribution of radio sources cannot be described by a spatial Poisson process. Instead, a good fit is provided by a compound Poisson distribution. The differential source counts are in good agreement with previous findings in deep fields at low radio frequencies and with simulated catalogues from the SKA design study sky and the Tiered Radio Extragalactic Continuum Simulation. The angular two-point correlation is $<10^{-2}$ at angular scales $> 1$ deg. Restricting the value added source catalogue to low-noise regions and a flux density threshold of 2 mJy provides our most reliable estimate of the angular two-point correlation. For smaller flux density thresholds systematic issues are identified, most likely related to the flux density calibration of the individual pointings. Based on the distribution of photometric redshifts of LoTSS sources and the Planck 2018 best-fit cosmological model, the theoretically predicted angular two-point correlation between 0.1 deg and 6 deg agrees with the measured clustering for a subsample of radio sources with redshift information. We find agreement with the expectation of large-scale statistical isotropy of the radio sky at the per cent level. The angular two-point correlation agrees well with the expectation of the cosmological standard model. (abbreviated)

Figures

Figures reproduced from arXiv: 1908.10309 by the authors.

**Figure 1.** The distribution of radio sources observed in the LoTSS-DR1 HETDEX spring field. Plotted are all individual sources (top), as well as the number counts per cell in Cartesian projection at HEALPix resolution Nside = 256 (bottom). Observed are nearly 325 000 sources within 58 pointings on the sky covering 424 square degrees. The positions of the five brightest radio sources in terms of integrated flux density are indi… view at source ↗

**Figure 2.** Left: Estimated point-source completeness for each of the 58 pointings in the HETDEX field as a function of flux density. Blue, green and red (dotted) lines indicate inner, outer and the five most incomplete pointings, respectively. Right: Mean point source completeness of all pointings (solid line) and after rejection of the five most incomplete pointings (dotted line). tests (most of them contain only noise) and i… view at source ↗

**Figure 3.** Top: Completeness of the LoTSS-DR1 catalogue per HEALPix cell. Bottom: Completeness of cells after applying a flux density threshold of 0.39 mJy, which corresponds to an overall point source completeness of 95%. sources to the total number of injected sources above a certain flux density threshold. In total we simulated 50 samples with 6000 sources each for each of the 58 pointings. The completeness of each point… view at source ↗

Figures from the paper (22 more)

**Figure 5.** Figure 5: LoTSS-DR1 HETDEX spring field masks: ‘mask p’ rejects all cells shown in dark blue and includes 53 pointings modelled by disks of radius 1.7 deg. Our default ‘mask d’ additionally rejects cells with less than five sources (yellow cells), see also text in Sec.3.4. For a…

**Figure 4.** Figure 4: Top: Source counts for each pointing within angular distance θ around the pointing center, normalized by covered area. Pointings are classified by position in the HETDEX field, with pointings on the edge (green), in the inner field (blue) and undersampled ones (red, do…

**Figure 6.** Figure 6: For comparison we also plot a Poisson distribution with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 6.** Figure 6: Histogram of source counts per cell (blue) and binned Poisson distribution with empirical mean (red line) from the LoTSS-DR1 radio source catalogue at Nside = 256, masked and including only cells with at least five sources (mask d). viously is not a good fit to the dat…

**Figure 8.** Figure 8: The three local rms noise masks. The red cells are included for an average noise < 0.07 mJy/beam in the HEALPix cells (‘mask 1’), red and yellow pixels are included for an average noise of < 0.14 mJy/beam (‘mask 2’) and red, yellow and light blue cells are included for…

**Figure 9.** Figure 9: Mock catalogue of random sources that are detectable at five times the local rms noise and masked with ‘mask d’. masking and higher flux density thresholds) and the demand for statistics (large number of radio sources). 4. Mock catalogues As discussed in Section 2.3, t…

**Figure 10.** Figure 10: Sample statistics of number counts in cells as a function of flux density threshold. Shown are the clustering parameter nc (variance over mean), which is expected to be one for the Poisson distribution, the skewness g1 and excess kurtosis g2 − 3 with error bars calcul…

**Figure 11.** Figure 11: Shown are the skewness (g1) and excess kurtosis (g2 − 3) of the masked LoTSS DR1 value-added source catalogue (mask d), also plotted are the expected moments of a Poisson and compound Poisson distribution. Errors bars for the data sample are computed from bootstrap s…

**Figure 12.** Figure 12: Histograms of LoTSS-DR1 counts-in-cell for the flux density thresholds 1, 2, 4 and 8 mJy. Also shown are the best-fit Poisson and compound Poisson distributions. 0 2 4 6 8 10 S [mJy] 1 0 1 2 3 4 5 Sample statistics Mock catalogue nc g1 g2 ¡ 3 [PITH_FULL_IMAGE:figures…

**Figure 13.** Figure 13: Clustering parameter and coefficients of skewness and kurtosis for a subsample of the mock catalogue, which matches the size of the value-added source catalogue. Error bars are computed from bootstrap sampling. analysis do include such corrections. Remaining discrepan…

**Figure 14.** Figure 14: Differential number counts per flux density interval of the masked LoTSS-DR1 value-added source catalogue for four different masks. Additionally the masked TGSS-ADR1 (147.5 MHz; this work, blue circle), the LOFAR Boötes field (Williams et al. 2016, orange triangle) an…

**Figure 17.** Figure 17: Differential source counts of the LoTSS-DR1 value-added sources masked with ‘mask z’ separated by redshift percentiles, z33 = 0.376 and z66 = 0.705. Additionally the differential source counts of all sources (‘All’) and of all sources with redshift information (‘Any z…

**Figure 15.** Figure 15: Top: Comparison of LoTSS-DR1 differential source counts using ‘mask d’ and SKADS 151 MHz and T-RECS ‘wide’ 150 MHz simulations. The grey band corresponds to a ±20% variation of the LoTSS flux density scale due to uncertainties in the flux density calibration. Bottom…

**Figure 16.** Figure 16: Number of radio sources as a function of available z for four different flux density thresholds, with error bars due to Poisson noise. Only sources with available redshift (‘z_best’) of the LoTSS-DR1 value added source catalogue after applying ‘mask z’ are considered …

**Figure 18.** Figure 18: Clustering parameter nc as function of flux density threshold and available redshift information based on the value ‘z_best’ from the LoTSS-DR1 value-added source catalogue after application of ‘mask z’. Top: We compare radio sources with and without redshift informa…

**Figure 19.** Figure 19: 6. Two-point statistics 6.1. The angular two-point correlation function In order to estimate the angular two-point correlation of radio sources we make use of the estimator proposed by Landy & Szalay (1993), wˆ(θ) = DD − 2DR + RR RR , (26) where DD, DR and RR denote …

**Figure 19.** Figure 19: Counts-in-cell map of the LoTSS-DR1 value-added source catalogue for S > 1.0 mJy and after applying ‘mask d’ (left) and ‘mask 1’ (right). 10 -1 10 0 10 1 µ [deg] 10 -4 10 -3 10 -2 10 -1 | ^w(µ)| Mask d 1 mJy 2 mJy 4 mJy 10 -1 10 0 10 1 µ [deg] 10 -4 10 -3 10 -2 10 -1 …

**Figure 20.** Figure 20: Angular two-point correlation of sources from the LoTSS-DR1 value-added source catalogue after masking with ‘mask d’ (top) and ‘mask 1’ (bottom) and at flux densities above 1, 2 and 4 mJy. Positive and negative values are shown with full and open symbols, respectively…

**Figure 21.** Figure 21: Comparison of two-point angular (auto-)correlation functions for ‘mask d’ for different random catalogues: mock catalogue based on LoTSS local rms noise (data-mock), homogeneous random catalogue accounting for survey geometry only (data-random), and the correlation of…

**Figure 24.** Figure 24: Angular two-point correlation function of sources from the LoTSS-DR1 value-added source catalogue with ‘mask 1’ and flux density threshold of 1 mJy and 2 mJy, for three regions namely ‘Left’, ‘Center’, and ‘Right’. ˆw(θ) for the non-partitioned region with 1 mJy and …

**Figure 25.** Figure 25: Comparison of the angular two-point correlation function estimated from the LoTSS-DR1 value-added source catalogue for radio sources with redshift information and theoretical expectations (solid lines) for the best-fit ΛCDM cosmological parameters from Planck, genera…

**Figure 26.** Figure 26: Angular two-point correlation function for three redshift bins z1,z2 and z3 for a flux density threshold of 2 mJy. The lines show the expectations for the cosmological standard model. Both panels use the Halofit option of CAMB sources, which accounts for the non-linea…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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