Overdispersed radio source counts and excess radio dipole detection

Overdispersed radio source counts and excess radio dipole detection

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  Overdispersed radio sourcecounts and excess radio dipole detection Lukas Böhme,1, ∗ Dominik J. Schwarz,1 Prabhakar Tiwari,2 Morteza Pashapour-Ahmadabadi,1 Benedict Bahr-Kalus,3, 4, 5 Maciej Bilicki,6 Catherine L. Hale,7 Caroline S. Heneka,8 and Thilo M. Siewert1, 9 1 Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany 2 Department of Physics, Guangdong Technion - Israel Institute of Technology, Shantou, Guangdong 515063, P.R. China 3 INAF – Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, Italy 4 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy 5 INFN – Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy 6 Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland 7 Astrophysics, Department of Physics, Denys Wilkinson Building, University of Oxford, Keble Road, Oxford, OX1 3RH, UK 8 Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 9 Evangelisches Klinikum Bethel gGmbH, Kantensiek 11, 33615 Bielefeld, Germany (Dated: September 23, 2025) The source count dipole from wide-area radio continuum surveys allows us to test the cosmological standard model. Many radio sources have multiple components, which can cause an overdispersion of the source counts distribution. We account for this effect via a new Bayesian estimator, based on the negative binomial distribution. Combining the two best understood wide-area surveys, NVSS and RACS-low, and the deepest wide-area survey, LoTSS-DR2, we find that the source count dipole exceeds its expected value as the kinematic dipole amplitude from standard cosmology by a factor of 3.67 ± 0.49 — a 5.4σ discrepancy. Introduction—The cosmological principle asserts that matter and radiation are isotropically and homoge- neously distributed, assuming us to be typical observers. The dominant contribution to the photon number den- sity in the Universe comes from the cosmic microwave background (CMB), a black body at T0 = 2.7 K, which is observed to be highly isotropic. However, its pri- mary deviation from isotropy is a temperature dipole with an amplitude of T1 = 3.4 mK [1, 2], attributed to the motion of the solar system [3, 4], with a speed of v = (369.82 ± 0.11) km/s [2]. The proper motion of an observer in an isotropic and homogeneous universe results in a dipole in the observed source counts [5]. For a flux-limited radio continuum survey, this anisotropy arises from (i) a Doppler shift in frequency ν, affecting the observed flux density Sν, modeled as Sν ∝ να , with spectral index α = α(ν), and (ii) aberration, causing source displacement towards the direction of motion. The kinematic source count dipole is given by [5] d = (2 + x[1 − α]) v c , (1) where x = x(ν) is the slope of the cumulative source count at the flux density limit, dN/dΩ (> Sν) ∝ S−x ν , v is the velocity, and c the speed of light. In addition, the observed source count dipole includes contributions from cosmic large-scale structures, known as the clustering dipole, and from shot noise due to the discrete nature of galaxy counts. Shot noise becomes less significant with a larger sample size. The clustering dipole depends on clustering strength, the growth factor, the redshift distribution of sources, and galaxy bias. In the flat ΛCDM model, the clustering dipole is typically smaller than the kinematic dipole if the sources are pre- dominantly at redshifts greater than 0.1 [6–11]. To measure the dipole anisotropy, a large-sky survey is required. Radio continuum surveys have been key due to their broad sky coverage and detection of radio sources, which are typically at higher redshifts. Recently, infrared quasar surveys have also contributed to such measure- ments. These surveys, using catalogs of radio sources and quasars, have detected a significant dipole [12–19], which is inconsistent with the size of the CMB-predicted kinematic dipole. A notable discrepancy of up to 5.7 σ [10] has been observed between velocities inferred from the CMB dipole and those from infrared sources though the dipole directions align at (αRA, δ) = (168◦ , −7◦ ) in equatorial coordinates [2]. In contrast, few studies [20, 21] find no discrepancy with the size of the CMB-predicted kinematic dipole. In [20], the combination of two catalogues cancels their in- dividual excess dipole amplitudes, as shown in [17]. The second study uses the sparsely sampled MeerKAT Ab- sorption Line Survey Data Release 2 [21]. The pointings exhibit systematic variation in source density of up to 5%, as well as systematic trends with declination, both of which have been accounted for by an empirical fit. Traditionally, studies have assumed that source counts follow a Poisson distribution, the standard model for shot noise, which works well for sparse data. However, high- resolution and high-sensitivity radio surveys challenge the independence of sources assumption due to overdis- persion from multi-component sources. This overdisper- sion is significantly better modeled using the negative binomial distribution [22], providing a more accurate rep- arXiv:2509.16732v1 [astro-ph.CO] 20 Sep 2025
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  2 resentation of thecounts and associated noise. In this letter, we present the first analysis of the ra- dio source count dipole that accounts for overdispersion, including the deepest wide-area radio survey, the Low Frequency Array (LOFAR; [23]) Two-metre Sky Survey Data Release 2 (LoTSS-DR2; [24]). Previous analyses of LoTSS-DR1 and LoTSS-DR2 excluded a Poissonian distribution with high statistical confidence, favoring a negative binomial distribution [25, 26]. We extend this analysis to five other wide-area radio continuum surveys. Data and theory—The following six surveys, sorted by increasing central frequency, are examined in this letter: (1) The LoTSS-DR2 survey covers 5,635 deg2 of the northern sky at 144 MHz, detecting nearly 4.4 million radio sources. It features a resolution of 6′′ and a me- dian sensitivity of 83 µJy beam−1 , with an astrometric accuracy of 0.2′′ . The point source completeness is es- timated at 95% for 1.1 mJy. We use the inner masked regions as defined in [27]. (2) The TIFR GMRT Sky Survey, TGSS, alternative data release 1 [28], conducted by the Giant Meterwave Radio Telescope (GMRT; [29]) covers the sky north of δ = −53◦ . It observes ∼ 0.62 million sources at a central frequency of 147 MHz with a resolution of 25′′ for δ > 19◦ , and 25′′ × 25′′ / cos(δ − 19◦ ) below. The median rms noise is 3.5 mJy beam−1 , while 50% point source completeness is expected to be reached at 25 mJy. We mask δ < −49◦ and δ > 80◦ and remove three under- or overdense regions, as seen in Fig. 2 in the End Matter. (3) The Rapid ASKAP Continuum Survey, RACS [30] is the first large sky survey with the Australian Square Kilometre Array Pathfinder (ASKAP; [31, 32]). The first data release is RACS-low at 887.5 MHz [33]. The RACS- low survey covers the extragalactic sky (|b| > 5◦ ) for δ ≤ +30◦ at a resolution of 25′′ and finds ∼ 2.1 million radio sources. It is estimated to be 95% point source complete at an integrated flux density of ∼ 3 mJy. (4) The RACS-mid survey is the second data release of RACS at 1367.5 MHz [34] and covers the sky at δ ≤ +49◦ . We use here the catalogue where images are convolved to a fixed 25′′ resolution which consists of 3 million radio sources. The estimated point source completeness is 95% at 1.6 mJy. For both RACS releases (low and mid) we mask δ < −78◦ and δ > +28◦ as well as three and one small fields, respectively. (5) The NRAO VLA Sky Survey, NVSS, [35] covers the sky for δ ≥ −40◦ at 1400 MHz and a resolution of 45′′ . The completeness is given as 99% at 3.4 mJy. We mask δ < −39◦ to prevent edge effects due to binning. (6) The Very Large Array Sky Survey, VLASS, [36, 37] observes the sky north of δ = −40◦ in S band (2–4 GHz) at a resolution of 2′′ .5. The point source completeness of 99.8% is reached at 3 mJy. A flux density correction of 10% is applied to the cataloged flux densities from the first epoch [36]. For masking we remove δ < −40◦ . Additionally, for all surveys we mask the galactic plane at latitudes |b| < 10◦ . Measurements are additionally af- fected by systematics, even above the completeness limit. To achieve a homogeneous sky coverage, flux density cuts up to an order of magnitude above the completeness limit are necessary [19, 38]. The counts-in-cells distribution of continuum radio sources above a certain flux density is expected to fol- low a homogeneous Poisson point process [39, 40]. The probability of finding N sources with an expected mean λ is given by the Poisson distribution: PP(N) = λN N! e−λ , (2) where the variance is σ2 P = λ. However, the Poisson model cannot describe the ob- served overdispersion in source counts, as the high- resolution radio sky includes not only independent point sources but also multiple components from extended sources. Source-finding algorithms struggle to associate multi-component sources, and may count components as individual sources [24, 41]. This results in overdispersion, where the observed variance of source counts exceeds the mean. To address this, we adopt a compound Poisson (Cox) process [42]. A brief overview of the model is pro- vided below; for details, see [25, 26]. In any cell i, the number of independent radio objects is Oi, with each object having Cji components, where j ∈ [1, . . . , Oi]. The total count of observed radio sources in cell i is Ni = Oi X j=1 Cji. (3) If Oi follows a Poisson distribution with mean λ, and Cji follows a logarithmic distribution Log(p) with p ∈ (0, 1) [43], this results in a negative binomial distribution. The probability of detecting Ni radio sources in cell i is [26] PNB(Ni) = Ni + r − 1 Ni pNi (1 − p)r , (4) where p encodes clustering, and r relates to the Poisson mean λ by r = λ(1−p)/p. The average number of compo- nents per object are given by the logarithmic distribution expectation: µLog = −1 ln(1 − p) p 1 − p . (5) The mean and variance of the negative binomial distri- bution are µNB = p r 1 − p , σ2 NB = p r (1 − p)2 . So far, this model describes a uniform Universe with random fluctuations due to shot noise. To include the ra- dio source count dipole, we adopt the Bayesian approach
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  3 TABLE I. Comparisonof counts-in-cells between best-fit Pois- son and negative binomial distributions as seen in Fig. 2 in the End Matter. Cells cover a sky area of ∼ 3.4 deg2 . Survey # Cells Smin[mJy] χ2 r,P χ2 r,NB µLog LoTSS-DR2 1267 5 2.25 0.76 1.70 ± 1.28 TGSS 8395 100 23.79 0.87 1.22 ± 0.55 RACS-low 7347 20 4.51 1.18 1.10 ± 0.33 RACS-mid 7365 20 2.80 0.70 1.07 ± 0.27 NVSS 8364 20 4.14 1.09 1.09 ± 0.32 VLASS 8369 10 40.9 0.90 1.32 ± 0.71 that was proposed for the Poisson distribution [19]. The source count dipole d is a variation of the expected source counts λi for cell i, λi = λ(1 + d cos θi), (6) with θi the angle between the observed cell i and the dipole direction. For the negative binomial distribu- tion the parameter r is directly linked to the direction- dependent quantity ri = r(1 + d cos θi). (7) To estimate the source count dipole, we maximize the log-likelihood of the negative binomial dipole distribution log L = X i log(Γ[Ni + ri]) − log(Γ[ri]) (8) + ri log(1 − p) + Ci , where Ci = Ni log p−log(Γ[Ni +1]), and Γ[·] denotes the gamma function. We use a Markov Chain Monte Carlo method to estimate r, d and the direction θ(αRA, δ), while p is inferred from the empirical mean and variance via p = 1 − b µ/c σ2. Table I lists the number of cells, reduced χ2 r values of the best-fit Poisson and negative binomial distribu- tions, and the inferred mean number of components µLog for each radio continuum surveys. Figure 2 in the End Matter displays the corresponding masked survey maps, counts-in-cells histograms, and comparisons between the two best-fit distributions. The negative binomial distri- bution provides a significant better fit to the overdis- persed data across all surveys. Results—We use the HEALPix[44] binning scheme with Nside = 32, yielding 12288 equal-area sky cells. Log-likelihoods are maximized using Bilby [45, 46] with the emcee sampler [47]. Estimator accuracy was verified through numerical simulations with 3 × 105 sources over the extragalactic sky (−20◦ δ 90◦ ), matching the source count and sky coverage of the surveys used. To find the expected dipole amplitude dexp from the in- ferred CMB dipole velocity, we measure x and α for each survey and flux density cut. x is measured over a narrow TABLE II. Expected dipole amplitude dexp using (1), the flux density cut, the measured spectral index α and the corre- sponding differential source count slope x. Errors for x are smaller than 10−3 and therefore not listed, but used in the error calculation of dexp. Survey Smin x α dexp (mJy) (×10−2 ) LoTSS-DR2 5 0.74 −0.73 ± 0.23 0.405 ± 0.021 TGSS 100 0.79 −0.76 ± 0.18 0.418 ± 0.018 RACS-low 20 0.85 −0.98 ± 0.24 0.454 ± 0.025 RACS-mid 20 0.88 −0.81 ± 0.30 0.443 ± 0.025 NVSS 20 0.88 −0.73 ± 0.23 0.435 ± 0.025 VLASS 10 0.99 −0.71a 0.456 ± 0.037 a taken from [36], which used the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; [48]) to calculate α. flux range (O(1 mJy)) around the flux density cut to en- sure a good fit. For α, a simple positional cross-matching method is applied between surveys, with a search radius equal to half the resolution of NVSS, 45′′ /2; a flux den- sity cut is applied to the survey for which α is calculated. The expected dipole amplitude is calculated using Eq. (1) and the results are summarized in Table II. More details can be found in Table IV in the End Matter. The novel estimator introduced in Eq. (8) is applied to each survey at different flux density cut. A uniform prior is used for both dipole direction and amplitude d, labeled as ‘Free’ in Table III. Restricting the direction to the CMB dipole direction is labeled as ‘CMB’. The ‘Free’ results fall into two categories: (i) ‘Problematic’: LoTSS- DR2, RACS-mid and VLASS show unreliable δ retrievals with TGSS yielding an unusually high amplitude; and (ii) Robust: RACS-low and NVSS provide stable and robust measurements, as listed in Table III under ‘Free’. Figure 3 in the End Matter illustrates δ behavior across flux density cuts. LoTSS-DR2 and VLASS tend to- ward the celestial poles, while RACS-mid shows strong flux-dependent variations in δ. For LoTSS-DR2, this is expected due to limited sky coverage and is confirmed via simulations with the survey mask. Using random mocks [27] and injecting a dipole as in Table 3, we find a pronounced variance of δ. VLASS consistently favors δ ≥ 60◦ , regardless of flux cut, indicating sys- tematic issues, likely from declination-dependent effects. As VLASS is based on Quick Look images, it lacks the precision of fully calibrated surveys. Dipoles near the poles can result from declination-dependent systematics in telescope sensitivity or calibration. For RACS-low and NVSS, we confirm previous results [19] showing that the excess dipole is about 3.5 times the expected CMB dipole amplitude. We confirm the increased dipole in TGSS, which is roughly ten times larger than the CMB expectation [16]. This excess likely stems, at least partly, from large-scale
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  4 systematic deviations influx density calibration [49]. However, an improved catalogue [50] does not yield a lower amplitude. Next, we constrain the estimator’s direction to the CMB dipole to measure the amplitude projected along that axis, applying robust flux density cuts for each sur- vey, well above the 95% completeness limits and in line with prior works. For LoTSS-DR2 and RACS-mid, the constrained estimator yields results consistent with the reported dipole excess of ≈ 3 × dexp. However, these values should be considered lower bounds, since the true dipole may deviate by several tens of degrees from the CMB dipole direction. This is evident in VLASS, where the constrained amplitude matches expectations, but the actual orientation differ by 60◦ − 80◦ (see Fig. 3 in the End Matter.) All measurements are repeated with the Poisson esti- mator [19] for which the results are also provided in Ta- ble III. Differences between the Poisson and negative bi- nomial estimators mainly appear in the error bars, which vary significantly depending on survey properties such as resolution and sensitivity. For instance, in LoTSS- DR2, the standard deviation under the negative bino- mial model is 0.44 × 10−2 , compared to 0.27 × 10−2 for the Poisson case, an increase of around 60%. As the final step of our analysis, we combine the two wide-area surveys, RACS-low and NVSS, which are the only ones yielding stable results with the uncon- strained estimator, both alone and together with the deeper LoTSS-DR2 survey (without fixing the dipole di- rection). As shown in Fig. 1 and detailed in Table III, the com- bined measurement from all three surveys yields a di- rection closely aligned with the CMB dipole (αRA, δ) = (165◦ ± 8◦ , −11◦ ± 11◦ ), with an angular separation ∆θ = (5 ± 10)◦ , and amplitude (3.67 ± 0.49) × dexp. Adding LoTSS-DR2 increases the discrepancy in ampli- tude from 4.8σ (using the Poisson estimator on RACS- low and NVSS [19]) to over 5.4σ with our negative bi- nomial estimator. Applying the negative binomial es- timator to RACS-low and NVSS alone (with the same parameters as [19]) reduces the tension to 4.5σ, with ∆θ = (8 ± 10)◦ , as expected due to the larger error bars. Conclusion—In this work, we develop a new radio dipole estimator based on the negative binomial distri- bution, motivated by the observed source count distri- bution. It is applied to six different radio surveys span- ning 120 MHz to 4 GHz, covering most of the northern or southern sky. Combining two wide-area surveys (RACS-low, NVSS) with the deeper LoTSS-DR2, our estimator robustly con- strains the dipole amplitude to (3.67 ± 0.49) × dexp, re- vealing a 5.4σ excess over the CMB-inferred kinematic dipole. The direction remains consistent with the CMB dipole within 1σ. This marks the first significant detec- 160 200 RA [°] 2 4 d/dexp 40 0 40 Dec [°] 160 200 RA [°] 40 0 40 Dec [°] NVSS+RACS-low LoTSS+NVSS+RACS-low FIG. 1. Corner plot for the combined dipole estimate from LoTSS-DR2, NVSS, and RACS-low. The amplitude is pre- sented in multiples of the expected dipole amplitude, with the green lines and dot representing the expected values based on the CMB dipole. tion of the radio dipole excess using radio surveys alone. We find that the multi-component nature of radio sources, captured by our estimator, is a key factor in measuring the cosmic radio dipole and can increase the amplitude uncertainty by up to 60%. Possible explanations for the excess radio source count dipole include greater-than-expected contamination from local sources, potentially producing a clustering dipole not predicted by ΛCDM [6, 7]. A local bulk flow ex- tending beyond ΛCDM expectations may also contribute [51–53]. Another possible explanation is systematics, exten- sively studied across many surveys (see Appendix B in the End Matter for references and details). Achieving precise flux density calibration over wide fields is chal- lenging and may introduce bias. An alternative strategy is sparse sky sampling via independent pointings, which yields a dipole consistent with the CMB expectation [21]; however, unlike the active galactic nuclei-dominated sur- veys in this work, that sample is dominated by star- forming galaxies. Survey geometry, whether sparse or wide, can itself bias dipole measurements and must be tested via simu- lations. Galactic synchrotron emission may affect sensi- tivity if not properly addressed during calibration. How- ever, it is unlikely that these systematics would be con- sistent across all radio frequencies or mimic results seen in infrared quasars studies [54].
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  5 TABLE III. Measurementresults of amplitude d and direction (αRA,δ) of the radio source count dipole and comparison between Poisson (P) and negative binomial distribution (NB). Smin is the applied minimum flux density cut at the survey’s frequency, while this value scaled to 144 MHz is given as S144 and calculated with a general spectral index of −0.75. The direction ‘Free’ refers to an unconstrained estimator, with the preferred direction given in the columns (αRA,δ). ‘CMB’ refers to the constrained estimator, where the direction is fixed to the CMB dipole direction. Values without error bars were fixed during parameter estimation. Survey ν Smin S144 N Direction d (P) d (NB) αRA (P) αRA (NB) δ (P) δ (NB) (MHz) (mJy) (mJy) (×10−2 ) (×10−2 ) (deg) (deg) (deg) (deg) LoTSS-DR2 144 5 5 376658 CMB 1.36+0.24 −0.30 1.39+0.45 −0.42 167.94 167.94 -6.94 -6.94 TGSS 147 100 102 234619 Free 5.33+0.34 −0.36 5.33+0.40 −0.42 140.7+3.5 −3.7 140.7+4.4 −4.4 1.5+4.6 −4.4 1.6+5.6 −5.2 100 CMB 4.51+0.30 −0.34 4.51+0.36 −0.40 167.94 167.94 -6.94 -6.94 RACS-low 888 20 78 330540 Free 1.78+0.26 −0.26 1.80+0.28 −0.30 191.1+9.7 −9.8 191.1+10.8 −10.8 5.8+13.8 −12.5 6.0+14.2 −14.1 20 CMB 1.70+0.24 −0.24 1.73+0.27 −0.30 167.94 167.94 -6.94 -6.94 RACS-mid 1367 20 108 237069 CMB 1.15+0.30 −0.30 1.15+0.33 −0.33 167.94 167.94 -6.94 -6.94 NVSS 1400 20 110 263125 Free 1.54+0.30 −0.34 1.54+0.36 −0.36 148.4+12.6 −12.5 148.1+13.9 −13.1 −10.3+14.7 −14.4 −9.5+16.0 −15.6 20 CMB 1.36+0.27 −0.30 1.36+0.30 −0.30 167.94 167.94 -6.94 -6.94 VLASS 3000 10 97.5 275516 CMB 0.55+0.27 −0.30 0.55+0.36 −0.30 167.94 167.94 -6.94 -6.94 Survey combination Direction d/dexp (P) d/dexp (NB) αRA (P) αRA (NB) δ (P) δ (NB) RACS-low + NVSS Free 3.55+0.46 −0.46 3.57+0.50 −0.50 175.4+8.3 −8.1 175.0+9.0 −8.9 −3.0+10.5 −10.5 −2.8+11.8 −11.3 LoTSS-DR2 + RACS-low + NVSS Free 3.92+0.46 −0.45 3.67+0.49 −0.49 156.1+6.8 −6.7 164.9+8.3 −8.0 −20.1+9.6 −8.7 −10.8+11.4 −10.8 A genuine discrepancy between the dipole amplitudes measured in the CMB and large-scale structure frames would have profound cosmological implications. Depend- ing on whether the cause is local or large-scale, it could challenge the cosmological principle itself. Upcoming large-area sky surveys such as LoTSS-DR3, LoLSS-DR2 (northern sky at 54 MHz), RACS-high [55], RACS-low-DR2, EMU [56, 57], and eventually the SKA surveys, along with wide-field spectroscopic follow-ups like the WEAVE-LOFAR project [58], will significantly improve our understanding of the origin of the radio dipole. Acknowledgments—We acknowledge discussions with Nathan Secrest, Sebastian von Hausegger, and Jonah Wagenveld who helped us to sharpen and develop our arguments. LB acknowledges support by the Studi- enstiftung des deutschen Volkes. MPA acknowledges support from the Bundesministerium für Bildung und Forschung (BMBF) ErUM-IFT 05D23PB1. BB-K ac- knowledges support from INAF for the project ‘Paving the way to radio cosmology in the SKA Observatory era: synergies between SKA pathfinders/precursors and the new generation of optical/near-infrared cosmologi- cal surveys’ (CUP C54I19001050001). MB is supported by the Polish National Science Center through grants no. 2020/38/E/ST9/00395 and 2020/39/B/ST9/03494. CLH acknowledges support from the Oxford Hintze Cen- tre for Astrophysical Surveys which is funded through generous support from the Hintze Family Charitable Foundation. CSH’s work is funded by the Volkswa- gen Foundation. CSH acknowledges additional sup- port from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excel- lence Strategy EXC 2181/1 - 390900948 (the Heidel- berg STRUCTURES Excellence Cluster). LOFAR is the Low Frequency Array designed and constructed by AS- TRON. It has observing, data processing, and data stor- age facilities in several countries, which are owned by various parties (each with their own funding sources), and which are collectively operated by the ILT founda- tion under a joint scientific policy. The ILT resources have benefited from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Uni- versité d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Depart- ment of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Tech- nology Facilities Council, UK; Ministry of Science and Higher Education, Poland; The Istituto Nazionale di As- trofisica (INAF), Italy. This scientific work uses data obtained from Inyarrimanha Ilgari Bundara/the Murchi-
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  6 son Radio-astronomy Observatory.We acknowledge the Wajarri Yamaji People as the Traditional Owners and native title holders of the Observatory site. CSIRO’s ASKAP radio telescope is part of the Australia Telescope National Facility (https://ror.org/05qajvd42). Opera- tion of ASKAP is funded by the Australian Govern- ment with support from the National Collaborative Re- search Infrastructure Strategy. ASKAP uses the re- sources of the Pawsey Supercomputing Research Cen- tre. Establishment of ASKAP, Inyarrimanha Ilgari Bun- dara, the CSIRO Murchison Radio-astronomy Observa- tory and the Pawsey Supercomputing Research Centre are initiatives of the Australian Government, with sup- port from the Government of Western Australia and the Science and Industry Endowment Fund. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. The VLA is run by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We used a range of python software packages during this work and the production of this manuscript, includ- ing Astropy [59–61], matplotlib [62], NumPy [63], SciPy [64], healpy [65], and HEALPix [66]. ∗ lboehme@physik.uni-bielefeld.de [1] A. A. Penzias and R. W. Wilson, “A Measurement of Excess Antenna Temperature at 4080 Mc/s.” Astrophys. J. 142, 419 (1965). [2] Planck Collaboration, N. Aghanim, Y. Akrami, F. Ar- roja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Bal- lardini, A. J. Banday, R. B. Barreiro, et al., “Planck 2018 results. I. Overview and the cosmological legacy of Planck,” Astron. Astrophys. 641 (2020). [3] J. M. Stewart and D. W. Sciama, “Peculiar Velocity of the Sun and its Relation to the Cosmic Microwave Back- ground,” Nat. 216, 748 (1967). [4] P. J. E. Peebles and D. T. Wilkinson, “Comment on the Anisotropy of the Primeval Fireball,” Phys. Rev. 174, 2168 (1968). [5] G. F. R. Ellis and J. E. Baldwin, “On the expected anisotropy of radio source counts,” Mon. Not. R. Astron. Soc. 206, 377 (1984). [6] P. Tiwari and A. Nusser, “Revisiting the NVSS number count dipole,” J. Cosmol. Astropart. Phys. 2016, 062 (2016). [7] C. A. P. Bengaly, T. M. Siewert, D. J. Schwarz, and R. Maartens, “Testing the standard model of cosmology with the SKA: The cosmic radio dipole,” Mon. Not. R. Astron. Soc. 486, 1350 (2019). [8] Y.-T. Cheng, T.-C. Chang, and A. Lidz, “Is the Radio Source Dipole from NVSS Consistent with the Cosmic Microwave Background and ΛCDM?” Astrophys. J. 965, 32 (2024). [9] O. T. Oayda and G. F. Lewis, “Testing the cosmological principle: on the time dilation of distant sources,” Mon. Not. R. Astron. Soc. 523, 667 (2023). [10] L. Dam, G. F. Lewis, and B. J. Brewer, “Testing the cos- mological principle with CatWISE quasars: A bayesian analysis of the number-count dipole,” Mon. Not. R. As- tron. Soc. 525, 231 (2023). [11] P. Tiwari, D. J. Schwarz, G.-B. Zhao, R. Durrer, M. Kunz, and H. Padmanabhan, “An Independent Mea- sure of the Kinematic Dipole from SDSS,” Astrophys. J. 975, 279 (2024). [12] C. Blake and J. Wall, “A velocity dipole in the distribu- tion of radio galaxies,” Nature 416, 150 (2002). [13] A. K. Singal, “Large peculiar motion of the solar sys- tem from the dipole anisotropy in sky brightness due to distant radio sources,” Astrophys. J. 742, L23 (2011). [14] C. Gibelyou and D. Huterer, “Dipoles in the Sky,” Mon. Not. R. Astron. Soc. 427, 1994 (2012). [15] M. Rubart and D. J. Schwarz, “Cosmic radio dipole from NVSS and WENSS,” Astron. Astrophys. 555, A117 (2013). [16] T. M. Siewert, M. Schmidt-Rubart, and D. J. Schwarz, “Cosmic radio dipole: Estimators and frequency depen- dence,” Astron. Astrophys. 653, A9 (2021). [17] N. J. Secrest, S. von Hausegger, M. Rameez, R. Mo- hayaee, and S. Sarkar, “A Challenge to the Standard Cosmological Model,” Astrophys. J. 937, L31 (2022). [18] V. Mittal, O. T. Oayda, and G. F. Lewis, “The cos- mic dipole in the Quaia sample of quasars: A Bayesian analysis,” Mon. Not. R. Astron. Soc. 527, 8497 (2024). [19] J. D. Wagenveld, H. R. Klöckner, and D. J. Schwarz, “The cosmic radio dipole: Bayesian estimators on new and old radio surveys,” Astron. Astrophys. 675, A72 (2023). [20] J. Darling, “The Universe is Brighter in the Direction of Our Motion: Galaxy Counts and Fluxes are Consis- tent with the CMB Dipole,” Astrophys. J. Lett. 931, L14 (2022). [21] J. D. Wagenveld, H. R. Klöckner, N. Gupta, S. Sekhar, P. Jagannathan, P. P. Deka, J. Jose, S. A. Balashev, D. Borgaonkar, A. Chatterjee, et al., “The MeerKAT Ab- sorption Line Survey Data Release 2: Wideband contin- uum catalogues and a measurement of the cosmic radio dipole,” Astron. Astrophys. 690, A163 (2024). [22] N. L. Johnson, A. W. Kemp, and S. Kotz, Univari- ate Discrete Distributions, 3rd ed. (Wiley, Hoboken, N.J, 2005). [23] M. P. van Haarlem, M. W. Wise, A. W. Gunst, G. Heald, J. P. McKean, J. W. T. Hessels, A. G. de Bruyn, R. Ni- jboer, J. Swinbank, R. Fallows, et al., “LOFAR: The LOw-Frequency ARray,” Astron. Astrophys. 556, A2 (2013). [24] T. W. Shimwell, M. J. Hardcastle, C. Tasse, P. N. Best, H. J. A. Röttgering, W. L. Williams, A. Botteon, A. Drabent, A. Mechev, A. Shulevski, et al., “The LO- FAR Two-metre Sky Survey. V. Second data release,” Astron. Astrophys. 659, A1 (2022). [25] T. M. Siewert, C. Hale, N. Bhardwaj, M. Biermann, D. J. Bacon, M. Jarvis, H. J. A. Röttgering, D. J. Schwarz, T. Shimwell, P. N. Best, et al., “One- and two-point source statistics from the LOFAR Two-metre Sky Survey first data release,” Astron. Astrophys. 643, A100 (2020). [26] M. Pashapour-Ahmadabadi, L. Böhme, T. M. Siewert, D. J. Schwarz, C. L. Hale, C. Heneka, P. Tiwari, and J. Zheng, “Cosmology from LOFAR Two-metre Sky Sur- vey Data Release 2: Counts-in-cells statistics,” Astron. Astrophys. 698, A148 (2025).
• 7.
  7 [27] C. L.Hale, D. J. Schwarz, P. N. Best, S. J. Nakoneczny, D. Alonso, D. Bacon, L. Böhme, N. Bhardwaj, M. Bilicki, S. Camera, et al., “Cosmology from LOFAR Two-metre Sky Survey Data Release 2: Angular clustering of radio sources,” Mon. Not. R. Astron. Soc. 527, 6540 (2023). [28] H. T. Intema, P. Jagannathan, K. P. Mooley, and D. A. Frail, “The GMRT 150 MHz all-sky radio survey. First al- ternative data release TGSS ADR1,” Astron. Astrophys. 598, A78 (2017). [29] G. Swarup, “Giant metrewave radio telescope (GMRT),” in IAU Colloq. 131: Radio Interferometry. Theory, Tech- niques, and Applications, Astronomical Society of the Pa- cific Conference Series, Vol. 19, edited by T. J. Cornwell and R. A. Perley (1991) pp. 376–380. [30] D. McConnell, C. L. Hale, E. Lenc, J. K. Banfield, G. Heald, A. W. Hotan, J. K. Leung, V. A. Moss, T. Mur- phy, A. O’Brien, et al., “The Rapid ASKAP Continuum Survey I: Design and first results,” Publ. Astron. Soc. Aust. 37, e048 (2020). [31] S. Johnston, R. Taylor, M. Bailes, N. Bartel, C. Baugh, M. Bietenholz, C. Blake, R. Braun, J. Brown, S. Chat- terjee, et al., “Science with ASKAP,” Exp. Astron. 22, 151 (2008). [32] A. W. Hotan, J. D. Bunton, A. P. Chippendale, M. Whit- ing, J. Tuthill, V. A. Moss, D. McConnell, S. W. Amy, M. T. Huynh, J. R. Allison, et al., “Australian square kilometre array pathfinder: I. system description,” Publ. Astron. Soc. Aust. 38, e009 (2021). [33] C. L. Hale, D. McConnell, A. J. M. Thomson, E. Lenc, G. H. Heald, A. W. Hotan, J. K. Leung, V. A. Moss, T. Murphy, J. Pritchard, et al., “The Rapid ASKAP Con- tinuum Survey Paper II: First Stokes I Source Catalogue Data Release,” Publ. Astron. Soc. Aust. 38, e058 (2021). [34] S. W. Duchesne, J. A. Grundy, G. H. Heald, E. Lenc, J. K. Leung, D. McConnell, T. Murphy, J. Pritchard, K. Rose, A. J. M. Thomson, et al., “The Rapid ASKAP Continuum Survey V: Cataloguing the sky at 1 367.5 MHz and the second data release of RACS-mid,” Publ. Astron. Soc. Aust. 41, e003 (2024). [35] J. J. Condon, W. D. Cotton, E. W. Greisen, Q. F. Yin, R. A. Perley, G. B. Taylor, and J. J. Broderick, “The NRAO VLA Sky Survey,” Astron. J. 115, 1693 (1998). [36] Y. A. Gordon, M. M. Boyce, C. P. O’Dea, L. Rudnick, H. Andernach, A. N. Vantyghem, S. A. Baum, J.-P. Bui, M. Dionyssiou, S. Safi-Harb, et al., “A Quick Look at the 3 GHz Radio Sky. I. Source Statistics from the Very Large Array Sky Survey,” Astrophys. J. Suppl. Ser. 255, 30 (2021). [37] M. Lacy, S. A. Baum, C. J. Chandler, S. Chatterjee, T. E. Clarke, S. Deustua, J. English, J. Farnes, B. M. Gaensler, N. Gugliucci, et al., “The Karl G. Jansky Very Large Array Sky Survey (VLASS). Science Case and Survey Design,” Publ. Astron. Soc. Pac. 132, 035001 (2020). [38] O. T. Oayda, V. Mittal, G. F. Lewis, and T. Murphy, “A Bayesian approach to the cosmic dipole in radio galaxy surveys: joint analysis of NVSS RACS,” Mon. Not. R. Astron. Soc. 531, 4545 (2024). [39] J. Neyman, E. L. Scott, and C. D. Shane, “The Index of Clumpiness of the Distribution of Images of Galaxies.” Astrophys. J. Suppl. Ser. 1, 269 (1954). [40] P. J. E. Peebles, The Large-Scale Structure of the Uni- verse, Princeton Series in Physics (Princeton University Press, 1980). [41] M. J. Hardcastle, M. A. Horton, W. L. Williams, K. J. Duncan, L. Alegre, B. Barkus, J. H. Croston, H. Dickin- son, E. Osinga, H. J. A. Röttgering, et al., “The LOFAR Two-Metre Sky Survey. VI. Optical identifications for the second data release,” Astron. Astrophys. 678, A151 (2023). [42] D. R. Cox, “Some Statistical Methods Connected with Series of Events,” Journal of the Royal Statistical Society: Series B (Methodological) 17, 129 (1955). [43] R. A. Fisher, A. S. Corbet, and C. B. Williams, “The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Popu- lation,” The Journal of Animal Ecology 12, 42 (1943). [44] Http://healpix.sourceforge.net. [45] G. Ashton, M. Hübner, P. D. Lasky, C. Talbot, K. Ack- ley, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov, et al., “BILBY: A User-friendly Bayesian Inference Library for Gravitational-wave Astronomy,” Astrophys. J. Suppl. Ser. 241, 27 (2019). [46] I. M. Romero-Shaw, C. Talbot, S. Biscoveanu, V. D’Emilio, G. Ashton, C. P. L. Berry, S. Coughlin, S. Galaudage, C. Hoy, M. Hübner, et al., “Bayesian in- ference for compact binary coalescences with BILBY: Validation and application to the first LIGO-Virgo gravitational-wave transient catalogue,” Mon. Not. R. Astron. Soc. 499, 3295 (2020). [47] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, “Emcee: The MCMC Hammer,” Publ. Astron. Soc. Pac. 125, 306 (2013). [48] R. H. Becker, R. L. White, and D. J. Helfand, “The FIRST Survey: Faint Images of the Radio Sky at Twenty Centimeters,” Astrophys. J. 450, 559 (1995). [49] P. Tiwari, S. Ghosh, and P. Jain, “The Galaxy Power Spectrum from TGSS ADR1 and the Effect of Flux Cal- ibration Systematics,” Astrophys. J. 887, 175 (2019). [50] N. Hurley-Walker, “A Rescaled Subset of the Alternative Data Release 1 of the TIFR GMRT Sky Survey,” arXiv e-prints , arXiv:1703.06635 (2017). [51] Y. Hoffman, D. Pomarède, R. B. Tully, and H. M. Cour- tois, “The dipole repeller,” Nat. Astron. 1, 0036 (2017). [52] R. Watkins, T. Allen, C. J. Bradford, A. Ramon, A. Walker, H. A. Feldman, R. Cionitti, Y. Al-Shorman, E. Kourkchi, and R. B. Tully, “Analysing the large-scale bulk flow using cosmicflows4: increasing tension with the standard cosmological model,” Mon. Not. R. Astron. Soc. 524, 1885 (2023). [53] Courtois, H. M., Dupuy, A., Guinet, D., Baulieu, G., Ruppin, F., and Brenas, P., “Gravity in the local uni- verse: Density and velocity fields using cosmicflows-4,” Astron. Astrophys. 670, L15 (2023). [54] N. J. Secrest, S. von Hausegger, M. Rameez, R. Mo- hayaee, S. Sarkar, and J. Colin, “A Test of the Cos- mological Principle with Quasars,” Astrophys. J. 908, L51 (2021). [55] S. W. Duchesne, K. Ross, A. J. M. Thomson, E. Lenc, T. Murphy, T. J. Galvin, A. W. Hotan, V. Moss, and M. T. Whiting, “The Rapid ASKAP Continuum Survey (RACS) VI: The RACS-high 1655.5 MHz images and cat- alogue,” Publ. Astron. Soc. Aust. 42, e038 (2025). [56] R. P. Norris, A. M. Hopkins, J. Afonso, S. Brown, J. J. Condon, L. Dunne, I. Feain, R. Hollow, M. Jarvis, M. Johnston-Hollitt, et al., “EMU: Evolutionary Map of the Universe,” Publ. Astron. Soc. Aust. 28, 215 (2011). [57] A. M. Hopkins, A. Kapinska, J. Marvil, T. Vernstrom, J. D. Collier, R. P. Norris, Y. A. Gordon, S. W. Duch-
• 8.
  8 esne, L. Rudnick,N. Gupta, et al., “The Evolutionary Map of the Universe: A new radio atlas for the south- ern hemisphere sky,” arXiv e-prints , arXiv:2505.08271 (2025), arXiv:2505.08271. [58] D. J. B. Smith, P. N. Best, K. J. Duncan, N. A. Hatch, M. J. Jarvis, H. J. A. Röttgering, C. J. Simpson, J. P. Stott, R. K. Cochrane, K. E. Coppin, et al., “The WEAVE-LOFAR Survey,” in SF2A-2016: Proceedings of the Annual meeting of the French Society of Astron- omy and Astrophysics, edited by C. Reylé, J. Richard, L. Cambrésy, M. Deleuil, E. Pécontal, L. Tresse, and I. Vauglin (2016) pp. 271–280. [59] Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, P. Greenfield, M. Droettboom, E. Bray, T. Aldcroft, M. Davis, A. Ginsburg, A. M. Price-Whelan, et al., “As- tropy: A community Python package for astronomy,” As- tron. Astrophys. 558, A33 (2013). [60] Astropy Collaboration, A. M. Price-Whelan, B. M. Sipőcz, H. M. Günther, P. L. Lim, S. M. Crawford, S. Conseil, D. L. Shupe, M. W. Craig, N. Dencheva, et al., “The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package,” Astron. J. 156, 123 (2018). [61] Astropy Collaboration, A. M. Price-Whelan, P. L. Lim, N. Earl, N. Starkman, L. Bradley, D. L. Shupe, A. A. Patil, L. Corrales, C. E. Brasseur, et al., “The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package,” Astrophys. J. 935, 167 (2022). [62] J. D. Hunter, “Matplotlib: A 2D Graphics Environ- ment,” Computing in Science and Engineering 9, 90 (2007). [63] S. van der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy Array: A Structure for Efficient Numerical Com- putation,” Computing in Science and Engineering 13, 22 (2011). [64] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peter- son, W. Weckesser, J. Bright, et al., “SciPy 1.0: Funda- mental algorithms for scientific computing in Python,” Nat. Methods 17, 261 (2020). [65] A. Zonca, L. Singer, D. Lenz, M. Reinecke, C. Rosset, E. Hivon, and K. Gorski, “Healpy: Equal area pixeliza- tion and spherical harmonics transforms for data on the sphere in Python,” JOSS 4, 1298 (2019). [66] K. M. Górski, E. Hivon, A. J. Banday, B. D. Wan- delt, F. K. Hansen, M. Reinecke, and M. Bartel- mann, “HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere,” Astrophys. J. 622, 759 (2005). [67] C. G. T. Haslam, C. J. Salter, H. Stoffel, and W. E. Wilson, “A 408-MHZ All-Sky Continuum Survey. II. The Atlas of Contour Maps,” Astron. Astrophys. Suppl. 47, 1 (1982). [68] M. Remazeilles, C. Dickinson, A. J. Banday, M.-A. Bigot- Sazy, and T. Ghosh, “An improved source-subtracted and destriped 408-mhz all-sky map,” Mon. Not. R. As- tron. Soc. 451, 4311 (2015).
• 9.
  9 LoTSS-DR2 159 395 Counts-in-cell TGSS 8 56 Counts-in-cell RACS-low 2279 Counts-in-cell 210 260 310 360 Counts-in-cell 0.000 0.005 0.010 0.015 0.020 PDF LoTSS-DR2 S5 mJy Poisson Neg. binom. Data 10 20 30 40 50 Counts-in-cell 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 PDF TGSS S100 mJy Poisson Neg. binom. Data 20 30 40 50 60 70 Counts-in-cell 0.00 0.01 0.02 0.03 0.04 0.05 0.06 PDF RACS-low S20 mJy Poisson Neg. binom. Data RACS-mid 14 62 Counts-in-cell NVSS 13 63 Counts-in-cell VLASS 6 72 Counts-in-cell 20 30 40 50 Counts-in-cell 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 PDF RACS-mid S20 mJy Poisson Neg. binom. Data 20 30 40 50 Counts-in-cell 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 PDF NVSS S20 mJy Poisson Neg. binom. Data 10 20 30 40 50 60 Counts-in-cell 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 PDF VLASS S10 mJy Poisson Neg. binom. Data FIG. 2. Maps (first and third row) and histograms (second and fourth row) of source counts from six radio continuum surveys and their best-fit Poisson and negative binomial distributions. Appendix A: Additional figures tables—This section includes additional figures and tables that provide detailed information on the improvement of the negative binomial distribution for radio surveys, the fitting of the spectral index, and the flux density dependence of the radio dipole direction. Figure 2 displays maps and histograms, along with the best-fit Poisson and negative binomial distributions for all six radio surveys. The maps show the applied masks and counts-in-cells for the corresponding flux density cut. The chosen flux density cuts guarantee high degree of completeness and that all six surveys are dominated by radio emission associated with active galactic nuclei. The histogram highlights the enhancement achieved by the negative binomial distribution in comparison to the Poisson distribution in describing the observed counts-in-cells distribution. Fitting parameters are provided in the main text, Table I. Table IV details the cross-matching results for survey pairs. To determine the spectral index of a given radio survey, the chosen mask and flux density cut are applied, and it is matched to another survey without a flux density cut. If multiple matches occur, the flux densities are summed, especially when cross-matching between low and high- resolution surveys. The table also shows the percentage of matches in the overlap of the two surveys. For example, matching LoTSS-DR2 with a 5 mJy flux density cut to NVSS results in 51% of LoTSS-DR2 sources matching 83% of NVSS sources in the overlap. The resulting spectral index is α = −0.73 ± 0.23. Figure 3 illustrates (αRA, δ) as a function of the flux density cut for all six radio surveys. As discussed in the main text, this helps determine which flux density cuts and surveys are considered robust and which are analyzed with the constrained estimator.
• 10.
  10 80 40 0 40 80 Dec LoTSS-DR2 5 mJy 10 mJy TGSS 100mJy 150 mJy 80 40 0 40 80 Dec RACS-low 10 mJy 20 mJy 30 mJy RACS-mid 10 mJy 20 mJy 30 mJy 100 200 300 RA 80 40 0 40 80 Dec NVSS 10 mJy 20 mJy 30 mJy 100 200 300 RA VLASS 10 mJy 20 mJy FIG. 3. Declination and right ascension of source count dipole for all six radio surveys for different flux density cuts using the unconstrained estimator. The green lines and dot represent the expected value based on the CMB dipole. TABLE IV. Percentage of matched radio sources for pairs of surveys and derived mean spectral index, both calculated in their overlap. The third column gives the percentage of matched sources for the first and second surveys, respectively. The given flux density cut is applied to the first survey only, and none applied to the second. Surveys Smin Percent matched α LoTSS-DR2–NVSS 5 51% / 83% −0.73 ± 0.23 TGSS–NVSS 100 97% / 16% −0.76 ± 0.18 RACS-low–NVSS 20 97% / 25% −0.98 ± 0.24 RACS-low–RACS-mid 20 99% / 19% −0.95 ± 0.23 RACS-mid–VLASS 20 81% / 23% −0.81 ± 0.30 FIRST–VLASSa - - −0.71 a taken from [36], which used the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; [48]) to calculate α. Appendix B: Systematic effects of radio surveys— The systematic effects regarding the large-scale structure of LoTSS-DR2 were discussed and studied in [24, 26, 27]. Simulations of the LoTSS-DR2 sky coverage and number density show that the dipole retrieval in declination has a small bias but is mainly dominated by a large variance. Simulations using the understood and modeled LoTSS- DR2 systematics, as described by random mocks [27], show that a retrieval of the dipole direction is not pos- sible, due to limited sky coverage. Additionally, we find no cross-correlation between the LoTSS-DR2 map and Galactic synchrotron emission as described by the 408 MHz all-sky Haslam map [67] from [68]. [19] showed that there also is no cross-correlation between the RACS-low number count maps and Galactic synchrotron emission. For NVSS, [35] provide an extensive discussion of obser- vational systematic effects. Systematic effects of NVSS with respect to estimators, survey geometry, flux density cuts, shot noise and more have been studied in [6, 15–17].