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arxiv: 1906.10744 · v1 · pith:BPB76EP2new · submitted 2019-06-25 · 🌌 astro-ph.CO

The angular scale of homogeneity in the Local Universe with the SDSS blue galaxies

F. Avila , C. P. Novaes , A. Bernui , E. de Carvalho , J. P. Nogueira-Cavalcante This is my paper

Pith reviewed 2026-05-25 15:46 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords angular homogeneitySDSS blue galaxiescosmological principlefractal correlation dimensionlocal universeLambdaCDMscaled counts in spherical caps
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The pith

Blue galaxies from the SDSS survey transition to homogeneity at an angular scale of 22.19 degrees

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

The paper measures the angular scale at which blue galaxies from the SDSS survey reach homogeneity in the local universe. It applies the scaled counts in spherical caps and the fractal correlation dimension to identify this transition at 22.19 degrees with an uncertainty of 1.02 degrees. The measurement agrees with results from HI sources in the ALFALFA catalogue after bias correction. The scale matches expectations from the LambdaCDM model. The findings supply additional evidence that the universe appears homogeneous on large scales, consistent with the Cosmological Principle.

Core claim

Through the scaled counts in spherical caps N(<θ) and the fractal correlation dimension D2(θ), the analysis of SDSS blue galaxies finds an angular scale of transition to homogeneity of 22.19 degrees with an uncertainty of 1.02 degrees. Comparison with HI extragalactic sources from the ALFALFA catalogue at similar redshifts shows excellent agreement after bias correction. Robustness tests on survey area size and a reduced sample produce similar scales. The observed scale agrees with LambdaCDM expectations and provides further evidence strengthening the validity of the Cosmological Principle.

What carries the argument

The scaled counts in spherical caps N(<θ) and fractal correlation dimension D2(θ), which track when the galaxy distribution becomes statistically uniform with increasing angular scale.

If this is right

  • The angular scale of transition to homogeneity for SDSS blue galaxies is 22.19 degrees plus or minus 1.02 degrees.
  • Blue galaxies and HI sources yield consistent homogeneity scales after bias correction.
  • The measured scale matches what is expected in the LambdaCDM scenario.
  • Robustness checks confirm that survey area size and sample reduction do not alter the identified transition scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Consistency between two independent tracers at the same redshift suggests the reported scale is not driven by the choice of galaxy population.
  • The method could be applied to deeper surveys to check whether the homogeneity scale changes with look-back time.
  • If the scale holds across multiple independent datasets, it tightens constraints on possible large-scale deviations from uniformity.

Load-bearing premise

The scaled counts in spherical caps and fractal correlation dimension methods correctly identify the physical transition to homogeneity without residual survey systematics or tracer bias that would shift the reported angle.

What would settle it

An independent measurement of the homogeneity transition scale using a different tracer or survey at z less than 0.06 that differs from 22.19 degrees by more than the stated uncertainty would falsify the reported value.

Figures

Figures reproduced from arXiv: 1906.10744 by A. Bernui, C. P. Novaes, E. de Carvalho, F. Avila, J. P. Nogueira-Cavalcante.

Figure 1
Figure 1. Figure 1: Colour-colour diagram of the SDSS galaxies, corrected by Galac￾tic and intrinsic extinctions. The horizontal and vertical plots show the bi￾modal distribution of g − r and u − r galaxy colours, respectively. where k(λ) = 1.17  −2.156 + 1.509 λ − 0.198 λ2 + 0.011 λ3  + 1.78 , (3) for 0.12 ≤ λ[µm] ≤ 0.63, and k(λ) = 1.17  −1.857 + 1.040 λ  + 1.78 , (4) for 0.63 ≤ λ[µm] ≤ 2.20 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 2
Figure 2. Figure 2: Cartesian projection of the SDSS sample selected for our analy￾ses; it contains a total of 35356 blue galaxies. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 z 0 500 1000 1500 2000 2500 3000 N [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Redshift distribution of the SDSS blue galaxies sample selected for our homogeneity analyses. In this section we briefly present the SCC method, calculated using the Landy-Szalay estimator, and the definition of the fractal correlation dimension, D2 (more details can be found in the section 3 of A18). We also describe the criterium used to determine the angular scale of transition to homogeneity, θH, for a… view at source ↗
Figure 4
Figure 4. Figure 4: Two-point angular correlation function for the selected sample of SDSS blue galaxies. The blue dots and error bars correspond to the average and standard deviation from the set of {ωj (θ), j = 1, ..., 20}. The grey lines correspond to best-fit of ωj (θ) data points obtained using each of the 20 random catalogues. 0 10 20 30 40 θ [degrees] 1.850 1.875 1.900 1.925 1.950 1.975 2.000 D2(θ) data points θH =22.1… view at source ↗
Figure 5
Figure 5. Figure 5: Polynomial fits of order five (grey curves) of the D j 2 (θ) data points measured using each of the 20 random catalogues through equation (6). Their intersection with the orange dot-dashed line determines the angu￾lar scale of transition to homogeneity for each random catalogue, θ j H , whose average and standard deviation furnishes the measured θH value. The aver￾age D2(θ) data points (blue dots) are pres… view at source ↗
Figure 6
Figure 6. Figure 6: Two-point angular correlation function calculated for the SDSS blue galaxies sample, and the ALFALFA sample analysed in A18. For each sample, the data points correspond to the average over the ωj (θ) calcu￾lated using each random catalogue. The graphic shows the θ range used to estimate the relative bias among the two samples. scaled counts-in-caps: N (2)(< θ) − 1 = b 2 t2/t1 [ N (1)(< θ) − 1 ], where b 2 … view at source ↗
Figure 7
Figure 7. Figure 7: Fractal correlation dimension analyses to find θH using twice the original sky area of the surveyed SDSS blue galaxies. This finding, θH = 22.80◦ ± 1.16◦, agrees well with the results summarised in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left panel: Redshift distribution of the reduced SDSS blue galaxies catalogue, with elements randomly removed in a way that the resulting sample has similar median and standard deviation parameters as those from the ALFALFA catalogue (see A18). Right panel: Fractal correlation dimension analyses of the reduced sample whose distribution is shown in the left panel, which results in the angular scale of homog… view at source ↗
read the original abstract

We probe the angular scale of homogeneity in the local Universe using blue galaxies from the SDSS survey as a cosmological tracer. Through the scaled counts in spherical caps, $ \mathcal{N}(<\theta) $, and the fractal correlation dimension, $\mathcal{D}_{2}(\theta)$, we find an angular scale of transition to homogeneity for this sample of $\theta_{\text{H}} = 22.19^{\circ} \pm 1.02^{\circ}$. A comparison of this measurement with another obtained using a different cosmic tracer at a similar redshift range ($z < 0.06$), namely, the HI extragalactic sources from the ALFALFA catalogue, confirms that both results are in excellent agreement (taking into account the corresponding bias correction). We also perform tests to asses the robustness of our results. For instance, we test if the size of the surveyed area is large enough to identify the transition scale we search for, and also we investigate a reduced sample of blue galaxies, obtaining in both cases a similar angular scale for the transition to homogeneity. Our results, besides confirming the existence of an angular scale of transition to homogeneity in different cosmic tracers present in the local Universe, show that the observed angular scale $\theta_{\text{H}}$ agrees well with what is expected in the $\Lambda$CDM scenario. Although we can not prove spatial homogeneity within the approach followed, our results provide one more evidence of it, strengthening the validity of the Cosmological Principle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript measures the angular homogeneity scale in the local Universe (z < 0.06) using blue galaxies from SDSS as tracers. Applying the scaled counts-in-spherical-caps statistic N(<θ) and the fractal correlation dimension D2(θ), the authors report a transition scale θ_H = 22.19° ± 1.02°. They find consistency with an independent measurement from the ALFALFA HI catalog after bias correction, and after performing area-size and subsample robustness tests they conclude that the result supports the Cosmological Principle and agrees with ΛCDM expectations.

Significance. If the reported θ_H is shown to be free of residual survey systematics, the work would add a useful cross-tracer confirmation (blue galaxies versus HI sources) of the homogeneity transition scale in the very local universe. The dual-statistic approach and explicit comparison to another catalog are positive features that would strengthen the evidential value for tests of large-scale homogeneity.

major comments (2)
  1. [Abstract (robustness tests)] Abstract (robustness tests paragraph): The area-size and subsample tests are reported only at summary level and do not include end-to-end mock catalogs that inject a known homogeneity scale, apply the actual SDSS footprint, magnitude limits, and blue-galaxy color selection, and then recover θ_H. Without such controlled simulations, it remains possible that mask or selection effects shift the apparent transition angle, which directly affects the central claim that θ_H = 22.19° ± 1.02° is the physical scale.
  2. [Abstract and results] Abstract and results: The uncertainty ±1.02° on θ_H is stated without any description of how the transition point is identified in N(<θ) or D2(θ), how errors are propagated, or whether bin-to-bin covariance is included. This omission prevents assessment of whether the quoted agreement with ALFALFA (after bias correction) and with ΛCDM is statistically meaningful.
minor comments (1)
  1. [Abstract] Typo: 'asses' should read 'assess' in the sentence 'we test if the size of the surveyed area is large enough to identify the transition scale we search for, and also we investigate a reduced sample of blue galaxies, obtaining in both cases a similar angular scale for the transition to homogeneity.'

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We respond to each major point below.

read point-by-point responses

  1. Referee: [Abstract (robustness tests)] Abstract (robustness tests paragraph): The area-size and subsample tests are reported only at summary level and do not include end-to-end mock catalogs that inject a known homogeneity scale, apply the actual SDSS footprint, magnitude limits, and blue-galaxy color selection, and then recover θ_H. Without such controlled simulations, it remains possible that mask or selection effects shift the apparent transition angle, which directly affects the central claim that θ_H = 22.19° ± 1.02° is the physical scale.

    Authors: We agree that end-to-end mocks with injected homogeneity scales, the SDSS footprint, and full selection would provide stronger validation against systematics. The area-size and subsample tests in the manuscript vary the surveyed region and galaxy selection and recover consistent θ_H values, indicating that mask and selection effects do not drive the result. We will expand the description of these tests in the revised text. Full mock suites are beyond the present scope. revision: partial

  2. Referee: [Abstract and results] Abstract and results: The uncertainty ±1.02° on θ_H is stated without any description of how the transition point is identified in N(<θ) or D2(θ), how errors are propagated, or whether bin-to-bin covariance is included. This omission prevents assessment of whether the quoted agreement with ALFALFA (after bias correction) and with ΛCDM is statistically meaningful.

    Authors: We will add an explicit description in the revised manuscript of how the transition scale is identified from the convergence of N(<θ) to 1 and D2(θ) to 2, the method used to obtain the quoted uncertainty, and whether bin-to-bin covariance enters the error estimate. This will allow direct evaluation of the statistical comparisons. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard measurement on independent data

full rationale

The paper applies the established scaled counts N(<θ) and fractal dimension D2(θ) methods to SDSS blue-galaxy data to measure the transition angle θ_H, then cross-checks the result against the independent ALFALFA HI catalog (after bias correction) and performs area-size and subsample robustness tests. None of these steps reduce the reported angle to a fitted parameter by construction, nor does the central claim rest on a self-citation chain or imported uniqueness theorem. The derivation chain is self-contained against external catalogs and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The measurement rests on the assumption that blue galaxies trace the underlying density field sufficiently well for the chosen statistics and that the transition angle identified by N(<θ) and D2(θ) corresponds to the physical homogeneity scale; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Blue galaxies constitute an unbiased enough tracer for the homogeneity-scale statistics at z < 0.06
    Invoked when interpreting the measured θ_H as a cosmological quantity and when applying bias correction to the ALFALFA comparison.
  • domain assumption The scaled counts and fractal dimension methods recover the true homogeneity scale when applied to a sufficiently large survey volume
    Tested only via area-size and subsample checks; central to claiming the reported angle is physical rather than artifactual.

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Reference graph

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