Revisiting the CMB homogeneity scale: low multipoles removal effect and extragalactic foreground masking

Carlos A. P. Bengaly; Diego Garcia Lambas; Facundo Toscano; Frode K. Hansen; Heliana E. Luparello; Jailson Alcaniz; Rodrigo S. Gon\c{c}alves; Xiaoyun Shao

REVIEW 3 major objections 5 minor 57 references

Revisiting the CMB homogeneity scale: low multipoles removal effect and extragalactic foreground masking

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that the mismatch between the CMB homogeneity scale and the ΛCDM model is not an independent anomaly: it is carried by the low observed quadrupole, and masking foregrounds from nearby galaxies does not remove it.

desk verdict A careful reanalysis of the CMB homogeneity scale whose cleanest result gets overshadowed by an unproven quadrupole claim; the low-multipole removal works, but the paper never isolates ell=2. read the letter →

arxiv 2506.00219 v2 pith:FYIB4O3H submitted 2025-05-30 astro-ph.CO

Xiaoyun Shao , Facundo Toscano , Diego Garcia Lambas , Rodrigo S. Gonçalves , Carlos A. P. Bengaly , Heliana E. Luparello , Frode K. Hansen , Jailson Alcaniz This is my paper

classification astro-ph.CO

keywords cosmicmicrowavebackgroundhomogeneityscaleCMBquadrupoleanomalyangularcorrelationfunctionindexLambda-CDMmodelextragalacticforegroundPlanckSMICAmap

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 pin down what causes the mismatch between the angular scale at which the cosmic microwave background becomes homogeneous and the prediction of the standard ΛCDM model. Working with Planck SMICA maps, a thousand ΛCDM simulations, and the homogeneity-index estimator $H(\theta)$, the authors find that cutting the lowest angular modes from both data and simulations erases the discrepancy: the derived scales $\theta_W$, $\theta_H$, and the slope $S_{90}$ move into agreement, with p-values rising from about 1% to 12–33%. They attribute this to the anomalously low CMB quadrupole, because the simulations carry substantial quadrupoles whose removal shifts them toward the data. They also test the new extragalactic foreground traced by nearby galaxies and find that masking those regions improves consistency only slightly and cannot resolve the disagreement. If the paper is right, the homogeneity-scale tension is not an independent challenge to ΛCDM but a symptom of the well-known quadrupole anomaly.

What carries the argument

The argument runs through the homogeneity-index chain adapted from earlier work: the two-point correlation function written as $W_\eta(\theta) = 1 + C(\theta)$, the scaled counts-in-spheres $P(<\theta)$, the homogeneity index $H(\theta) = 2\,d\ln P(<\theta)/d\ln\Omega(<\theta)$, and the derived quantities — the crossing scale $\theta_W$ where $W_\eta=1$, the homogeneity scale $\theta_H$ where $H=0$, and the slope $S_{90}$ at $\theta=90^\circ$. The decisive comparison is between the Planck SMICA map and 1000 synthetic ΛCDM maps, with data variance estimated from 128 (and 256) jackknife sky regions and disagreement quantified by the p-value of each observed estimator in the simulation distribution. Low-multipole removal is implemented by estimating the $a_{\ell m}$ coefficients from low-resolution maps through a $\chi^2$ fit with a mode-coupling matrix, then rebuilding maps that keep only $\ell \geq 2$ or $\ell \geq 5$; the three foreground masks respectively block large spiral galaxies, higher-redshift extensions of that foreground, and the most nearby galaxies.

What would settle it

Construct maps that excise only the quadrupole — remove $\ell = 2$ while keeping $\ell = 0,1,3,4,5,\dots$ — from both the SMICA map and the ΛCDM simulations, and recompute $\theta_W$, $\theta_H$, and $S_{90}$. The paper's claim predicts the p-values jump above the ~16% no-tension threshold with only this mode removed; if they stay near 1–3%, the improvement is a collective low-multipole effect and the quadrupole attribution fails. A complementary check is to replace the simulated $\ell=2$ coefficients with the observed SMICA quadrupole values and verify that the synthetic homogeneity scales move to the observed ones.

Watch

Extended reading notes

Core claim

The paper's central claim is that the homogeneity-scale discrepancy in the CMB is caused by the low value of the observed quadrupole, not by foreground emission associated with nearby galaxies. On the full SMICA map the homogeneity estimators $\theta_W$ and $\theta_H$ sit at p-values near 1% relative to the ΛCDM simulations, a mild tension by the paper's own calibration, while the three foreground masks — L2023, H2023, and a new Neargal mask built from the Nearby Galaxy Catalogue — leave the estimators essentially unchanged, with $\theta_H$ p-values of about 1–3%. Removing the lowest multipoles from both data and simulations, however, raises the p-values to 12–33%, inside the regime the paper treats as consistent with the model. The direction of the effect carries the argument: the observed quadrupole is already very small, so cutting it barely changes the data, whereas the simulated quadrupoles are substantial and their removal shifts the simulated homogeneity scales toward the observed values. The authors read this as direct evidence that the low homogeneity scale is the quadrupole anomaly appearing in a different observable.

Load-bearing premise

The conclusion that the quadrupole in particular drives the tension rests on an untested premise, since Section 3.3 asserts that removing $\ell=2$ suffices while the maps actually analyzed either keep the quadrupole in (the $\ell \geq 2$ case) or remove it together with four other low modes (the $\ell \geq 5$ case).

Editorial extensions

If this is right

The homogeneity-scale discrepancy in $\theta_W$, $\theta_H$, and $S_{90}$ disappears once the lowest multipoles are removed from both data and simulations, with p-values rising from about 1% to 12–33%.
Masking the new extragalactic foreground traced by nearby galaxies does not resolve the discrepancy; at best it improves consistency marginally, with $\theta_H$ p-values staying near 3% on the full map.
The improvement comes mainly from the simulations: the data quadrupole is already small, so removing it barely changes the data, whereas cutting the substantial simulated quadrupoles pulls the ΛCDM homogeneity scales down toward the observed values.
With the low multipoles cut, the reduced $\chi^2$ for both $W_\eta(\theta)$ and $H(\theta)$ drops to roughly 1 for the data against ΛCDM, whether or not the Neargal mask is also applied.
If the paper is right, the homogeneity scale adds no independent tension to ΛCDM beyond the quadrupole anomaly itself.

Reading between the lines

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

A sharper test would isolate $\ell = 2$: the maps analyzed here either keep the quadrupole in (the $\ell \geq 2$ case) or remove it together with the monopole, dipole, octopole, and $\ell = 4$ modes (the $\ell \geq 5$ case), so the specific attribution of the effect to the quadrupole alone is an interpretation, not a measured contrast.
If an $\ell = 2$-only test confirms the attribution, the homogeneity scale stops being a separate cosmological constraint: any mechanism that explains the low quadrupole — modified initial conditions, a physical foreground, or non-standard geometry — would automatically reconcile the large-angle CMB with ΛCDM.
The same analysis could be run on the WMAP data and on the other Planck component-separation maps; the paper's claim predicts the same jump in p-values across all of them, whereas a foreground origin would predict method-dependent behavior.

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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

The paper does something useful: it checks whether the reported CMB homogeneity-scale tension survives low-multipole removal and foreground masking. The p-values for theta_W, theta_H, and S90 under these cuts are new, and the negative result on nearby-galaxy masking is a legitimate data point. The jackknife and 1000-map simulation pipeline is standard and appears competently executed. I buy the broad conclusion that removing the lowest multipoles brings the SMICA map and Lambda CDM into much better agreement. That is worth saying clearly.

The soft spot is the abstract's and Section 4's claim that the quadrupole specifically is responsible. That claim is not supported by the analysis as presented. The maps labeled ell>=2 actually remove monopole and dipole only; the quadrupole is retained. The ell>=5 maps remove ell=0-4 simultaneously, so they cannot separate the quadrupole from the octopole and hexadecapole. Section 3.3 asserts that removing the quadrupole is sufficient, but there is no map that removes only ell=2. Tables 1-3 show that removing monopole and dipole already lifts p-values to 18-23%, and the additional change from ell>=2 to ell>=5 is modest. So the improvement could come from monopole/dipole removal, from the combined ell=0-4 band, or from something else. The paper needs an explicit ell=2-only (or ell>=3) cut before it can attribute anything to the quadrupole. This is a labeling error in the text, not a fabrication, but it is load-bearing for the advertised result.

There is also a mild circularity concern: removing modes that are known to be anomalous and then finding improved agreement is partly a restatement of the known low-quadrupole anomaly. That does not sink the paper, but it tempers the novelty. The foreground-masking results are clean and the negative result is honest.

I would send this to a serious referee. The numerical machinery is fine and the low-multipole removal effect is real; the paper just overclaims the quadrupole attribution. A revision that adds an ell>=3 cut or rewrites the conclusions to say 'low multipoles' rather than 'the quadrupole' would fix most of the problem. The chi-square definition is also a bit under-specified, but that is a minor fix.

Recommended verdict: major revision, not rejection. The paper is for people who work on CMB anomalies and homogeneity-scale tests; they will get a useful dataset of p-values even if the interpretive claim needs to be dialed back.

Referee Report

3 major / 5 minor

Summary. The paper revisits the claimed discrepancy between the observed CMB homogeneity scale and the ΛCDM prediction, using the angular two-point correlation function, the homogeneity index H(θ), and the slope S90 at θ=90°. The analysis uses the Planck SMICA map with the common mask, 1000 synthetic ΛCDM maps, and jackknife resampling. Two modifications are studied: removal of low multipoles (labeled ℓ≥2 and ℓ≥5) from both data and simulations, and application of three foreground masks (L2023, H2023, and a new Neargal mask). The authors report that low-multipole removal substantially improves data-simulation agreement and conclude that the quadrupole anomaly is the main driver of the homogeneity-scale tension, while foreground masking alone does not resolve the discrepancy.

Significance. If the quadrupole attribution were established, this would be a useful result connecting the homogeneity-scale tension to the well-known low-ℓ CMB anomalies, and the negative foreground-mask result would also be informative. The paper has concrete strengths: it uses 1000 synthetic maps, jackknife uncertainties, multiple estimators, and several masks, and it reports p-values and reduced chi-squared values in a transparent tabular form. However, the central claim is currently not established because the maps labeled ℓ≥2 do not remove the quadrupole under the paper's own harmonic convention, so the specific attribution to the quadrupole is unsupported.

major comments (3)

[Sec. 3.1, Table 1/3, Figs. 2/5] The maps labeled ℓ≥2 are obtained by removing 'the first two multipoles', which in the harmonic expansion of Eq. (2.1) are ℓ=0 and ℓ=1; the quadrupole ℓ=2 is therefore retained in the ℓ≥2 maps, while the ℓ≥5 maps remove ℓ=0,1,2,3,4 simultaneously. Since Table 3 shows that the ℓ≥2 cut alone already raises p-values from 1% to 18–23% (for example, θ_H from 1.0% to 23.0%), the improvement cannot be attributed specifically to the quadrupole; it may be driven by monopole/dipole removal. The abstract and Section 4 claim that the quadrupole 'in particular' drives the effect, but this is not supported by the displayed cuts. Please add an analysis with a genuine ℓ≥3 map (removing ℓ=0,1,2) or an ℓ=2-only excision and compare it with the ℓ≥2 and ℓ≥5 cases.
[Sec. 3.3, first paragraph] The statement that 'the removal of the quadrupole (ℓ=2) is sufficient' is not backed by the displayed analysis: no map isolates ℓ=2. The ℓ≥5 removal includes ℓ=3 and ℓ=4, so the additional improvement from ℓ≥2 to ℓ≥5 could be due to the octopole and hexadecapole instead of the quadrupole. At minimum, a ℓ≥3 map is needed to isolate the quadrupole contribution, and the text should be revised to describe the actual cuts (ℓ<2 and ℓ<5) rather than 'quadrupole removal'.
[Sec. 3.1, Tables 3 and 5] The p-values are computed after removing multipoles that were selected because the full-map statistics are anomalous. These are conditional, post-selection probabilities; the paper does not account for the fact that the low multipoles were chosen a posteriori. Consequently, improved p-values after the cuts are expected if the anomaly resides in the removed modes, but they do not by themselves quantify the evidence for the specific claim that the quadrupole amplitude is anomalously low. Please add a direct test of the quadrupole amplitude (for example, comparing observed C_2 with the ΛCDM distribution) or a likelihood comparison that includes the selection, or explicitly frame the results as a decomposition rather than a significance test.

minor comments (5)

[Sec. 2.3] The text says the CMB map is downgraded to Nside=4 but then refers to estimating a_lm from ℓ=0 to ℓ=4·(Nside=64); this is contradictory and should be clarified, including the actual maximum multipole used.
[Eq. (2.13)] The definition of P(<θ) appears malformed in the typeset equation; the ratio of the observed integrated correlation to the Poisson expectation should be written explicitly with distinct symbols for the numerator and denominator.
[Eqs. (3.1)-(3.2)] In Eq. (3.2), Δ_i is already divided by σ_W(i); if the matrix C_{ij} in Eq. (3.1) is the full covariance, the normalization is double-counted. Please specify whether C is the correlation matrix or define Δ_i without the pre-division.
[Tables 3, 5, and 8] The method used to compute the p-values should be stated explicitly, including whether they are one-sided or two-sided and whether they are the fraction of 1000 simulations with statistics more extreme than the data.
[Throughout] The wording 'first two multipoles' and 'first five multipoles' is ambiguous and has led to the quadrupole misidentification; please consistently define the cuts as ℓ<2 and ℓ<5, and fix the repeated typo 'remotion' (e.g., Section 2.3 title and the abstract).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the low-multipole removal is a self-contained sensitivity analysis; the quadrupole attribution is under-supported (no ℓ=2-only cut) but this is a labeling gap, not a circular derivation.

full rationale

The paper's derivation chain is: define W_eta(theta) = 1 + C(theta) with C(theta) = sum_l (2l+1)/(4pi) C_l P_l (Eqs. 2.4 and 2.10), define H(theta) from W_eta (Eqs. 2.12-2.13), then remove low multipoles from both the Planck SMICA map and 1000 LambdaCDM synthetic maps and compare theta_W, theta_H, S90 and p-values. This is a legitimate sensitivity analysis: the improvement in p-values from 1% to 18-32% when the first two or first five multipoles are removed is an empirical outcome, not forced by the definitions. The conclusion that low multipoles contribute to the homogeneity-scale discrepancy is supported by the comparison. The stronger claim that the quadrupole specifically is responsible is not established because the maps labelled ell>=2 remove ell=0 and 1, and the ell>=5 maps remove ell=0-4, so no map isolates ell=2 (Sections 2.3 and 3.3). This is a support/labeling gap, not circularity: no parameter is fitted from the target observable, no self-citation supplies the central premise, and the foreground-mask results using the authors' earlier masks explicitly fail to resolve the discrepancy, so they are not load-bearing. Under the strict definition of circularity, the paper's quantitative sensitivity analysis is self-contained; score 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to the target result in a numerical sense; the multipole cuts and mask thresholds are hand-chosen analysis choices. The paper introduces no new particles, forces, or entities; the extragalactic foreground is carried over from prior work (L2023, H2023, H2025).

free parameters (2)

Low-multipole removal thresholds (ell<2 and ell<5) = ell<2 and ell<5
The central analysis is organized around these hand-chosen cuts, motivated by the known low-ell power deficit. They are not fitted to the homogeneity data, but no ell<3 cut is shown, so the quadrupole is not isolated.
Neargal mask thresholds (galaxy radius > 7.5 kpc, 2 degree mask radius) = r>7.5 kpc, 2 deg
Hand-chosen selection to define the most nearby galaxy mask; affects the masking analysis but not the low-multipole conclusion.

assumptions (5)

standard math Spherical harmonic expansion and Legendre relation C(theta)=sum (2ell+1)/(4pi) C_ell P_ell(cos theta)
Eq. (2.4); standard background for relating power spectrum and correlation function.
domain assumption The galaxy-survey homogeneity index H=2 d ln N(<theta)/d ln Omega can be adapted to the CMB by replacing counts with W_eta(theta)=1+C(theta)
Eqs. (2.8)-(2.13); imported from CQG2022 with no new derivation; the mapping assumes the CMB temperature field behaves like a point process in this estimator.
domain assumption Lambda CDM synthetic maps generated from Planck PR3 best-fit spectrum provide a fair cosmic variance benchmark for masked, multipole-filtered maps
Sections 2.1 and 2.3; the p-values and chi-squared comparisons depend on this.
domain assumption The L2023, H2023 and Neargal masks trace the regions affected by the extragalactic foreground
Section 2.3; if the masks miss part of the foreground, the conclusion that masking cannot remove the discrepancy is weakened.
standard math Jackknife resampling with 128 or 256 regions estimates the intrinsic sample variance of the cut-sky estimators
Eqs. (2.14)-(2.15); standard resampling assumption.

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Cite this review

Pith. "Pith review of Revisiting the CMB homogeneity scale: low multipoles removal effect and extragalactic foreground masking." pith.science (2026). https://pith.science/paper/FYIB4O3H

@misc{pith2026250600219,
  author       = {Pith},
  title        = {Pith review of: Revisiting the CMB homogeneity scale: low multipoles removal effect and extragalactic foreground masking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYIB4O3H}},
  note         = {Machine review of arXiv:2506.00219}
}

abstract

The Cosmic Microwave Background (CMB) reaches homogeneity at relatively modest angular scales compared to the expectation of the standard $\Lambda$CDM model revealing an important challenge to the theoretical predictions. We analyze this inconsistency through the homogeneity scale $H$ and the slope of the homogeneity index at $\theta = 90^\circ$. We find that the removal of low multipoles, in particular the quadrupole, from both the data and the $\Lambda$CDM synthetic CMB maps, significantly improve the consistency between models and observations. This adds to indications of the relevant contribution of the low value of the CMB quadrupole to the observed anomalies in the homogeneity scale. Due to the presence of a new extragalactic foreground in the CMB maps, we have performed statistical analyses with different masking taking into account the regions mostly affected. In particular we consider galaxies in the local neighborhood which are expected to affect more significantly the large angular scales. We find that by masking these regions, the analysis cannot solve the discrepancy between the observations and the $\Lambda$CDM model in spite of a small improvement of their mutual consistency. The studies with both foreground masking and low-$\ell$ removed CMB maps show similar results than those of the full CMB map indicating that the main discrepancy between theory and observations is associated to the quadrupole anomaly and may require more exhaustive analysis.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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