Accessing the homogeneity scale with 21 cm intensity mapping surveys

Bruno B. Bizarria; Camila P. Novaes; Carlos A. Wuensche; Felipe Avila; Gabriel A. S. Silva; Helissa H. da Costa; Rahima Mokeddem

REVIEW 3 major objections 4 minor 3 cited by

Accessing the homogeneity scale with 21 cm intensity mapping surveys

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper shows that for any redshift there is a maximum telescope beam width beyond which 21 cm intensity-mapping data is smoothed too much to reveal the cosmic homogeneity scale, and draws the accessible/inaccessible boundary for current

desk verdict Useful first forecast of the 21cm IM homogeneity scale, but the RSD handling is wrong and the quantitative boundary may shift. read the letter →

arxiv 2511.13931 v2 pith:R3SJEWID submitted 2025-11-17 astro-ph.CO

Bruno B. Bizarria , Camila P. Novaes , Felipe Avila , Rahima Mokeddem , Helissa H. da Costa , Carlos A. Wuensche , Gabriel A. S. Silva This is my paper

classification astro-ph.CO

keywords homogeneityscale21cmintensitymappingCosmologicalPrinciplebeamsmoothingcorrelationdimensiontwo-pointfunctionradiotelescopesurveyslarge-scalestructure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper addresses a simple question: can 21 cm intensity mapping surveys measure the cosmic homogeneity scale — the distance at which the Universe becomes statistically uniform — given that a telescope beam smooths the signal? Using a Gaussian beam model and the correlation dimension D_2(r), the authors show that wider beams artificially homogenize small scales and systematically lower the recovered homogeneity scale R_H. They show that a maximum beam width σ_max(z) exists for every redshift, beyond which the transition to homogeneity is completely erased, defining an inaccessible region in the beam-width–redshift plane. Applying the criterion to seven single-dish radio telescopes, they find that the small-dish instruments ASKAP and MeerKAT cannot recover the homogeneity scale at z ≳ 0.6, while BINGO, GBT, FAST, GMRT, and Parkes remain in the accessible region across their frequency bands. This is the first quantitative forecast of the instrumental requirements for measuring cosmic homogeneity with 21 cm intensity mapping.

What carries the argument

The load-bearing element is the Gaussian beam damping function B(k, μ) = exp[−k²(1−μ²)R_beam²/2] multiplying the matter power spectrum, which yields a beam-convolved two-point correlation function ξ_obs(r) through a spherical-Bessel integral. From ξ_obs the correlation dimension is obtained as D_2(r) = 3 [1 + ξ_obs]/[1 + ξ̄_obs], and the homogeneity scale R_H is the radius where D_2 reaches 2.97, the standard 1-percent threshold. The maximum beam width is defined by the condition R_H(σ_max) = R_min(σ_max), where R_min is the local minimum of D_2; this sharp criterion is what converts an instrument's beam width and observing frequency band into a yes/no statement about whether a survey can re

What would settle it

Run an end-to-end 21 cm intensity-mapping simulation with a known input homogeneity scale and a realistic beam profile (e.g., with sidelobes), then measure D_2 with the same pipeline. If the transition to homogeneity is still recovered for beam widths larger than the Gaussian-model σ_max(z), the criterion is too conservative; if it disappears even for σ below σ_max(z), the Gaussian model is too optimistic.

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Extended reading notes

Core claim

The central claim is that the correlation dimension D_2(r), built from the beam-convolved two-point correlation function, falls to a minimum R_min that rises with beam width while the recovered homogeneity scale R_H falls, so the two meet at a critical width σ_max(z). At that width the artificial smoothing erases the transition to homogeneity entirely, making R_H undetectable. Fitting this boundary as σ_max(z) = A + B/(z + C), the authors partition the σ–z parameter space into accessible and inaccessible regions and evaluate seven current and upcoming single-dish instruments. Only the small-dish telescopes ASKAP and MeerKAT cross the boundary at high redshift (z ≳ 0.6); the others could in p

Load-bearing premise

The forecast rests on modelling the beam as a single Gaussian with purely transverse damping and no noise, foregrounds, or frequency-dependent structure — the authors note a realistic beam with sidelobes would homogenize the signal more strongly, so the predicted σ_max is probably an optimistic upper bound.

Editorial extensions

If this is right

ASKAP and MeerKAT will not be able to measure the homogeneity scale at z ≳ 0.6, since their wide beams push them into the inaccessible region.
For accessible surveys, the recovered R_H is systematically shifted downward by beam smoothing, so future measurements must model the beam to correct this bias.
The fitted σ_max(z) curve provides a concrete specification for future 21 cm instruments: a survey that wants to test the Cosmological Principle needs angular resolution better than this limit.
The framework is tracer-agnostic and extends to other line intensity mapping surveys, as the authors state explicitly.
Because realistic beams with sidelobes are expected to homogenize more strongly, instruments sitting near the Gaussian boundary should be treated as higher-risk for this measurement.

Reading between the lines

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

A corollary the authors leave implicit: any 'detection' of a homogeneity scale from maps whose beam exceeds σ_max should be read as a feature of beam smoothing, not cosmic structure — the paper's boundary gives a quick check for such claims.
Since the recovered R_H decreases monotonically with beam width within the accessible region, combining measurements from different dishes could, in principle, be used to fit out the beam-induced shift — an extension not pursued here.
The σ_max(z) curve could be converted into an explicit survey-selection metric: for a given dish diameter and frequency band, compute σ(z) from equation 1 and compare against Eq. 9 before committing observing time.
The natural next experiment is an end-to-end simulation with a realistic (sidelobe-inclusive) beam; the paper's model predicts the inaccessible region would grow, which is a falsifiable outcome.

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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,

Here's the take. The paper introduces a genuinely new quantity: σ_max(z), the beam width beyond which the homogeneity scale cannot be recovered in 21cm intensity mapping. That's the first quantitative forecast of this kind. The framework is simple, the code is public, and the spherical-Bessel integration is checked against Hankl to 2%. The qualitative conclusion that a single Gaussian beam erases the transition to homogeneity above some width is solid, and the application to single-dish surveys gives a useful boundary in the σ–z plane.

The main soft spot is the treatment of redshift-space distortions. The paper drops the Kaiser factor from the observed power spectrum and justifies it by saying it's a constant amplitude factor. That's true for the monopole without a beam, but not with the beam: Eq. 3 gives a μ-dependent damping, so the μ integral in Eq. 4 does not factor. The radial modes (μ≈1) survive the beam and are exactly the modes that the Kaiser factor boosts. The effective damping kernel changes shape, not just amplitude. That could shift the RH(σ) crossing and the σ_max(z) boundary in Fig. 4. The authors may argue the shift is small, but they haven't shown it. The statement in the text is incorrect, and the forecast rests on that.

There are two smaller issues. Table 2 promises best-fit parameters for both cosmologies but only lists ΛCDM; the ΛCDMν row is missing. And the paper sets \bar{T}_HI b_HI = 1, but D2 is not invariant to that normalization: the D2=2.97 threshold is absolute, so the amplitude affects RH(σ) and hence σ_max. That should be discussed, or at least the dependence shown. The idealized beam model and lack of foregrounds/noise are acknowledged; those are less concerning because they would make the inaccessible region larger, so the qualitative conclusion is conservative.

The citation pattern is fine. The relevant BAO and beam-forecasting literature is covered. No sign of self-promotion.

Recommendation: this deserves peer review. The central idea is sound and the paper is clear. A referee should ask the authors to include the Kaiser factor or demonstrate its effect is negligible, fix the missing table row, and address the amplitude dependence. If those are handled, it would be a useful addition to the IM forecasting literature.

Referee Report

3 major / 4 minor

Summary. The paper models the two-point correlation function (2pCF) of 21 cm intensity mapping under a Gaussian telescope beam (Eqs. 1–4), converts it to the correlation dimension D2(r) via the counts-in-spheres formalism (Eqs. 5–8), and defines the maximum beam width σ_max(z) as the point where D2(R_min) just reaches the homogeneity threshold 2.97, i.e., where R_H(σ_max)=R_min(σ_max). This yields an accessible/inaccessible split of the σ–z plane. The authors fit σ_max(z) with a three-parameter formula (Eq. 9, Table 2) and compare the beam widths of seven single-dish instruments (Fig. 4). They find that MeerKAT and ASKAP are unable to probe R_H at z≳0.6, while the other instruments remain in the accessible region. The numerical 2pCF integration is validated to 2% against the Hankl package, and the public code is provided.

Significance. If the model were complete, this would be a valuable first step: it gives a clean criterion for whether a single-dish 21 cm instrument can in principle preserve the transition-to-homogeneity signal, and it delivers a concrete forecast that can be updated as more realistic instrumental models become available. The conceptual existence of a maximum beam width is robust to many simplifications, and the paper is honest about the idealized nature of the Gaussian beam. The strength of the contribution is the framework and the instrument comparison, not the precise numerical boundary. However, the quantitative σ_max(z) curve and the derived instrument rankings are conditional on at least one demonstrably incomplete piece of physics (redshift-space distortions), so the main quantitative deliverable is not yet supported.

major comments (3)

[Section 2, Eq. (4)] The sentence 'Redshift space distortions were not considered, since the Kaiser effect introduces a constant amplitude factor in the power spectrum and 2pCF' is not valid once the beam is μ-dependent. With the Kaiser factor the observed power spectrum is P_obs(k,μ) = B^2(k,μ) (1+βμ^2)^2 Tbar^2 b^2 P_m(k), and B^2(k,μ)=exp[−k^2(1−μ^2)R_beam^2/2] does not factor out of the μ integral in Eq. (4). The radial (μ≈1) modes, which are the least damped by the beam, are enhanced relative to the transverse modes, changing the shape of ξ_obs(r), D2(r), and hence the R_H(σ)=R_min(σ) crossing that defines σ_max(z) and the Fig. 4 rankings. Please include RSD (or a quantitative argument for why they are negligible for the D2 threshold), and recompute the forecasts.
[Section 4, Fig. 4] The 'Accessible' region is labeled as if all instruments inside it can 'have the potential to measure' R_H, but the calculation contains no noise, no foregrounds, and no systematics. The criterion σ<σ_max only guarantees that the beam does not by itself erase the homogeneity transition in the ideal signal. A real survey could still fail to detect R_H because of SNR, foreground cleaning residuals, or calibration errors. The qualitative existence of a maximum beam width survives, but the instrument-by-instrument conclusions in Fig. 4 are stronger than the model supports. Please either add noise/foregrounds or reframe the boundary as a 'signal-preservation limit' rather than a detectability limit.
[Section 4, Eq. (9) and Table 2] The σ_max(z) curve is computed from the same model (Eqs. 4–8) and then fitted with a three-parameter function; there is no external benchmark or error propagation from the model inputs into the fit. This is acceptable for a forecast, but it means Table 2 and Fig. 4 are essentially an interpolation of the idealized model output. The authors do note that realistic beams would enlarge the inaccessible region, but the magnitude of the shift is unquantified. I ask the authors to state explicitly that σ_max is an upper limit under the Gaussian-beam, no-RSD assumptions, and, if possible, to estimate the systematic uncertainty from the beam model.

minor comments (4)

[Section 2, after Eq. (4)] j0 is the zeroth-order spherical Bessel function, not the 'first-order Bessel function' as written.
[Table 2] The text says the parameters of Eq. (9) are shown for the two cosmologies, but Table 2 lists only the ΛCDM row. The ΛCDMν parameters used for the orange dashed curve in Fig. 4 are missing and should be added.
[Conclusions] Typographical issues: 'bellow' should be 'below', and 'and and' appears in the Data Availability statement.
[Section 2, validation paragraph] The 2% agreement with Hankl validates the numerical transform of the no-RSD integral, not the physical model. This is fine, but the sentence in the abstract and conclusions should not imply the validation covers the beam/RSD treatment.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained; no load-bearing circularity found.

full rationale

The paper's central quantity, σmax(z), is derived in a forward chain: Equation 4 defines the beam-damped two-point correlation function from the matter power spectrum and a Gaussian beam; Equation 7 converts it to the scaled counts-in-spheres; Equation 8 gives the correlation dimension; R_H(σ) is the scale where D2 = 2.97, and σmax is defined by R_H = R_min. Each quantity is a function of the previous one, and no step fits the target result from data that already contains it. Equation 9 is a three-parameter interpolation of the already-computed σmax(z) curve, not a fit to external observations, so it does not constitute a "prediction" that is forced by construction. The cited self-work (Avila et al. 2018, 2019; Novaes et al. 2022) is used for context or methodology and is not load-bearing for the existence or value of σmax. The numerical check against Hankl validates the transform, not the physical assumptions. The main limitations—idealized Gaussian beam, neglected RSD, and the setting of T_HI b_HI to unity—are modeling assumptions that affect accuracy; they do not make the derivation circular. In particular, the statement that the Kaiser effect introduces only a constant amplitude factor is questionable once the μ-dependent beam (Eq. 3) is included, and the amplitude T_HI b_HI does not cancel in D2 (Eq. 8), but these are correctness risks rather than circular steps. No reduction of a claimed prediction to its own input was found.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities. The forecast relies on a Gaussian beam model, a fixed D2 threshold, and a 3-parameter fit to the model's own σ_max(z) curve; the central 'prediction' is therefore a repackaging of the model assumptions rather than an externally anchored measurement.

free parameters (4)

A (Eq. 9) = -0.1050 ± 0.0034
Best-fit parameter for σ_max(z)=A+B/(z+C) using the ΛCDM model's computed σ_max curve; ΛCDMν fit is claimed but not tabulated.
B (Eq. 9) = 0.4283 ± 0.0028
Best-fit parameter for the σ_max(z) parametrization; fitted to the model's own σ_max values.
C (Eq. 9) = -0.0030 ± 0.0011
Best-fit parameter for the σ_max(z) parametrization; fitted to the model's own σ_max values.
Homogeneity threshold = 2.97
Chosen D2 threshold (1% condition from Scrimgeour et al. 2012); arbitrary but standard, and directly sets where RH and hence σ_max are defined.

assumptions (7)

domain assumption Gaussian beam model with σ from Eq. 1
Assumes telescope beam is a pure Gaussian with FWHM conversion; the authors note realistic sidelobes would strengthen homogenization.
domain assumption Beam damping is multiplicative in Fourier space (Eqs. 2-3)
Assumes the observed power spectrum is the true spectrum times B^2, with no frequency dependence of the beam on k∥.
domain assumption Halofit for the nonlinear matter power spectrum
Nonlinear P(k) affects small scales and the D2 minimum; no uncertainty from this choice is propagated.
domain assumption Fiducial ΛCDM and ΛCDM+ν cosmological parameters
Inputs from Planck (2020) and Ivanov et al. (2020); σ_max(z) may depend on the assumed cosmology.
domain assumption Intensity pixels can be treated as discrete tracers in counts-in-spheres
Follows Camacho-Quevedo & Gaztañaga (2022); assumes the galaxy-count formalism applies to IM pixels.
domain assumption D2 threshold fixed at 2.97
Standard 1% homogeneity convention; arbitrary but enables comparison with previous RH studies.
domain assumption Redshift-space distortions can be neglected as a constant amplitude
The paper claims Kaiser introduces a constant amplitude, but this does not cancel in D2 = 3(1+ξ)/(1+ξ̄); see red flag.

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Cite this review

Pith. "Pith review of Accessing the homogeneity scale with 21 cm intensity mapping surveys." pith.science (2026). https://pith.science/paper/R3SJEWID

@misc{pith2026251113931,
  author       = {Pith},
  title        = {Pith review of: Accessing the homogeneity scale with 21 cm intensity mapping surveys},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3SJEWID}},
  note         = {Machine review of arXiv:2511.13931}
}

abstract

The homogeneity scale, $R_{\rm H}$, offers a fundamental test of the Cosmological Principle, yet it has not yet been measured with 21cm intensity mapping surveys. A key limitation for such a measurement is the telescope beam, which artificially smooths the observed signal. We quantify this effect using the two-point correlation function and the correlation dimension, $\mathcal{D}_2(r)$, to model how beam convolution suppresses intrinsic clustering. For any given redshift $z$, we identify a maximum beam width, $\sigma_{\rm max}(z)$, beyond which the homogeneity scale cannot be recovered. This limit defines an inaccessible region in the $\sigma \times z$ parameter space, where $R_{\rm H}$ is erased by beam smoothing. Applying this framework to several current and upcoming radio telescopes, we assess their ability to probe $R_{\rm H}$. Our results provide the first quantitative forecast of the instrumental requirements for measuring the cosmic homogeneity scale with 21cm IM, and establish a theoretical basis for future observational applications.

Figures

Figures reproduced from arXiv: 2511.13931 by the authors.

**Figure 1.** Illustration of the impact of different beam widths, 𝜎, on the 2pCF at a fixed redshift, 𝑧 = 0.8. The black dashed line corresponds to the 𝜎 = 0 case (no beam). All cases assume the fiducial ΛCDM cosmology. (a) at small separations, the beam smooths the clustering signal, reducing the amplitude; the suppressed information is redistributed to larger scales, where the amplitude increases; and (b) the BAO peak becomes… view at source ↗

**Figure 3.** Detectability of the transition scale, 𝑅H, for varying beam widths (𝜎), illustrated for the fixed redshift, 𝑧 = 0.8. The black solid (orange dashed) line represents 𝑅 ΛCDM H as affected by 𝜎 considering the fiducial ΛCDM (ΛCDM𝜈) model. The solid dark and dashed light grey lines represent 𝑅H (𝜎max ) = 𝑅min (𝜎max ), with 𝑅min (𝜎max ) calculated for ΛCDM and ΛCDM𝜈 models, respectively. The blue and green regions repre… view at source ↗

**Figure 4.** Detectability of 𝑅H in the 𝜎 × 𝑧 parameter space. The black solid (orange dashed) line represents 𝜎max (𝑧) as fitted by equation 9 for the ΛCDM (ΛCDM𝜈) model. The coloured lines show the 𝜎(𝑧) × 𝑧 for several 21 cm IM instruments. Their redshift (frequency) ranges are summarised in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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1 extracted references · cited by 3 Pith papers

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P., Bernui A., de Carvalho E., 2018, J

Abdalla E., et al., 2022, Journal of High Energy Astrophysics, 34, 49 Aghanim N., et al., 2020, A&A, 641, A6 Akrami Y., et al., 2020, A&A, 641, A7 Amiri M., et al., 2023, ApJ, 947, 16 Amiri M., et al., 2024, ApJ, 963, 23 Anderson C., et al., 2018, MNRAS, 476, 3382 Avila F., Novaes C. P., Bernui A., de Carvalho E., 2018, J. Cosmology Astropart. Phys., 2018...

arXiv 2022

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Reviewed August 3, 2026 · model on record in the stance chip above.

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P., Bernui A., de Carvalho E., 2018, J

Abdalla E., et al., 2022, Journal of High Energy Astrophysics, 34, 49 Aghanim N., et al., 2020, A&A, 641, A6 Akrami Y., et al., 2020, A&A, 641, A7 Amiri M., et al., 2023, ApJ, 947, 16 Amiri M., et al., 2024, ApJ, 963, 23 Anderson C., et al., 2018, MNRAS, 476, 3382 Avila F., Novaes C. P., Bernui A., de Carvalho E., 2018, J. Cosmology Astropart. Phys., 2018...

arXiv 2022