Gravitational Lensing Effects by Galaxy Clusters on Ionised Bubble Size Distribution during the Epoch of Reionisation

Di Wu; Huanyuan Shan; Nan Li; Zhenghao Zhu

REVIEW 3 major objections 4 minor 1 references

Gravitational Lensing Effects by Galaxy Clusters on Ionised Bubble Size Distribution during the Epoch of Reionisation

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Foreground galaxy clusters can inflate the apparent abundance of large ionised bubbles during reionisation, by 219% to 832% at z=14.

desk verdict Novel, honest first pass at cluster-lensing contamination of the EoR bubble-size distribution; the qualitative result is probably right, but the abstract's headline percentages sit in a noise-dominated tail and aren't yet supported. read the letter →

arxiv 2603.11371 v2 pith:IZFM7XB5 submitted 2026-03-11 astro-ph.CO

Di Wu , Nan Li , Huanyuan Shan , Zhenghao Zhu This is my paper

classification astro-ph.CO

keywords gravitationallensingEpochofReionisationionisedbubblesbubblesizedistribution21-cmsignalgalaxyclustersmagnificationbias

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 argues that gravitational lensing by galaxy clusters in the foreground of the Epoch of Reionisation systematically distorts the measured size distribution of ionised hydrogen bubbles. Combining ray-traced cluster lenses with several reionisation simulations, it finds that the apparent number of large bubbles (R > 15 cMpc) increases by 219% in a faint-galaxy-dominated model and by 832% in a bright-galaxy-dominated model at redshift 14, while small bubbles remain unchanged. If correct, cluster lensing is a non-negligible systematic that must be modelled before interpreting 21-cm bubble statistics from SKA-era surveys. The effect fades as reionisation proceeds and bubbles grow past the typical lensing cross-section.

What carries the argument

The argument runs on a multiple-lens-plane ray-tracing simulation: cluster-scale dark matter haloes are placed along the line of sight with masses sampled from a halo mass function and ellipticities from a redshift-dependent distribution, each modelled as an elliptical truncated dark-matter density profile; light rays are deflected plane by plane with the mean convergence subtracted. The source side is a set of reionisation light cones from semi-numerical and published simulations. Bubble sizes are measured by the Mean Free Path method, which shoots random rays through ionised regions and uses their path lengths to define a size distribution; the comparison of lensed and unlensed BSDs from 5

What would settle it

Rerun the same lensing pipeline with a deflector light cone taken from a cosmological N-body simulation that includes clustering, filaments and subhaloes, and compare the large-bubble enhancement at z=14; if the enhancement does not reproduce or exceed the Poisson-halo value, the central claim is unsupported.

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Extended reading notes

Core claim

The central discovery is that gravitational lensing by foreground galaxy clusters changes the apparent bubble size distribution (BSD) of the Epoch of Reionisation in a size-dependent way: it boosts the count of large ionised bubbles but leaves small ones untouched. Using a multiple-lens-plane ray-tracing pipeline with deflectors drawn from a halo mass function and modelled as elliptical truncated dark-matter haloes, the paper measures lensed and unlensed BSDs with a ray-shooting size estimator. Across all source models, the large-bubble enhancement is strongest at z=12–14 when ionised bubbles are small relative to cluster Einstein radii, reaching a 219% increase in the apparent number of R>1

Load-bearing premise

The lens population is built from isolated elliptical haloes placed at random positions with no clustering, filaments, subhaloes or external shear, so the computed enhancements rest on how well this simplified deflector population represents real lines of sight; the authors acknowledge this makes their numbers a lower limit.

Editorial extensions

If this is right

SKA-era 21-cm surveys that infer reionisation progress from bubble size distributions must include cluster lensing as a forward-modelled systematic; otherwise the apparent abundance of large bubbles will be overestimated.
Small bubbles (down to ~2 cMpc) appear robust to lensing, so conclusions about small-scale ionisation topology from BSDs are unlikely to be biased by foreground clusters.
The lensing enhancement is strongest at the earliest observable stages (z ≈ 12–14) and weakens by z ≈ 9, meaning early-EoR observations need the largest corrections.
A directional projection test reported in the paper shows only minor lensing-induced changes along the line of sight, suggesting a possible route to statistically correct for lensing or to recover unlensed bubble statistics.

Reading between the lines

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

Because the deflector model treats haloes as isolated and unclustered, the reported 219% and 832% enhancements are probably lower limits; including filaments, subhaloes and correlated line-of-sight structure should strengthen the effect (the paper itself says its assumptions lead to a lower limit).
If the effect holds, it should also imprint on other size-dependent EoR observables, such as the brightness-temperature power spectrum at large scales, because magnification remaps the ionisation field; a direct test would be to measure the 21-cm power spectrum in fields behind known massive clusters versus blank fields.
A natural calibration experiment is to simulate lensing through an N-body-based light cone with realistic large-scale structure and compare enhancement factors to the Poisson-halo values; a discrepancy would quantify the missing clustering contribution.

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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

First things first: this is the first direct computation I know of that puts cluster lensing into the EoR bubble-size distribution, and the qualitative claim is probably right — lensing inflates the apparent abundance of large ionised bubbles at z≈12–14 while leaving small bubbles alone. I'd trust that part. It holds across all five source models, and the machinery is clean: lensed and unlensed BSDs come from identical maps run through the identical pipeline, with 50 deflector realisations and proper convergence tests on both ray-tracing resolution and MFP iteration counts. The deflector population is validated against the Tinker mass function and ellipticity distributions. The authors also flag their own simplifications — uniform deflector placement, single-halo TNFW profiles, no subhaloes — and honestly call the result a lower limit. No circularity: every input is external.

The soft spot is the abstract's headline numbers, and the stress-test note lands. The 219% and 832% figures are ratios in the extreme tail of the BSD (R>15 cMpc), exactly where the paper's own convergence test (Fig. 10) shows the largest relative scatter. That test was run on HIGH-RES at z=14, not on the EOS models the headline numbers come from. And the unlensed EOS BSD comes from a single light cone — no error bar on the denominator — while the lensed numerator is averaged over 50 realisations. With 10^6 MFP iterations and tail probabilities near 10^-6, the unlensed tail count might be a handful of ray hits; a ratio built on that is fragile. The authors even acknowledge Poisson uncertainty tied to the rarity of large bubbles in §5.2. So the qualitative direction is likely robust, but the specific percentages in the abstract are not yet supported.

One more thing: the arXiv abstract promises a directional-projection test suggesting a route to recovering unlensed statistics, but that analysis does not appear in the body. Either it's a remnant of an earlier version or it got lost; either way the abstract overclaims. No code or data shipped either — just "available upon request."

Who should read this: anyone forecasting what SKA-era 21-cm observations can say about the BSD, and lensing people thinking about magnification bias on extended sources. It deserves a serious referee. I'd send it to review, asking that the tail ratios get real error bars or softer wording, that the z=14 iteration test be run on the EOS models, and that the directional-projection claim be reinstated or cut from the abstract.

Referee Report

3 major / 4 minor

Summary. The paper studies gravitational lensing by foreground galaxy clusters as a potential systematic for the ionised-bubble size distribution (BSD) during the Epoch of Reionisation. The authors build Monte Carlo lens light cones from the Tinker halo mass function and elliptical truncated-NFW profiles, combine them with five EoR source models (three 21CMFAST resolution runs and the EOS Faint/Bright Galaxies models), and compare lensed and unlensed BSDs measured with the Mean Free Path method on identical maps. They report that lensing enhances the abundance of large bubbles (R > 15 cMpc) by 219% and 832% at z = 14 for the EOS Faint and Bright Galaxies models, leaves small bubbles nearly unchanged, and produces only minor directional projection changes. Resolution and MFP iteration convergence tests are presented.

Significance. If the central result is quantitatively robust, cluster-lensing magnification is a previously neglected systematic for interpreting 21-cm bubble statistics in the SKA era. The paper has clear strengths: the core comparison is controlled (lensed and unlensed BSDs are measured from identical source maps), deflector realisations are numerous (50), the deflector population is built from standard public ingredients (Tinker HMF, TNFW profiles), and explicit ray-tracing resolution and MFP iteration tests are shown. The authors are also transparent about the simplified deflector model and describe it as a lower limit. There is no circularity: no target quantity is fitted, and the lensing calculation is driven by externally specified inputs. The main weakness is that the headline percentages rest on the extreme tail of the BSD, where the paper's own convergence tests show the largest scatter and where the unlensed sample is a single light cone per EOS model.

major comments (3)

[§5.4, Fig. 10] The MFP iteration-convergence test at z = 14 is performed only for the HIGH-RES model; the EOS Faint and Bright Galaxies models, which produce the headline 219% and 832% numbers, are not given the high-iteration test at z = 14 (the z = 9 EOS tests do not cover z = 14). Figure 10 itself shows the largest relative difference at the largest bubble sizes, exactly the R > 15 cMpc tail used for the abstract's percentages. Please rerun the EOS z = 14 slices with i = 10^8 (or otherwise demonstrate convergence in the tail) and quote unlensed/lensed tail counts, or restrict the abstract to the qualitative direction of the effect.
[Table 1, §3.1, §3.3, §5.2] The unlensed EOS BSDs are derived from a single light cone per source model, while the lensed BSDs are averaged over 50 deflector realisations. No Poisson or cosmic-variance uncertainty is quoted for the unlensed tail. If the R > 15 cMpc bin contains only a few MFP ray hits, the reported 219% and 832% enhancements are ratio-noise dominated. The authors themselves note in §5.2 that Poisson uncertainty is significant for rare large bubbles. Please report the number of MFP rays (or bubbles) in the tail, add Poisson/bootstrap error bars or multiple EOS source light cones, and state the statistical significance of the headline percentages.
[§5.3, Eqs. (16)–(18)] The deflector population is uniform and single-halo, with no clustering, subhaloes, or external shear. The authors correctly describe this as a lower limit, but the abstract and conclusions frame the effect as an 'unavoidable systematic' for SKA-era measurements. As written, the quantitative estimates are conditional on a minimal deflector model. A concrete sensitivity test — for example, adding line-of-sight halo contributions as in Li et al. (2019) or a clustering-boosted halo population — would substantiate the claim that the true effect is larger, rather than merely model-dependent.

minor comments (4)

[§5.1, Fig. 8] The quantity R is described as 'bubble size' but the units (cMpc) appear only in the figure axes; please define R and its units explicitly in the text.
[Fig. 9 caption] The caption uses both δp/p_{2.5} and p_2.5, and refers to 'PDF*' without defining it; please standardise the notation.
[§5.4] The phrase 'At z_s > 12 and z_s = 9' is ambiguous; it should be 'At z_s = 12 and 14, and at z_s = 9'.
[Table 1] For the EOS models, the resolution is listed but not the simulation box size or light-cone transverse scale; since box-size effects on large bubble statistics are discussed in §5.2, these values should be given in the table or nearby text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lensing-BSD comparison is a forward simulation whose inputs (Tinker mass function, TNFW profiles, external EOS/21CMFAST source boxes) are external to the output.

full rationale

The derivation chain is a forward multi-plane ray-tracing calculation. Source slices come from external EOS data and 21CMFAST (Table 1); deflector realizations are Monte-Carlo draws from the Tinker et al. (2008) mass function, TNFW profiles, and Hopkins et al. (2005) ellipticity distributions (Sect. 3.2). The lensed BSD is obtained by ray-tracing these inputs and measuring with the MFP method; the unlensed BSD is the same map without deflection (Sect. 3.3). No quantity appearing in the output, such as the 219%/832% tail ratios, is used to set or fit any input parameter, and the MFP estimator is not calibrated to the lensed-minus-unlensed difference. The only self-citation, Li et al. (2019), is used in Sect. 5.3 to argue qualitatively that ignoring line-of-sight haloes makes the computed effects a lower limit; this is peripheral rather than load-bearing, and it is not used to normalize or derive the reported percentages. The paper itself flags the deflector model and single-light-cone statistics as limitations (Sects. 5.2 and 5.3), and its own convergence tests show the largest scatter at the largest bubble sizes (Sect. 5.4); these are soundness/robustness concerns, not circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper adds no new physical entities. Its central result rests on standard cosmological and reionisation-model assumptions plus two explicit deflector-population simplifications (uniform positions, single-halo TNFW profiles). The main free parameters are ionisation-model inputs that set the intrinsic bubble topology; the 2 cMpc MFP cutoff bounds the small-bubble portion of the claim.

free parameters (3)

Ionising efficiency ζ = ζ=20 (EOS Faint), ζ=200 (EOS Bright), ζ=30 (LOW/MID/HIGH-RES)
Input parameter of the 21CMFAST excursion-set model controlling the ionising photon budget; it determines the intrinsic bubble topology before lensing. Chosen per model from literature, not constrained by BSD or lensing data (Table 1).
Minimum host halo virial temperature T_vir (M_min) = 2×10^4 K (Faint), 2×10^5 K (Bright), 5×10^4 K (LOW/MID/HIGH-RES)
Sets the minimum mass of haloes hosting ionising sources; strongly affects the bubble-size distribution. Chosen per model, not fitted to the target result (Table 1).
MFP shortest travel distance (resolution floor) = 2 cMpc
Analysis cutoff chosen conservatively in Sect. 5.4; defines the claimed lower bound down to which small bubbles are unaffected by lensing. Below this scale the lensing effect is unresolved, so part of the central claim is conditional on this choice.

assumptions (6)

domain assumption ΛCDM cosmology with Planck 2018 parameters (Ω_m=0.3111, Ω_Λ=0.6889, h=0.6766, n_s=0.9665)
Used for deflector/source distances and lensing geometry; not derived. The EOS source models use Planck 2015 parameters, introducing a small unquantified mismatch (Sect. 3.1).
domain assumption 21CMFAST excursion-set reionisation model with recombination and unsaturated spin temperature (Eq. 2)
The source light cones and intrinsic BSDs depend on this semi-numerical formalism (Furlanetto et al. 2004; Mesinger et al. 2016); its accuracy relative to full radiative transfer is not re-derived here.
domain assumption TNFW mass profile with truncation radius r_t=2.6 r_200 (Eqs. 11–15)
All deflectors are modelled as single truncated NFW haloes; subhaloes and external shear are omitted. This is acknowledged in Sect. 5.3 as a simplification.
ad hoc to paper Uniform angular distribution of deflectors
Monte Carlo halo positions are drawn uniformly in angle, ignoring halo clustering and filaments; explicitly stated in Sect. 5.3 as a simplification.
standard math Multiple-lens-plane ray tracing with mass-sheet subtraction (Eqs. 9–10, 19–20)
Standard gravitational lensing formalism is adopted; the mean convergence κ_mean is subtracted per plane so ray tracing is driven by density fluctuations, which affects the absolute magnification convention.
domain assumption MFP method with x_HI<0.5 boundary defines bubble size
BSD is measured with the mean free path algorithm (Mesinger & Furlanetto 2007) using the Tools21cm implementation; the 2 cMpc slice thickness and ray-length cut set the resolution floor (Sect. 4).

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Cite this review

Pith. "Pith review of Gravitational Lensing Effects by Galaxy Clusters on Ionised Bubble Size Distribution during the Epoch of Reionisation." pith.science (2026). https://pith.science/paper/IZFM7XB5

@misc{pith2026260311371,
  author       = {Pith},
  title        = {Pith review of: Gravitational Lensing Effects by Galaxy Clusters on Ionised Bubble Size Distribution during the Epoch of Reionisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZFM7XB5}},
  note         = {Machine review of arXiv:2603.11371}
}

read the original abstract

The statistical properties of ionisation structures during the Epoch of Reionisation (EoR) provide valuable insights into the formation of the first stars and galaxies. However, size distributions of ionisation structures can be affected by gravitational lensing from foreground massive structures such as galaxy clusters. To quantify the impact of cluster lensing on the ionised Bubble Size Distribution (BSD), we performed a series of multiple-lens-plane simulations combining light cones of clusters with source light cones based on different ionisation models. The deflector population is generated with a Monte Carlo method guided by the halo mass function and empirical scaling relations, and the deflectors are modelled with truncated Navarro-Frenk-White (TNFW) profiles. Source light cones are produced semi-numerically or taken directly from the Evolution of 21 cm Structure (EOS) project. Using the Mean Free Path method, we measure both unlensed and lensed BSDs. We find that gravitational lensing increases the apparent abundance of large bubbles while leaving small bubbles nearly unchanged across all source models considered. In particular, for the EOS faint-galaxies model, the apparent number of bubbles with R > 15 cMpc increases by 219% at z = 14; for the EOS bright-galaxies model, it increases by 832% under the same conditions. Moreover, a directional projection test shows only minor lensing-induced changes in line-of-sight direction, suggesting a possible route to recovering unlensed bubble statistics. Above all, lensing introduces unavoidable systematics into BSD measurements that should be carefully taken into account for relevant studies in the SKA era.

Figures

Figures reproduced from arXiv: 2603.11371 by the authors.

**Figure 1.** Demonstration of a slice in the redshift direction of one HIGH-RES simulation. The upper panel shows the evolution of the 𝑥HI (neutral hydrogen fraction). The middle panel displays the 21 cm differential brightness temperature. The lower panel presents the global 21 cm differential brightness temperature evolution [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Number counts of ionised bubbles as a function of projected area of bubble for simulation slices at redshifts 𝑧= 9 (left), 𝑧= 12 (middle), and 𝑧= 14 (right), respectively. Bubbles were identified via a Friends-of-Friends (FoF) algorithm (Iliev et al. 2006) using an ionisation fraction threshold of 0.5. The HIGH-RES resolves the highest abundance of small-scale ionised regions across all three epochs, highlighting th… view at source ↗

**Figure 3.** [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Comparison between the mass functions of the deflectors in our light cones (red points) and the theoretical outcomes given by Tinker et al. (2008) (blue lines) at redshifts 𝑧= 0.5, 1.5, and 2.5, from left to right, respectively. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 …

**Figure 5.** Figure 5: Comparison between the PDFs of projected mass ellipticity 𝑒 of the deflectors in our light cone (orange bars) and theoretical outcomes of the mean 𝑒 evolution given by Hopkins et al. (2005) with Gaussian dispersion 𝜎𝑒 = 0.16 (blue lines) at redshifts 𝑧= 0.5, 1.5, and 2…

**Figure 6.** Figure 6: A showcase about how lensing changes the morphology of the ionised bubble of HIGH-RES at redshift 𝑧 = 12. The panels from left to right are an unlensed 𝑥HI map and the corresponding lensed one. The red segments at the bottom-right of each panel correspond to 50 arcsec,…

**Figure 7.** Figure 7: A schematic diagram of using the MFP method to measure the BSD. This picture is a cartoon diagram of a two-dimensional 𝑥HI map during EoR, we randomly select starting points in ionised regions of this map where 𝑥HI < 0.5 (white regions) and shoot rays in random directi…

**Figure 8.** Figure 8: Bubble Size Distribution (BSD) for lensed and unlensed cases at redshifts 𝑧 = 14, 𝑧 = 12, and 𝑧 = 9, from top to bottom. Solid lines indicate unlensed BSD and dashed lines represent lensed BSD. Different colours represent the results of different ionisation models. The…

**Figure 9.** Figure 9: The influence of ray-tracing resolution on BSD. These panels show lensed and unlensed BSD of HIGH-RES when ray-tracing resolution is set to 5, 7.5, 10, and 15 arcsec at redshifts 12 and 14, respectively. We also show the unlensed BSD when the resolution is 2.5 arcsec a…

**Figure 10.** Figure 10: Robustness test of iteration counts in the MFP method for BSD measurement at 𝑧𝑠 = 14 and 𝑧𝑠 = 12 for HIGH-RES model. The upper panels present the standard implementation with 𝑖 = 106 iterations in the MFP algorithm (solid curves) and the realisation with 𝑖 = 108 itera…

**Figure 11.** Figure 11: Robustness test of iteration counts in the MFP method for BSD measurement at 𝑧𝑠 = 9 for the EOS Bright galaxies and Faint galaxies model. The upper panels present the standard implementation with 𝑖 = 108 iterations in the MFP algorithm (solid curves) and the realisati…

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