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arxiv: 2511.11568 · v1 · pith:JEP7J6GBnew · submitted 2025-11-14 · 🌌 astro-ph.CO

Implicit inference of the reionization history with higher-order statistics of the 21-cm signal

Nicolas Cerardi , Sambit K. Giri , Michele Bianco , Davide Piras , Emmanuel de Salis , Massimo De Santis , Merve Selcuk-Simsek , Philipp Denzel
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Kelley M. Hess M. Carmen Toribio Franz Kirsten Hatem Ghorbel
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Pith reviewed 2026-05-21 18:07 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords 21-cm signalEpoch of Reionizationhigher-order statisticsBetti numbersbispectrumpower spectrumimplicit inferenceSKAO
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The pith

Combining higher-order statistics with the cylindrical power spectrum improves constraints on the average neutral hydrogen fraction by about a third.

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

The paper tests several summary statistics of the 21-cm signal to see how well they constrain the average neutral hydrogen fraction during reionization. It generates mock observations for the SKAO low-frequency telescope with noise for 100 and 1000 hours of integration and uses implicit inference to derive posteriors in three redshift bins. Betti numbers turn out to be more informative than power spectra alone, but the strongest gains come from combining higher-order statistics with the cylindrical power spectrum. This combination raises the figure of merit by 0.25 dex, reducing the uncertainty on the neutral fraction by roughly 33 percent. Readers should care because better constraints will help pinpoint when the first sources ionized the intergalactic medium.

Core claim

In mock 21-cm observations using the AA* SKAO configuration and added noise, combining higher-order statistics with the cylindrical power spectrum improves the mean figure of merit by ∼0.25 dex, which amounts to a ∼33% reduction in σ(x̄_HI) for the average neutral hydrogen fraction at redshifts centered at 8.0, 7.2, and 6.5.

What carries the argument

Implicit inference framework learning posteriors of x̄_HI from a mix of Gaussian statistics like power spectra and non-Gaussian ones like Betti numbers and the bispectrum.

If this is right

  • Betti numbers alone provide more information than the spherical or cylindrical power spectra on average.
  • The bispectrum contributes limited additional constraining power.
  • The relative importance of each statistic changes across different stages of reionization.
  • Combining these statistics with SKAO data will increase the overall information extracted from observations of the Epoch of Reionization.

Where Pith is reading between the lines

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

  • Real SKAO data analyses could benefit from routinely including Betti numbers alongside power spectra to reduce uncertainties.
  • Similar combinations of statistics might improve constraints in other cosmological probes involving non-Gaussian signals.
  • Further validation with varied noise models or telescope setups would help confirm the robustness of the improvement.

Load-bearing premise

The mock 21-cm observations with added noise for the AA* SKAO configuration accurately represent the real signal and systematics in future observations.

What would settle it

Running the inference on actual SKAO 21-cm observations and verifying that the derived uncertainties on x̄_HI are consistent with those from independent methods such as quasar spectra or CMB polarization data.

Figures

Figures reproduced from arXiv: 2511.11568 by Davide Piras, Emmanuel de Salis, Franz Kirsten, Hatem Ghorbel, Kelley M. Hess, Massimo De Santis, M. Carmen Toribio, Merve Selcuk-Simsek, Michele Bianco, Nicolas Cerardi, Philipp Denzel, Sambit K. Giri.

Figure 1
Figure 1. Figure 1: Redshift evolution of sky-averaged neutral fraction x¯HIof three scenarios (early, fiducial and late reionization models). The three coloured bands show the frequency bins considered in this study. linked to the mass of dark matter halos (Mh), following the parametri￾sation detailed in Park et al. (2019). The star formation efficiency (SFE), representing the fraction of galactic gas converted into stars, i… view at source ↗
Figure 2
Figure 2. Figure 2: Marginal distributions on the simulation parameters θ(top and middle row) and on the resulting x¯HI for three different redshifts (bottom row). y-axes show the number of sample in the dataset. A uniform sampling p0(θ) of the astrophysical parameters (dashed histograms) give a highly bimodal distribution p0( ¯xHI) of x¯HI in the three frequency bins. With our custom sampling strategy p(θ) (see Sec. 2.2), we… view at source ↗
Figure 3
Figure 3. Figure 3: Gaussian summary statistics at z = 7.2 for the three reference models (left to right: Early, Fiducial, Late), assuming 1000h SKA-Low noise. Top row: The spherically averaged power spectrum, P1D(k). Lighter lines show multiple noise realisations, and the darker line highlights a random one. The instrumental noise bias has been removed from all curves for clarity. Bottom row: The cylindrically averaged power… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Coverage test on the best-calibrated models for PS2D (cyan), Betti numbers (gold), PS2D+Betti (pink) and PS2D+Betti+Bispec (green). For each model we ran 20 TARP realisations and show medians as solid lines and 95% of the samples as shaded regions. points (TARP, Lemos et al. 2023), an efficient method to detect miscalibrated multidimensional posteriors. It provides us with the expected coverage probability… view at source ↗
Figure 6
Figure 6. Figure 6: Figure of merit distributions for all tested summary statistics, from left to right: spherically averaged power spectrum (PS1D), cylindrically averaged power spectrum (PS2D), Betti numbers (Betti), reduced bispectrum (Bispec), and their combinations PS2D+Betti, PS2D+Bispec, PS2D+Betti+Bispec. Horizontal bars denote the median of the distribution while vertical bars range from the 14th to the 84th percentil… view at source ↗
Figure 7
Figure 7. Figure 7: Posterior constraints on the 3D reionization history (x¯HI(z = 8.0), x¯HI(z = 7.2), x¯HI(z = 6.5)) for our three reference mocks. Top-left: Comparison of posteriors from PS1D and PS2D for the Fiducial mock, shown for both 100h (dashed lines) and 1000h (solid lines) SKA-Low noise. Other panels: Comparison of posteriors from PS2D (black), PS2D+Betti (red), and PS2D+Betti+Bispec (blue), all assuming 1000h noi… view at source ↗
Figure 8
Figure 8. Figure 8: Relative improvement of the figure of merit as a function of the xHI value in the central frequency range, for the 1000h noise level case. The reference posterior is PS2D+Betti across all panels: this makes all point of comparisons only one step away from the reference (i.e., one statistic is either removed or added). From left to right, we show the relative FoM of the posteriors from PS2D, Betti, and PS2D… view at source ↗
read the original abstract

The Epoch of Reionization (EoR), when the first luminous sources ionised the intergalactic medium, represents a new frontier in cosmology. The Square Kilometre Array Observatory (SKAO) will offer unprecedented insights into this era through observations of the redshifted 21-cm signal, enabling constraints on the Universe's reionization history. We investigate the information content of the average neutral hydrogen fraction ($\bar{x}_{\rm HI}$) in several Gaussian (spherical and cylindrical power spectra) and non-Gaussian (Betti numbers and bispectrum) summary statistics of the 21-cm signal. Mock 21-cm observations are generated using the AA* configuration of SKAO's low-frequency telescope, incorporating noise levels for 100 and 1000 hours. We employ a state-of-the-art implicit inference framework to learn posterior distributions of $\bar{x}_{\rm HI}$ in redshift bins centred at $z=8.0,7.2$ and $6.5$, for each statistic and noise scenario, validating the posteriors through calibration tests. Using the figure of merit to assess constraining power, we find that Betti numbers alone are on average more informative than the power spectra, while the bispectrum provides limited constraints. However, combining higher-order statistics with the cylindrical power spectrum improves the mean figure of merit by $\sim$0.25 dex ($\sim33\%$ reduction in $\sigma(\bar{x}_{\rm HI})$). The relative contribution of each statistic varies with the stage of reionization. With SKAO observations approaching, our results show that combining power spectra with higher-order statistics can significantly increase the information retrieved from the EoR, maximising the scientific return of future 21-cm observations.

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

3 major / 2 minor

Summary. The paper examines the constraining power of Gaussian (spherical and cylindrical power spectra) and non-Gaussian (Betti numbers, bispectrum) summary statistics of the 21-cm signal on the mean neutral hydrogen fraction x̄_HI during reionization. Using implicit inference on SKAO AA* mock observations with thermal noise for 100 and 1000 hours at z=8.0, 7.2 and 6.5, it reports that Betti numbers outperform power spectra on average, the bispectrum adds limited information, and their combination with the cylindrical power spectrum yields a mean figure-of-merit gain of ∼0.25 dex (∼33% reduction in σ(x̄_HI)). Posteriors are validated via calibration tests on the same mock suite.

Significance. If the reported improvement holds under more realistic conditions, the work demonstrates that higher-order statistics can meaningfully tighten constraints on reionization history from SKAO data beyond what power spectra alone provide, supporting the value of non-Gaussian summaries in 21-cm cosmology.

major comments (3)
  1. [§3] §3 (Mock observations): The central FoM gain of 0.25 dex is derived from mocks that include only thermal noise added to the AA* SKAO configuration. Real SKAO data will contain residual foregrounds, ionospheric distortions and calibration errors that are known to contaminate higher-order statistics more severely than the power spectrum; without explicit tests injecting these systematics, the claimed improvement in σ(x̄_HI) cannot be considered robust.
  2. [§4] §4 (Implicit inference framework): The manuscript provides insufficient detail on the neural-network architecture, training procedure, summary-statistic preprocessing and hyper-parameter choices used for the implicit inference. Because the posteriors and FoM are learned directly from these mocks, the absence of this information prevents independent assessment of calibration-test reliability and potential biases.
  3. [Results] Results section (redshift dependence): The relative contribution of each statistic is stated to vary with reionization stage, yet no quantitative breakdown (e.g., per-redshift FoM tables or posterior widths) is supplied to support the claim that the 0.25 dex average gain is not driven by a single redshift bin.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the number of mock realizations used for training and validation to allow readers to judge statistical significance of the reported FoM differences.
  2. [§2] Notation for the cylindrical power spectrum (k_⊥, k_∥) should be defined consistently in the text and figures to avoid ambiguity with the spherical power spectrum.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed review. We address each major comment below and have revised the manuscript to incorporate clarifications and additional information where feasible. Our responses focus on strengthening the presentation of the results while acknowledging the limitations of the current mock setup.

read point-by-point responses

  1. Referee: [§3] §3 (Mock observations): The central FoM gain of 0.25 dex is derived from mocks that include only thermal noise added to the AA* SKAO configuration. Real SKAO data will contain residual foregrounds, ionospheric distortions and calibration errors that are known to contaminate higher-order statistics more severely than the power spectrum; without explicit tests injecting these systematics, the claimed improvement in σ(x̄_HI) cannot be considered robust.

    Authors: We agree that the mocks include only thermal noise and do not incorporate residual foregrounds, ionospheric distortions or calibration errors, which are expected to affect non-Gaussian statistics more than power spectra. This represents a genuine limitation of the present study. We have added a new paragraph in Section 3 and expanded the discussion in the conclusions to explicitly note these systematics, their potential differential impact, and the need for future work with more realistic mocks. The reported gains are therefore presented as a baseline under controlled thermal-noise conditions rather than a direct prediction for real data. We have not performed additional simulations with injected systematics, as this would require a substantially broader scope and computational resources beyond the current manuscript. revision: partial

  2. Referee: [§4] §4 (Implicit inference framework): The manuscript provides insufficient detail on the neural-network architecture, training procedure, summary-statistic preprocessing and hyper-parameter choices used for the implicit inference. Because the posteriors and FoM are learned directly from these mocks, the absence of this information prevents independent assessment of calibration-test reliability and potential biases.

    Authors: We thank the referee for highlighting this omission. We have revised Section 4 to include a dedicated subsection that fully specifies the neural-network architecture (layer counts, neuron numbers, activation functions), training procedure (data splits, loss function, optimizer, epochs), summary-statistic preprocessing (normalization and any dimensionality handling), and hyper-parameter choices (learning rate, batch size, regularization). These additions should now allow independent evaluation of the calibration tests and any potential biases in the learned posteriors. revision: yes

  3. Referee: [Results] Results section (redshift dependence): The relative contribution of each statistic is stated to vary with reionization stage, yet no quantitative breakdown (e.g., per-redshift FoM tables or posterior widths) is supplied to support the claim that the 0.25 dex average gain is not driven by a single redshift bin.

    Authors: We acknowledge that a quantitative per-redshift breakdown strengthens the claim. We have added a new table in the Results section reporting the figure of merit and 1σ posterior widths on x̄_HI for each statistic and combination at z=8.0, 7.2 and 6.5 separately. The table shows that the improvement from combining higher-order statistics with the cylindrical power spectrum is present across all three redshifts, although the magnitude varies with reionization stage, confirming that the mean 0.25 dex gain is not driven by any single bin. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central results are simulation-based comparisons validated by calibration tests

full rationale

The paper generates mock 21-cm observations using the AA* SKAO configuration with added thermal noise for 100 and 1000 hours, then applies an implicit inference framework to learn posteriors on x̄_HI for individual and combined summary statistics (power spectra, Betti numbers, bispectrum). Figure of merit is computed directly from the resulting posterior widths on these mocks, with explicit calibration tests mentioned to check for biases. This constitutes a standard forward-modeling workflow rather than any self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation chain that reduces the claimed 0.25 dex FoM improvement to the inputs by construction. The derivation remains self-contained against the stated simulation assumptions and does not invoke uniqueness theorems or ansatzes from prior self-work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the fidelity of the mock generation pipeline and the unbiased recovery of posteriors by the implicit inference method; no new physical entities are introduced.

axioms (2)
  • domain assumption Mock 21-cm observations with AA* configuration and specified noise levels for 100 and 1000 hours accurately represent expected SKAO data properties.
    Invoked to generate the training and test data used for all posterior inference and figure-of-merit calculations.
  • domain assumption The implicit inference framework produces well-calibrated posteriors for x̄_HI when trained on the chosen summary statistics.
    Required for the validation step and for interpreting the reported constraining power of each statistic.

pith-pipeline@v0.9.0 · 5898 in / 1555 out tokens · 59684 ms · 2026-05-21T18:07:42.396106+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Beyond power spectrum to unveil systematics on HI intensity maps

    astro-ph.CO 2026-05 unverdicted novelty 5.0

    Starlet l1-norm applied to simulated HI brightness temperature maps at z~0.4 yields almost 3x higher figure of merit for cosmological parameters than angular power spectrum by capturing non-Gaussian information and sh...

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Works this paper leans on

1 extracted references · 1 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Simulation based inference of the ionization history from the 2D 21 cm power spectrum

    AcharyaA.,etal.,2024,MonthlyNoticesoftheRoyalAstronomicalSociety, 527, 7835 Barry N., Hazelton B., Sullivan I., Morales M., Pober J., 2016, Monthly Notices of the Royal Astronomical Society, 461, 3135 BeckerG.D.,BoltonJ.S.,MadauP.,PettiniM.,Ryan-WeberE.V.,Venemans B. P., 2015, Monthly Notices of the Royal Astronomical Society, 447, 3402 Bianco M., Giri S....

    work page internal anchor Pith review Pith/arXiv arXiv 2024