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A New Constraint on the Optical Depth from the Reionization History Independent of CMB Large-Scale E-Mode Polarization

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

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Pith's one-line read The paper claims that the cosmic optical depth can be pinned down to τ=0.0552 using reionization history combined with CMB data that exclude the large-scale E-mode polarization, sidestepping the systematic worries attached to that polarizat

desk verdict Useful tau from CMB-without-lowE plus reionization data, but the independence claim is weakened by a GP prior mean from Planck lowE; needs a robustness test. read the letter →

arxiv 2601.09644 v2 pith:2PXPF3LS submitted 2026-01-14 astro-ph.CO astro-ph.GA

Yuta Kageura , Masami Ouchi , Fumihiro Naokawa , Hiroya Umeda , Akinori Matsumoto , Yuichi Harikane , Minami Nakane , Tran Thi Thai This is my paper · ORCID
classification astro-ph.COastro-ph.GA
keywords cosmicopticaldepthreionizationhistoryneutralhydrogenfractionGaussianprocessregressionCMBE-modepolarizationbaryonacousticoscillationsneutrinomasssumdynamicaldarkenergy
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 sets out to measure the CMB optical depth τ without relying on the large-scale E-mode polarization, which is vulnerable to foreground and instrumental systematics. It combines the latest measurements of the neutral hydrogen fraction x_HI(z) from Lyman-α forest and damping-wing absorption with CMB power spectra that exclude that large-scale polarization, and obtains τ=0.0552+0.0019−0.0026 (stat) +0.0075−0.0049 (sys). The precise τ breaks the degeneracy between τ and the matter density Ωm, exposing a 2.4σ tension with baryon acoustic oscillation results and yielding a 95% upper limit of Σmν<0.055 eV that supports normal neutrino mass ordering. A mild 2.2σ tension between that cosmological cap and the oscillation-based lower limit is presented as an independent hint of physics beyond ΛCDM.

What carries the argument

The central object is the Gaussian-process reconstruction of the neutral-hydrogen fraction x_HI(z), using a sigmoid link function and a radial-basis-function kernel with hyperparameters set by marginal likelihood. A prior mean μ(z)=z−7.67 is adopted, where 7.67 is the reionization midpoint inferred from CMB analyses that include the large-scale E-mode data the paper claims to avoid; this choice imports information from those low-E data. The reconstructed history is then integrated in the Thomson-scattering formula for τ, making τ a derived parameter rather than a fitted one.

What would settle it

Take the same x_HI(z) data and run the Gaussian-process reconstruction with a prior mean that contains no low-E information, such as a constant or a mean fitted from the neutral-fraction data alone; if the resulting τ moves outside 0.0552±0.0026 (or the systematic range), then the claimed independence does not hold.

Watch

Extended reading notes

Core claim

On its own terms, the paper discovers that the optical depth to reionization can be determined to ~4% precision without any large-scale E-mode polarization data, provided the reionization history is known. Using a Gaussian-process reconstruction of x_HI(z) from z≈5 to 14 and a Thomson-scattering integral, the analysis yields τ=0.0552+0.0019−0.0026, a value consistent with (but more precise than) the result that uses the E-mode data. This τ breaks the τ–Ωm degeneracy, and the resulting (H0rd, Ωm) is in 2.4σ tension with BAO measurements, confirming the mild ΛCDM discrepancy. The same combination with BAO gives Σmν<0.055 eV at 95% confidence, which favors normal mass ordering and, combined wit

Load-bearing premise

The 'independent' τ measurement leans on a Gaussian-process prior whose mean μ(z)=z−7.67 is taken from the large-scale E-mode-derived reionization midpoint; if that prior is replaced or is wrong, the τ value and the derived tensions could shift.

Editorial extensions

If this is right

  • τ can be measured at ~0.002 precision without using the uncertain large-scale E-mode polarization, so any systematic bias in that polarization measurement cannot be the sole driver of the CMB–BAO tension.
  • The τ–Ωm degeneracy is broken, giving Ωm=0.3165±0.0061 and H0=67.17±0.44 km/s/Mpc, both in mild 1.8σ tension with BAO and in 2.4σ tension in the (H0rd, Ωm) plane.
  • The derived upper limit Σmν<0.0550 eV (95%) strongly favors normal mass ordering and, when combined with oscillation results, disfavors quasi-degenerate neutrino spectra.
  • The 2.2σ tension between the cosmological upper limit and the oscillation lower bound is reached without low-E data, pointing toward either BAO systematics or new physics such as neutrino decay.
  • Precision on As, ns, and σ8 also improves by factors of 3–5 because the τ degeneracies are removed.

Reading between the lines

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

  • The claim of 'independence' is only partial: the Gaussian-process prior mean encodes the low-E-derived midpoint z=7.67, so a version that conditions only on the x_HI data (e.g., with a flat or data-driven mean) would test how much of the τ constraint is actually carried by the low-E information.
  • The same pipeline is ready-made for future 21-cm reionization measurements or deeper high-redshift galaxy damping-wing samples, which would shrink the systematic uncertainty that currently dominates the error budget.
  • If the neutrino-mass tension persists with more precise BAO data, it may constrain exotic mechanisms like neutrino decay or negative-mass effective models; the paper explicitly leaves a negative-mass prior for future work.
  • The 2.4σ CMB–BAO tension and the dynamical dark energy fit (w0=−0.55, wa=−1.40) are consistent with previous results, suggesting the low-E polarization data are not responsible for the apparent ΛCDM discrepancy.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper determines the CMB optical depth τ using Planck CMB power spectra that exclude the large-scale E-mode polarization (CamSpec l≥30 TT/TE/EE, Commander low-ell TT, ACT+Planck lensing) combined with a Gaussian-process reconstruction of the reionization history x_HI(z) from Lyα forest, damping-wing, LAE, and JWST/Subaru data. The central result is τ=0.0552+0.0019−0.0026 (stat.) +0.0075/−0.0049 (sys.), which the authors claim is independent of the CMB large-scale E-mode polarization. They further use this τ to break the τ–Ωm degeneracy, report a 2.4σ tension with DESI DR2 BAO in the H0rd–Ωm plane, and derive a 95% upper limit Σmν<0.0550 eV from CMB+BAO+x_HI(z). The statistical machinery is plausible and the τ value is consistent with previous estimates, but the headline independence is weakened because the Gaussian-process prior mean is fixed to μ(z)=z−7.67, where 7.67 is the Planck TTTEEE+lowE+lensing reionization midpoint. The paper does not quantify how much of the constraining power comes from this prior, so the central claim requires a robustness test.

Significance. If the central result survives an alternative choice of the Gaussian-process prior mean, the paper would provide a valuable route to τ that does not rely on low-ell E-mode polarization, with direct implications for the CMB–BAO tension and neutrino masses. The paper has clear strengths: it carefully selects x_HI measurements to avoid duplication, incorporates recent JWST/Subaru constraints, uses publicly available CMB likelihoods, and gives an explicit Gaussian-process framework with MCMC sampling. However, because the GP prior mean imports the very lowE information the analysis claims to avoid, the strict independence claim is not established. The systematic-error treatment is also insufficiently specified to assess the robustness of the reported τ and the derived tension. These issues are load-bearing for the paper's main conclusions, so the manuscript needs substantial revision rather than minor polishing.

major comments (3)
  1. [Appendix A, Eq. (A2)] The GP prior mean is μ(z)=z−7.67, with 7.67 explicitly the reionization midpoint from Planck TTTEEE+lowE+lensing. Through the sigmoid (A1) this is not a weak centering: it imposes a fairly specific monotonic history, strongest where x_HI data are sparse (z≳9–10). This prior enters the first term of Eq. (1) before the CMB (w/o lowE) likelihood, so the final τ=0.0552 and its uncertainty may partly inherit the lowE information the paper says it avoids. Please rerun the analysis with (i) μ=0, (ii) μ(z)=z−z_mid with z_mid marginalized over a broad prior, or (iii) z_mid constrained only from CMB without lowE, and report the shift in τ and in the BAO tension. Without such a test, the independence claim is not supported.
  2. [Sec. 4.5] The systematic-error treatment is not reproducible as written. The statement 'we propagate a ~7% modeling uncertainty in the x_HI values' does not specify whether this is a coherent shift of all points, the assumed correlation, or how the quoted τ=0.0552+0.0075−0.0049 was obtained. Since the systematic error exceeds the statistical error and the paper itself lists additional unquantified biases (e.g., intrinsic-spectrum evolution, simulation resolution), the robustness of the central τ and of the 2.4σ BAO tension is not demonstrated. Please provide the explicit propagation calculation, including the covariance matrix for the x_HI modeling uncertainty, and show its effect on the H0rd–Ωm contours.
  3. [Sec. 2.1 and Sec. 3] The adopted constraint x_HI>0.999 (1σ lower limit) at z=15 is an assumption, not a measurement, and is used to enforce a monotonic reionization history. It enters the GP posterior before the CMB fit. While its direct contribution to the τ integral is small, its role as an anchor for the GP at high redshift and for the monotonicity assumption is not tested. Please add a sensitivity test that relaxes or removes this point, or substitutes a less restrictive prior, and report the effect on τ.
minor comments (4)
  1. [Eq. (2)] The factor '×100 km/s' and the appearance of h^2 inside the redshift integral are dimensionally unconventional; a brief note explaining the unit convention would help readers verify Eq. (2).
  2. [Fig. 1 caption] The caption says the brown dotted lines indicate 'the redshift corresponding to instantaneous reionization with τ=0.09 and τ=0.0552' and then notes z=7.73 differs from z_mid=7.19. This is confusing because z_mid=7.19 is not the instantaneous-reionization redshift; please clarify which quantity each line represents.
  3. [Sec. 4.1 and Eq. (11)] The text describes the first term of Eq. (11) as the Gaussian-process posterior, but the notation p(x_HI(z⋆)|{x_HI}) is used both for the GP posterior and for the fitted values in Eq. (1). Please distinguish the GP training posterior from the 10-point fitting variables.
  4. [Sec. 4.3] When computing χ2 with Eq. (11), please state explicitly how the DESI covariance matrix is obtained and whether it includes the BBN prior on ωb used in the reported BAO-only Ωm and H0 values.

Circularity Check

1 steps flagged · score 5.0 of 10

Claimed independence from low-ell E-mode is compromised: the GP prior mean is fixed by Planck TTTEEE+lowE+lensing, and the impact is unquantified.

  1. fitted input called prediction [Appendix A, Eq. (A2) and surrounding text; enters Eq. (1) and produces Eq. (3)]
    "For the mean function of the Gaussian process prior, we adopt μ(z) = z − 7.67 to reflect the monotonically increasing nature of x_HI. The value 7.67 corresponds to the midpoint of reionization derived from the Planck TTTEEE+lowE+lensing results (Planck Collaboration et al. 2020a)."

    The GP prior mean is centered on the reionization midpoint obtained from Planck TTTEEE+lowE+lensing — the very large-scale E-mode data the paper claims to exclude. Through the sigmoid Eq. (A1), this prior fixes the midpoint of the reconstructed x_HI(z) where data are sparse. The posterior in Eq. (A5) uses this prior, Eq. (1) carries it into the CMB (w/o lowE) fit, and Eq. (2) converts it into τ. Therefore the headline τ = 0.0552 in Eq. (3), and the subsequent BAO-tension and neutrino-mass claims, partially inherit the lowE information that the paper says it avoids. The paper does not quantify how much of the constraining power comes from this prior or test robustness to removing/replacing it.

full rationale

The core derivation is not fully circular: the x_HI(z) constraints are external astrophysical measurements, and the CMB likelihood explicitly excludes low-ell EE. However, the claimed independence from low-ell E-mode polarization is partially undermined by the GP prior mean. Appendix A sets μ(z)=z−7.67, with 7.67 explicitly taken from Planck TTTEEE+lowE+lensing. This prior enters the reionization-history posterior used in the joint likelihood, then propagates through the τ integral. Because the paper presents the result as 'independent of the large-scale E-mode polarization' and does not quantify the prior's influence, this is a genuine, load-bearing circular step. Self-citations to Kageura et al. 2025, Umeda et al. 2025a,b, and Nakane et al. 2024 are data papers with independent observational content, so the overall score is moderate rather than high. The numerical impact of the prior is unknown: the x_HI data at z=5–8 are strong, so the final τ may be mostly data-driven, but the paper does not demonstrate that the prior is negligible, especially at z≳9 where the data are sparse.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central τ measurement rests on a small set of fitting parameters—GP hyperparameters, ten x_HI values, and an imported lowE-based prior offset—plus astrophysical assumptions about helium reionization and the high-z neutral fraction. No new particles or forces are introduced.

free parameters (4)
  • GP hyperparameters s and r = s=0.83, r=0.076
    Fitted by maximizing the marginalized likelihood against the x_HI data; they control the amplitude and correlation length of the reconstructed reionization history.
  • GP prior mean offset 7.67 = 7.67
    Imported from Planck TTTEEE+lowE+lensing; centers the prior on a lowE-derived reionization midpoint and partially compromises the claimed independence.
  • x_HI(z1..z10) = posterior values
    Ten neutral-fraction values at log-spaced redshifts between z=5 and z=15 are direct fitting parameters in the CMB likelihood.
  • 7% systematic shift on x_HI = 7%
    Chosen from a comparison of standard versus extreme ionized-bubble models in Kageura et al. (2025) and applied coherently to all x_HI values.
assumptions (5)
  • domain assumption Gaussian process with RBF kernel and sigmoid transform adequately represents reionization history
    The smooth, monotonic x_HI(z) curve is a modeling choice not derived from first principles; it affects the integrated τ.
  • ad hoc to paper x_HI>0.999 at z=15 (1σ lower limit)
    Assumed to anchor the high-redshift neutral fraction and ensure monotonicity, following Elbers (2025).
  • domain assumption Helium reionization schedule η=1 for z>3.5 and η=2 for z≤3.5
    Taken from Planck 2020 and used in the τ integral; alters the electron budget.
  • domain assumption CMB likelihoods excluding low-ell EE are unbiased
    The analysis relies on CamSpec, Commander low-ell TT, and ACT lensing being free of systematics that would bias τ.
  • domain assumption The literature x_HI constraints are independent and not overfit by the GP
    The GP hyperparameters are fitted to the same x_HI data used in the cosmological inference, creating a potential overfitting loop.

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Cite this review

Pith. "Pith review of A New Constraint on the Optical Depth from the Reionization History Independent of CMB Large-Scale E-Mode Polarization." pith.science (2026). https://pith.science/paper/2PXPF3LS

@misc{pith2026260109644,
  author       = {Pith},
  title        = {Pith review of: A New Constraint on the Optical Depth from the Reionization History Independent of CMB Large-Scale E-Mode Polarization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PXPF3LS}},
  note         = {Machine review of arXiv:2601.09644}
}
abstract

Recent studies report a mild discrepancy between BAO and CMB measurements within the $\Lambda$CDM framework. This discrepancy could be explained if the optical depth $\tau$ inferred from the CMB large-scale E-mode polarization is underestimated, which may be biased by foreground-subtraction or instrumental systematics. In this work, we present a determination of $\tau$ independent of the large-scale E-mode polarization, using the latest measurements of the redshift evolution of the neutral hydrogen fraction $x_\mathrm{HI}(z)$, which is constrained by Lyman-$\alpha$ forest and damping-wing absorption measurements at $z\sim5$--$14$, based on ground-based optical and JWST observations. Combining $x_\mathrm{HI}(z)$ with the Planck CMB power spectra excluding the large-scale E-mode polarization, we obtain $\tau=0.0552^{+0.0019}_{-0.0026}(\mathrm{stat.})^{+0.0075}_{-0.0049}(\mathrm{sys.})$, where the systematic uncertainty accounts for possible absorption-modeling effects in the inference of $x_\mathrm{HI}(z)$. This constraint is consistent with previous CMB results including the large-scale E-mode polarization. With this measurement, we resolve the degeneracy in the $\tau$--$\Omega_{\rm m}$ plane and find a $2.4\sigma$ tension with the DESI DR2 BAO results, thereby confirming the claimed mild discrepancy suggestive of physics beyond $\Lambda$CDM. Finally, we derive an upper limit on the sum of neutrino masses, $\Sigma m_\nu < 0.0550\,(0.0717)\,{\rm eV}$ at the 95\%\,(99\%) confidence level. This limit favors the normal mass ordering and, when combined with the lower limits from neutrino oscillation experiments, yields a further constraint, $\Sigma m_\nu = 0.0594_{-0.0007}^{+0.0113}\,{\rm eV}$. However, the cosmological upper limit and the oscillation-based lower limit show a mild $2.2\sigma$ tension, providing an independent indication of possible physics beyond $\Lambda$CDM.

Figures

Figures reproduced from arXiv: 2601.09644 by the authors.

Figure 1
Figure 1. Cosmic reionization history reconstructed via Gaussian process regression. The black line represents the median reionization history, while the gray shaded regions show the 68% (dark gray) and 95% (light gray) posterior ranges. We also present xH i estimates based on various methods used in this work: Lyα damping-wing absorption of GRBs (blue squares; Totani et al. 2006, 2014; Fausey et al. 2025), QSOs (orange circl… view at source ↗
Figure 2
Figure 2. Constraints on cosmological parameters and τ for the ΛCDM model. The orange contours represent constraints derived from CMB Planck TTTEEE and Planck+ACT lensing and cosmic reionization history, excluding low-l EE. The blue and black contours indicate constraints obtained from CMB (w/o lowE) and CMB (w/ lowE), respectively. The dark and light contours show the 68% and 95% regions, respectively [PITH_FULL_IMAGE:figur… view at source ↗
Figure 3
Figure 3. Comparison of our τ values with those from previ￾ous studies. The orange circles show our τ values from CMB (w/o lowE) and CMB (w/o lowE)+xH i(z). The purple and blue points indicate τ values from WMAP (Spergel et al. 2003; Bennett et al. 2013) and Planck (Planck Collaboration et al. 2014, 2016b, 2020a), respectively. The cyan circles show results from ACT (Louis et al. 2025) and SPT (Cam￾phuis et al. 2025) combined… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Constraints on H0 and Ωm for the ΛCDM model. The blue, black, and orange contours show results based on CMB (w/o lowE), CMB (w/ lowE), and CMB (w/o lowE)+xH i(z), respectively. The green region for Ωm indicates constraints from DESI DR2 BAO measurements, while the gree…
Figure 5
Figure 5. Figure 5: Constraints on H0rd and Ωm for the ΛCDM model from DESI DR2 BAO (green), CMB (w/o lowE) (blue), and CMB (w/o lowE)+xH i(z) (orange). The result including reionization history in this work is consistent with the CMB (w/ lowE) constraint and is in 2.4σ tension with the D…
Figure 6
Figure 6. Figure 6: Constraints on σ8 and S8 for the ΛCDM model from HSC-Y3 (black), KiDS-Legacy (pink), DES Y3 (green), CMB (w/o lowE) (blue), and CMB (w/o lowE)+xH i(z) (orange). The S8 value obtained from CMB (w/o lowE)+xH i(z) is 2.7σ higher than the DES Y3 constraint, while remaining…
Figure 7
Figure 7. Figure 7: Marginalized posterior distributions of the (w0-wa) parameters from CMB (w/o lowE) (blue), +BAO (green), and +xH i(z) (orange). The dashed lines indicate w0 = −1 and wa = 0. The ΛCDM model is disfavored at 2.1σ significance by CMB (w/o lowE)+BAO+xH i(z), con￾sistent wi…
Figure 8
Figure 8. Figure 8: Marginalized posterior distributions of the (Σmν-τ ) parameters from CMB (w/o lowE) (blue), +BAO (green), and +xH i(z) (orange). A 95% (99%) upper limit of Σmν < 0.0550 (0.0717) eV is obtained from CMB (w/o lowE)+BAO+xH i(z). A similar upper limit of Σmν < 0.0649 (0.09…
Figure 9
Figure 9. Figure 9: Marginalized posterior distributions of Σmν. An upper limit of Σmν < 0.0550 (0.0717) eV (95% (99%) limit) is obtained from CMB (w/o lowE)+BAO+xH i(z) (orange). The blue (pink) line indicates the likelihood of Σmν from neutrino oscillation experiments for normal (invert…

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Forward citations

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