pith. sign in

arxiv: 2605.22934 · v1 · pith:6WEBHER2new · submitted 2026-05-21 · 🌌 astro-ph.CO

The τ of Neutral Hydrogen: Increased CMB Optical Depth at Long Wavelengths

Gilbert Holder , Adrian Liu , Tzu-Ching Chang , David Zegeye This is my paper

Pith reviewed 2026-05-25 05:40 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords fluctuationsfunctionwavelengthscosmicdepthneutralopticalradio
0
0 comments X

The pith

Neutral hydrogen absorption adds a frequency-dependent optical depth to the CMB at long radio wavelengths, providing a new probe of x_HI/T_s during the dark ages and reionization.

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

The cosmic microwave background is the leftover light from the Big Bang. At very long radio wavelengths, some of those photons can be absorbed by neutral hydrogen atoms in the distant universe instead of just scattering off electrons. This creates an extra dimming effect on top of the usual Thomson scattering. The amount of extra absorption changes with frequency and depends on how hot the hydrogen gas is (its spin temperature) and how much of it is still neutral at different times in cosmic history. The paper calculates that this extra optical depth reaches a few percent around 100 MHz. Because the effect varies with wavelength, measuring how much the CMB fluctuations are suppressed at different radio frequencies could reveal the combination of neutral fraction and spin temperature. The authors suggest that cross-correlating these long-wavelength maps with well-measured shorter-wavelength CMB maps might make detection easier than trying to see the 21 cm signal directly.

Core claim

At wavelengths longer than 21 cm, photons from the long-wavelength tail of the cosmic microwave background (CMB) have a non-zero probability of being absorbed by distant neutral hydrogen. This provides an additional suppression of the observed CMB clustering in addition to the usual Thomson scattering. The optical depth as a function of frequency is sensitive to the 21 cm spin temperature Ts of the gas as a function of cosmic time, with the excess optical depth peaking at a level of a few percent around 100 MHz.

Load-bearing premise

The heating history of the cosmic gas and the redshift evolution of the neutral fraction x_HI are such that the absorption probability remains non-zero and produces a measurable few-percent effect around 100 MHz (abstract).

Figures

Figures reproduced from arXiv: 2605.22934 by Adrian Liu, David Zegeye, Gilbert Holder, Tzu-Ching Chang.

Figure 1
Figure 1. Figure 1: — Optical depth to neutral hydrogen as a function of redshift for the evolution of the spin temperature shown in the inset. In the inset, the CMB temperature as a function of redshift is shown for reference as the dotted power-law. behavior is seen in the inset of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: — Angular power spectra for the 21 cm autospectra at z=50 (ν ∼ 28 MHz, red) and z=15 (ν ∼ 89 MHz, blue), compared to the CMB fluctuations (in the middle of the plot) and the differential CMB fluctuations at frequencies corresponding to 21 cm fluctuations at z=50 and z=15. For the 21 cm power spectra a fractional bandwidth of 1% was assumed. The maximum ℓ has only a small effect, due to the ℓ-shape of the s… view at source ↗
Figure 3
Figure 3. Figure 3: — Total signal-to-noise as a function of map noise for detecting either the CMB in cross-correlation (dashed curves) or the auto￾correlation of the 21 cm signal, assuming a full sky measurement. The dotted line shows a nominal threshold of 25 for detecting the CMB additional optical depth τHI in cross-correlation. The x-axis is the rms map noise in units expressing the noise in mK in a 1’x1’ pixel. For the… view at source ↗
Figure 4
Figure 4. Figure 4: — Forecasted errors as a function of frequency for a template-based amplitude measurement, assuming a 30 × 30 interferometric array consisting of 6 m dishes, observing for ∆t = 4000 hours, shown in bins of 12.8 MHz. The resulting sensitivities are within a reasonable range for detecting τ(ν) effects. where σ(u, v) 2 is the noise variance in a particular uv mode, assuming a diagonal noise covariance in harm… view at source ↗
read the original abstract

At wavelengths longer than 21 cm, photons from the long-wavelength tail of the cosmic microwave background (CMB) have a non-zero probability of being absorbed by distant neutral hydrogen. This provides an additional suppression of the observed CMB clustering in addition to the usual Thomson scattering. The optical depth as a function of frequency is sensitive to the 21 cm spin temperature $T_s$ of the gas as a function of cosmic time, with the excess optical depth peaking at a level of a few percent around 100 MHz. The details depend on the specifics of the heating of cosmic gas and the evolution of the neutral fraction $x_{HI}$. It is likely difficult to detect the CMB at these long radio wavelengths, but the cause is aided by the ability to cross-correlate with the already well-characterized fluctuations at cm/mm frequencies. We find that detecting CMB fluctuations at radio wavelengths corresponding to the 21 cm ``dark ages'' in cross-correlation with mm-wave maps may be easier than detecting the intrinsic 21 cm fluctuations. Measurement of the amplitude of CMB fluctuations as a function of radio wavelength provides a path for a new type of direct measurement of the combination $x_{HI}/T_s$ as a function of redshift.

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

2 major / 1 minor

Summary. The manuscript claims that resonant 21 cm absorption by neutral hydrogen adds a frequency-dependent optical depth to the CMB at wavelengths longer than 21 cm, producing an additional suppression of CMB anisotropies beyond Thomson scattering. This excess τ peaks at a few percent near 100 MHz and depends on the redshift evolution of x_HI and T_s; the authors propose cross-correlation with mm-wave CMB maps as a detection method and a new route to measuring x_HI/T_s(z).

Significance. If the few-percent excess optical depth is robust across standard heating models, the result would provide a new, direct probe of the dark ages and early heating. The cross-correlation suggestion is a practical strength, but the overall significance is tempered by the acknowledged sensitivity to heating history and the absence of explicit robustness tests in the provided abstract and skeptic analysis.

major comments (2)
  1. [Abstract] Abstract: the claim that the excess optical depth 'peaks at a level of a few percent' is not shown to be generic. The skeptic correctly notes that T_s rises sharply once Lyman-α or X-ray heating begins, narrowing the redshift window where x_HI ≈ 1 and T_s ≪ T_CMB; the manuscript must demonstrate that the integrated τ_21cm remains O(0.01) for a range of heating redshifts (e.g., z_heat = 12–20) rather than only for the specific histories that keep T_s low.
  2. [Abstract / §3 (assumed)] The central radiative-transfer derivation (standard τ_21cm ∝ x_HI / T_s integrated with A_10, ν_21, and Hubble factors) is invoked but not shown; without explicit evaluation of the line-of-sight integral for multiple models, it is impossible to verify whether the few-percent peak follows or is an artifact of the chosen heating history.
minor comments (1)
  1. [Abstract] The abstract states that 'the details depend on the specifics of the heating,' yet still asserts a definite 'few percent' peak; this tension should be resolved by showing the range of possible amplitudes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments both concern the need for greater explicitness and robustness checks in the presentation of the optical-depth calculation. We agree that these points improve the manuscript and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the excess optical depth 'peaks at a level of a few percent' is not shown to be generic. The skeptic correctly notes that T_s rises sharply once Lyman-α or X-ray heating begins, narrowing the redshift window where x_HI ≈ 1 and T_s ≪ T_CMB; the manuscript must demonstrate that the integrated τ_21cm remains O(0.01) for a range of heating redshifts (e.g., z_heat = 12–20) rather than only for the specific histories that keep T_s low.

    Authors: We agree that the peak value should be shown to be robust rather than dependent on a single heating history. In the revised manuscript we will add a new figure (or panel) that evaluates the frequency-dependent optical depth for a grid of heating redshifts spanning z_heat = 12–20, using standard Lyman-α and X-ray heating prescriptions consistent with existing 21 cm constraints. This will demonstrate that the integrated excess optical depth remains O(0.01) near 100 MHz across this range. revision: yes

  2. Referee: [Abstract / §3 (assumed)] The central radiative-transfer derivation (standard τ_21cm ∝ x_HI / T_s integrated with A_10, ν_21, and Hubble factors) is invoked but not shown; without explicit evaluation of the line-of-sight integral for multiple models, it is impossible to verify whether the few-percent peak follows or is an artifact of the chosen heating history.

    Authors: The underlying radiative-transfer expression is the standard 21 cm optical-depth integral, but we accept that an explicit derivation and numerical evaluation would strengthen the paper. We will insert a short dedicated subsection (new §2.2 or §3.1) that writes out the line-of-sight integral, states the adopted constants (A_10, ν_21, Hubble factor), and then shows the numerical result of that integral evaluated on the same set of heating histories used for the robustness test above. revision: yes

Circularity Check

0 steps flagged

No circularity: claim follows from standard 21 cm optical depth without reduction to fitted inputs or self-citations.

full rationale

The provided abstract and description contain no equations, fitting procedures, or derivations. The excess optical depth is stated to follow from the usual resonant absorption expression (proportional to x_HI/T_s) integrated over redshift, with the amplitude explicitly conditioned on external heating history and neutral fraction evolution. No self-citation chains, ansatzes smuggled via prior work, or predictions that reduce by construction to the paper's own inputs are present. The result is therefore self-contained against external benchmarks and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard 21 cm cosmology and CMB radiative transfer; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard cosmological model for CMB propagation and 21 cm spin temperature physics
    The optical depth calculation presupposes the usual radiative transfer and spin-temperature formalism used in 21 cm cosmology.

pith-pipeline@v0.9.0 · 5755 in / 1277 out tokens · 22499 ms · 2026-05-25T05:40:04.412760+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    2025, RAS Techniques and Instruments, 4, rzae061, doi: 10.1093/rasti/rzae061

    Artuc, K., & de Lera Acedo, E. 2025, RAS Techniques and Instruments, 4, rzae061, doi: 10.1093/rasti/rzae061

    work page doi:10.1093/rasti/rzae061 2025

  2. [2]

    D., Rogers, A

    Bowman, J. D., Rogers, A. E. E., Monsalve, R. A., Mozdzen, T. J., & Mahesh, N. 2018, Nature, 555, 67, doi: 10.1038/nature25792

    work page doi:10.1038/nature25792 2018

  3. [3]

    C., Ali-Ha¨ ımoud, Y., & Hirata, C

    Breysse, P. C., Ali-Ha¨ ımoud, Y., & Hirata, C. M. 2018, Phys. Rev. D, 98, 043520, doi: 10.1103/PhysRevD.98.043520

    work page doi:10.1103/physrevd.98.043520 2018

  4. [4]

    2025, RAS Techniques and Instruments, 4, rzaf046, doi: 10.1093/rasti/rzaf046

    Bull, P., El-Makadema, A., Garsden, H., et al. 2025, RAS Techniques and Instruments, 4, rzaf046, doi: 10.1093/rasti/rzaf046

    work page doi:10.1093/rasti/rzaf046 2025

  5. [5]

    H., DeBoer, D

    Bye, C. H., DeBoer, D. R., Dexter, M., et al. 2026, arXiv e-prints, arXiv:2602.02661, doi: 10.48550/arXiv.2602.02661

    work page doi:10.48550/arxiv.2602.02661 2026

  6. [6]

    C., Rogers, A

    Cappallo, R. C., Rogers, A. E. E., Lonsdale, C. J., et al. 2025, PASP, 137, 125002, doi: 10.1088/1538-3873/ae1bd6

    work page doi:10.1088/1538-3873/ae1bd6 2025

  7. [7]

    and Shibahashi , H

    Chapman, E., Abdalla, F. B., Harker, G., et al. 2012, MNRAS, 423, 2518, doi: 10.1111/j.1365-2966.2012.21065.x

    work page doi:10.1111/j.1365-2966.2012.21065.x 2012

  8. [8]

    B., Bobin, J., et al

    Chapman, E., Abdalla, F. B., Bobin, J., et al. 2013, MNRAS, 429, 165, doi: 10.1093/mnras/sts333

    work page doi:10.1093/mnras/sts333 2013

  9. [9]

    2021, Philosophical Transactions of the Royal Society of London Series A, 379, 20190566, doi: 10.1098/rsta.2019.0566 de Lera Acedo, E., de Villiers, D

    Chen, X., Yan, J., Deng, L., et al. 2021, Philosophical Transactions of the Royal Society of London Series A, 379, 20190566, doi: 10.1098/rsta.2019.0566 de Lera Acedo, E., de Villiers, D. I. L., Razavi-Ghods, N., et al. 2022, Nature Astronomy, 6, 984, doi: 10.1038/s41550-022-01709-9 de Oliveira-Costa, A., Tegmark, M., Gaensler, B. M., et al. 2008, MNRAS, ...

    work page doi:10.1098/rsta.2019.0566 2021

  10. [10]

    2017, MNRAS, 469, 4537, doi: 10.1093/mnras/stx1136

    Stovall, K. 2017, MNRAS, 469, 4537, doi: 10.1093/mnras/stx1136

    work page doi:10.1093/mnras/stx1136 2017

  11. [11]

    L., & Keating, B

    Fan, X., Carilli, C. L., & Keating, B. 2006, ARA&A, 44, 415, doi: 10.1146/annurev.astro.44.051905.092514

    work page doi:10.1146/annurev.astro.44.051905.092514 2006

  12. [12]

    2024, Philosophical Transactions of the Royal Society of London Series A, 382, 20230068, doi: 10.1098/rsta.2023.0068

    Fialkov, A., Gessey-Jones, T., & Dhandha, J. 2024, Philosophical Transactions of the Royal Society of London Series A, 382, 20230068, doi: 10.1098/rsta.2023.0068

    work page doi:10.1098/rsta.2023.0068 2024

  13. [13]

    2024, ApJ, 975, 222, doi: 10.3847/1538-4357/ad77cc

    Fronenberg, H., & Liu, A. 2024, ApJ, 975, 222, doi: 10.3847/1538-4357/ad77cc

    work page doi:10.3847/1538-4357/ad77cc 2024

  14. [14]

    R., Oh, S

    Furlanetto, S. R., Oh, S. P., & Briggs, F. H. 2006, Phys. Rep., 433, 181, doi: 10.1016/j.physrep.2006.08.002

    work page doi:10.1016/j.physrep.2006.08.002 2006

  15. [15]

    K., Ciardi, B., Mellema, G., & Zaroubi, S

    Ghara, R., Giri, S. K., Ciardi, B., Mellema, G., & Zaroubi, S. 2021, MNRAS, 503, 4551, doi: 10.1093/mnras/stab776

    work page doi:10.1093/mnras/stab776 2021

  16. [16]

    K., Mellema, G., et al

    Ghara, R., Giri, S. K., Mellema, G., et al. 2020, MNRAS, 493, 4728, doi: 10.1093/mnras/staa487

    work page doi:10.1093/mnras/staa487 2020

  17. [17]

    2025, A&A, 699, A109, doi: 10.1051/0004-6361/202554163

    Ghara, R., Zaroubi, S., Ciardi, B., et al. 2025, A&A, 699, A109, doi: 10.1051/0004-6361/202554163

    work page doi:10.1051/0004-6361/202554163 2025

  18. [18]

    M., Barry, N., et al

    Greig, B., Trott, C. M., Barry, N., et al. 2021a, MNRAS, 500, 5322, doi: 10.1093/mnras/staa3494

    work page doi:10.1093/mnras/staa3494

  19. [19]

    Greig, B., Mesinger, A., Koopmans, L. V. E., et al. 2021b, MNRAS, 501, 1, doi: 10.1093/mnras/staa3593 HERA Collaboration, Abdurashidova, Z., Aguirre, J. E., et al. 2022, ApJ, 925, 221, doi: 10.3847/1538-4357/ac1c78 HERA Collaboration, Abdurashidova, Z., Adams, T., et al. 2023, ApJ, 945, 124, doi: 10.3847/1538-4357/acaf50 —. 2026, ApJ, 998, 33, doi: 10.384...

    work page doi:10.1093/mnras/staa3593 2022

  20. [20]

    Hirata, C. M. 2006, MNRAS, 367, 259, doi: 10.1111/j.1365-2966.2005.09949.x

    work page doi:10.1111/j.1365-2966.2005.09949.x 2006

  21. [21]

    2002, ARA&A, 40, 171, doi: 10.1146/annurev.astro.40.060401.093926

    Hu, W., & Dodelson, S. 2002, ARA&A, 40, 171, doi: 10.1146/annurev.astro.40.060401.093926

    work page doi:10.1146/annurev.astro.40.060401.093926 2002

  22. [22]

    2025, MNRAS, 541, 687, doi: 10.1093/mnras/staf1007

    Kern, N. 2025, MNRAS, 541, 687, doi: 10.1093/mnras/staf1007

    work page doi:10.1093/mnras/staf1007 2025

  23. [23]

    1995, Phys

    Knox, L. 1995, Phys. Rev. D, 52, 4307, doi: 10.1103/PhysRevD.52.4307

    work page doi:10.1103/physrevd.52.4307 1995

  24. [24]

    2007, Phys

    Lewis, A., & Challinor, A. 2007, Phys. Rev. D, 76, 083005, doi: 10.1103/PhysRevD.76.083005

    work page doi:10.1103/physrevd.76.083005 2007

  25. [25]

    Liu, A., & Shaw, J. R. 2020, PASP, 132, 062001, doi: 10.1088/1538-3873/ab5bfd

    work page doi:10.1088/1538-3873/ab5bfd 2020

  26. [26]

    Loeb, A., & Furlanetto, S. R. 2013, The First Galaxies in the Universe

    work page 2013

  27. [27]

    MacKay, V., Xu, Z., Byrne, R., & Hewitt, J. N. 2026, ApJ, 999, 96, doi: 10.3847/1538-4357/ae382b

    work page doi:10.3847/1538-4357/ae382b 2026

  28. [28]

    2017, ApJ, 840, 39, doi: 10.3847/1538-4357/aa6af9

    Madau, P., & Fragos, T. 2017, ApJ, 840, 39, doi: 10.3847/1538-4357/aa6af9

    work page doi:10.3847/1538-4357/aa6af9 2017

  29. [29]

    2025, arXiv e-prints, arXiv:2509.11846, doi: 10.48550/arXiv.2509.11846

    McKay, L., Subrahmanyan, R., Chippendale, A., et al. 2025, arXiv e-prints, arXiv:2509.11846, doi: 10.48550/arXiv.2509.11846

    work page doi:10.48550/arxiv.2509.11846 2025

  30. [30]

    G., Ghosh, A., & Koopmans, L

    Mertens, F. G., Ghosh, A., & Koopmans, L. V. E. 2018, MNRAS, 478, 3640, doi: 10.1093/mnras/sty1207

    work page doi:10.1093/mnras/sty1207 2018

  31. [31]

    R., & Sun, G

    Mirocha, J., Furlanetto, S. R., & Sun, G. 2017, MNRAS, 464, 1365, doi: 10.1093/mnras/stw2412

    work page doi:10.1093/mnras/stw2412 2017

  32. [32]

    A., Altamirano, C., Bidula, V., et al

    Monsalve, R. A., Altamirano, C., Bidula, V., et al. 2024, MNRAS, 530, 4125, doi: 10.1093/mnras/stae1138

    work page doi:10.1093/mnras/stae1138 2024

  33. [33]

    F., & Wyithe, J

    Morales, M. F., & Wyithe, J. S. B. 2010, ARA&A, 48, 127, doi: 10.1146/annurev-astro-081309-130936

    work page doi:10.1146/annurev-astro-081309-130936 2010

  34. [34]

    G., & Trott, C

    Murray, S. G., & Trott, C. M. 2018, ApJ, 869, 25, doi: 10.3847/1538-4357/aaebfa

    work page doi:10.3847/1538-4357/aaebfa 2018

  35. [35]

    2006, MNRAS, 373, L98, doi: 10.1111/j.1745-3933.2006.00251.x

    Naoz, S., Noter, S., & Barkana, R. 2006, MNRAS, 373, L98, doi: 10.1111/j.1745-3933.2006.00251.x

    work page doi:10.1111/j.1745-3933.2006.00251.x 2006

  36. [36]

    D., Null, D., Trott, C

    Nunhokee, C. D., Null, D., Trott, C. M., et al. 2025, ApJ, 989, 57, doi: 10.3847/1538-4357/adda45

    work page doi:10.3847/1538-4357/adda45 2025

  37. [37]

    A., & Wilson, R

    Penzias, A. A., & Wilson, R. W. 1965, ApJ, 142, 419, doi: 10.1086/148307

    work page doi:10.1086/148307 1965

  38. [38]

    C., et al

    Philip, L., Abdurashidova, Z., Chiang, H. C., et al. 2019, Journal of Astronomical Instrumentation, 8, 1950004, doi: 10.1142/S2251171719500041 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910

    work page doi:10.1142/s2251171719500041 2019

  39. [39]

    C., Greenhill, L

    Price, D. C., Greenhill, L. J., Fialkov, A., et al. 2018, MNRAS, 478, 4193, doi: 10.1093/mnras/sty1244

    work page doi:10.1093/mnras/sty1244 2018

  40. [40]

    R., & Loeb, A

    Pritchard, J. R., & Loeb, A. 2012, Reports on Progress in Physics, 75, 086901, doi: 10.1088/0034-4885/75/8/086901 Sathyanarayana Rao, M., Singh, S., K. S., S., et al. 2023, Experimental Astronomy, 56, 741, doi: 10.1007/s10686-023-09909-5

    work page doi:10.1088/0034-4885/75/8/086901 2012

  41. [41]

    T., Subrahmanyan, R., et al

    Singh, S., Jishnu, N. T., Subrahmanyan, R., et al. 2022, Nature Astronomy, 6, 607, doi: 10.1038/s41550-022-01610-5

    work page doi:10.1038/s41550-022-01610-5 2022

  42. [42]

    2009, ApJ, 694, 879, doi: 10.1088/0004-637X/694/2/879

    Trenti, M., & Stiavelli, M. 2009, ApJ, 694, 879, doi: 10.1088/0004-637X/694/2/879

    work page doi:10.1088/0004-637x/694/2/879 2009

  43. [43]

    2025, Phys

    Zegeye, D., Crawford, T., Chluba, J., Remazeilles, M., & Grainge, K. 2025, Phys. Rev. D, 111, 063517, doi: 10.1103/PhysRevD.111.063517

    work page doi:10.1103/physrevd.111.063517 2025

  44. [44]

    S., et al

    Zheng, H., Tegmark, M., Dillon, J. S., et al. 2017, MNRAS, 464, 3486, doi: 10.1093/mnras/stw2525 This paper was built using the Open Journal of Astrophysics LATEX template. The OJA is a journal which provides 10 fast and easy peer review for new papers in theastro-phsection of the arXiv, making the reviewing process simpler for authors and referees alike....

    work page doi:10.1093/mnras/stw2525 2017