Constraining the Hubble parameter with the 21 cm brightness temperature signal in a universe with inhomogeneities

A. S. Majumdar; Shashank Shekhar Pandey; Subhadeep Mukherjee

REVIEW 4 major objections 4 minor 1 cited by

Constraining the Hubble parameter with the 21 cm brightness temperature signal in a universe with inhomogeneities

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a backreaction-averaged universe makes the 21 cm absorption trough about 40 mK deeper for H0=67 than for H0=73, a difference that does not appear in ΛCDM.

desk verdict A transparent but premature forward-modeling paper: the 21 cm H0 sensitivity is real in the model, but the observable mapping is unproven and the post hoc scaling choice carries the whole claim. read the letter →

arxiv 2505.22219 v2 pith:SBO4JVK7 submitted 2025-05-28 astro-ph.CO gr-qc

Subhadeep Mukherjee , Shashank Shekhar Pandey , A. S. Majumdar This is my paper

classification astro-ph.COgr-qc

keywords 21cmcosmologyHubbletensionbackreactioncosmologicalaveragingbrightnesstemperaturecosmicdawnstructureformationsupernovaconstraints

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 the depth of the global 21 cm absorption trough can act as a probe of the Hubble tension, provided the universe's matter distribution is treated with a backreaction averaging scheme rather than as a homogeneous fluid. The authors model the cosmos as overdense and underdense regions with separate expansion histories, calibrate the effective matter density with supernova distance data, and choose the scaling exponent that reproduces the volume fraction of overdense regions in N-body structure-formation simulations. With these calibrated parameters, a low $H_0=67\ \mathrm{km/s/Mpc}$ value produces a trough of roughly $-320\ \mathrm{mK}$ at $z\approx18$, while a high $H_0=73\ \mathrm{km/s/Mpc}$ value gives roughly $-280\ \mathrm{mK}$. Standard $\Lambda$CDM changes the signal by only a few mK between the same two values, so the paper presents this gap as an observational signature of inhomogeneous backreaction. If the claim holds, a precise measurement of the cosmic dawn spectrum could weigh in on whether the Hubble tension comes from hidden systematics, new physics, or the way cosmic voids and clusters alter the averaged expansion.

What carries the argument

The load-bearing object is the backreaction-averaged Hubble parameter $H_D(z)$ of a domain divided into overdense and underdense subdomains, each with its own scale factor and expansion rate. Backreaction here is the averaged effect of differences in local expansion and shear, quantified by the kinematical backreaction term $Q_D$, and curvature is tracked by the averaged Ricci scalar $\langle R\rangle_D$. The system is closed by power-law ansatze $Q_F\propto a_F^{n}$ and $\langle R\rangle_F\propto a_F^{n}$ linked through the integrability condition; the choice $n_1=-1.7$ for the underdense region and $n_2=-2$ for the overdense region is selected because it matches the overdense volume fraction from N-body simulations. This $H_D(z)$ enters the 21 cm optical depth as $\tau\propto 1/H_D(z)$, so a suppressed expansion history directly deepens the absorption feature.

What would settle it

Run the same two-domain backreaction model through a full 21 cm light-cone radiative-transfer code that computes the optical depth region by region from local gas density, spin temperature, and peculiar velocity gradients; if the predicted gap of about 40 mK between $H_0=67$ and $H_0=73$ disappears or reverses, the central claim is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the global 21 cm brightness temperature at $15\lesssim z\lesssim30$ becomes a sensitive function of the present Hubble rate once the averaged expansion of an inhomogeneous universe is used in place of the standard homogeneous Hubble parameter. In the calibrated two-domain model, the overdense region is assigned a Friedmann-like scaling $n_2=-2$ with vanishing backreaction, while the underdense region uses $n_1=-1.7$ for the simultaneous scaling of backreaction and curvature; this choice reproduces the simulated growth of the overdense volume fraction. The best-fit effective matter density from the supernova fit is $\Omega_{D0}^m = 0.104\pm0.056$. Because the 21 cm optical depth scales as $\tau\propto 1/H_D(z)$, the lower expansion history associated with $H_0=67$ deepens the absorption trough to about $-320$ mK at $z\approx18$, compared with about $-280$ mK for $H_0=73$, and shifts the minimum by $\Delta z\sim3$. The paper emphasizes that this $H_0$ dependence is absent in $\Lambda$CDM, where the same parameter change shifts $T_{21}$ by a few mK at most, and that a three-domain extension with an intermediate-density ambient region modifies the trough's depth and position but preserves the $H_0$ dependence.

Load-bearing premise

The calculation assumes that substituting the averaged expansion rate of the backreaction model into the standard homogeneous 21 cm brightness-temperature formula gives the true global signal, even though the real signal is built from local gas densities and radiative transfer through dense and empty regions.

Editorial extensions

If this is right

A global 21 cm spectrum covering $z\simeq15$–$30$ can in principle distinguish low from high $H_0$ values in this model, because the predicted trough is about 40 mK deeper and shifted by $\Delta z\sim3$ for the lower value.
The absence of this gap in standard $\Lambda$CDM means a measured $H_0$-dependent trough would provide the claimed observational signature that averaged inhomogeneities, not only dark energy, shape the background expansion at cosmic dawn.
The calibrated model prefers an effective matter density near $\Omega_{D0}^m\simeq0.104$, noticeably below the standard value; if the model is right, low-redshift matter-density probes should converge to similarly low values.
Because the three-domain extension preserves the qualitative result, the signature is robust to at least one natural generalization of the domain partitioning.
The predicted trough position lies in the band targeted by global 21 cm experiments, so forthcoming observations can confront the model without needing high angular resolution.

Reading between the lines

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

Editorial inference: the same averaged $H_D(z)$ that deepens the 21 cm trough also changes angular-diameter distances, so a joint fit of this class of models to supernovae, baryon acoustic oscillations, and the 21 cm spectrum could test whether one backreaction history resolves the Hubble tension across all probes simultaneously.
Editorial inference: the quantitative depths are computed by substituting the averaged expansion into homogeneous formulas; a light-cone radiative-transfer calculation using local gas densities and spin temperatures inside voids and clusters would show whether the 40 mK gap survives line-of-sight averaging, and is a natural next step.
Editorial inference: if future experiments detect a deep trough at $z\approx18$ with the predicted $H_0$ dependence, competing explanations such as baryon–dark matter cooling or an excess radio background would still need to be excluded, since they can also deepen the 21 cm absorption feature.
Editorial inference: the preferred scaling exponent $n_1=-1.7$ for underdense regions could be calibrated further with void statistics and cosmic shear, providing an independent test of the model that does not rely on 21 cm data.

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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,

The genuinely new thing here is the application of Buchert backreaction scaling laws to the 21 cm brightness temperature. That combination is not in the cited literature, and the claimed H0 sensitivity of the trough—deeper for H0=67 than for H0=73, with a redshift shift—is a concrete, falsifiable-sounding prediction. The paper is also transparent about its machinery: the MCMC calibration to Union 2.1, the comparison to N-body volume fractions, and the three-domain robustness check are all shown. Credit where due: the authors are not hiding the ball, and the robustness appendix is a good-faith attempt.

But the central quantitative claim does not hold up as stated. The 21 cm brightness temperature in Eq. (1)-(2) is a homogeneous formula. They substitute the volume-averaged Buchert expansion HD(z) into that formula and call it the global signal. In a universe with order-unity density contrasts, the sky-averaged 21 cm signal is a line-of-sight average over regions with very different nHI, Ts, and velocity-gradient terms. Their own premise—that the underdense region dominates the volume at 91%—makes the proxy especially suspect, since the optical depth is weighted by nHI, which lives mostly in overdense regions. The peculiar-velocity gradient term is dropped because it is "local," but if the model's whole point is that density contrasts are large, that term can be of order H in overdense regions. Without a derivation of the averaged observable analogous to the covariant distance relations they use for SNe, the ~40 mK difference is not established.

The second problem is the post hoc selection of the scaling exponent n1. They scan n1 from -2 to -1, fit Omega_m from SNe for each, then pick n1=-1.7 because it matches the N-body volume fraction. That is a fair way to calibrate a model, but it means the output is not a parameter-free prediction. And the curves in Fig. 3 have no error bars from the MCMC posteriors, so even internally we cannot see whether the 40 mK difference is significant.

Is the paper worthless? No. The mechanism is physically sane: lower H gives higher optical depth, and the H0-dependence is genuinely absent in standard LCDM. If someone later does the proper inhomogeneous line-of-sight treatment, this paper would be the obvious reference for the idea. But as submitted, the headline result is unsupported.

Verdict: I would accept it for peer review—it is a serious, if flawed, attempt at a new observable—but I would expect the referees to require either a proper averaging derivation or a much more hedged claim. The paper is for readers working on backreaction and 21 cm theory; it will not convince a skeptic without the missing derivation.

Recommendation: send it to referees, but make sure at least one referee is deeply familiar with both Buchert averaging and 21 cm radiative transfer.

Referee Report

4 major / 4 minor

Summary. The paper constructs a two-domain Buchert backreaction model (overdense/underdense regions) with power-law scaling solutions for backreaction and curvature, calibrates the effective matter density parameter Omega_m^D0 against Union 2.1 supernovae via MCMC, and selects the scaling exponent n1 by comparing the model's overdense volume fraction with N-body simulation data. The authors then compute the global 21 cm brightness temperature T21(z) by inserting the volume-averaged Hubble parameter H_D(z) into the standard homogeneous brightness-temperature formula, obtaining a deeper absorption trough for H0 = 67 km/s/Mpc (about -320 mK at z ~ 18) than for H0 = 73 km/s/Mpc (about -280 mK), and claim that this H0 dependence is a clear observational signature of backreaction that is absent in Lambda CDM.

Significance. If the central claim were established, the 21 cm absorption trough would be a novel probe of the Hubble tension and of the backreaction of cosmic inhomogeneities. The paper has some genuine strengths: the MCMC calibration to supernova data is concrete, the comparison with N-body volume fractions gives a physical anchor to the scaling exponents, and the three-domain extension in Appendix A is a useful robustness check. However, the key quantitative result -- the ~40 mK difference between the two H0 values in Fig. 3 -- rests on substituting the averaged Hubble parameter into a local homogeneous observable without deriving the appropriate averaged 21 cm signal, and it is presented without any propagated uncertainty from the poorly constrained Omega_m^D0 and the post hoc choice of n1. These issues are load-bearing for the paper's central claim.

major comments (4)

[Sec. II A, Sec. V, Eqs. (1)-(2), Fig. 3] The global 21 cm brightness temperature is computed by substituting the volume-averaged Hubble parameter H_D(z) into the homogeneous formula for T21 and the optical depth tau, but the paper does not derive this as the correct sky-averaged observable in a universe with order-unity density contrasts. The optical depth depends nonlinearly on the local neutral hydrogen density nHI, the local spin temperature Ts, and the local velocity gradient H(z) + (1+z) delta_r v_r. The paper drops the peculiar-velocity term in Eq. (2) as 'local,' yet in overdense regions, which dominate the 21 cm absorption because nHI is much larger there, this term is typically of order H and cannot be neglected. Since <nHI/H> is not equal to <nHI>/<H>, the substitution H -> H_D(z) in Eqs. (1)-(2) is not justified, and the predicted ~40 mK difference between H0 = 67 and H0 = 73 in Fig. 3 is not established.
[Sec. V, Table I and Fig. 3] The central prediction in Fig. 3 is presented as a deterministic curve with no uncertainty band. The best-fit Omega_m^D0 = 0.104 ± 0.056 from Table I has a relative error of roughly 50%, and the scaling exponent n1 = -1.7 is selected post hoc by visual comparison to N-body volume fractions without any quantitative goodness-of-fit criterion. No error propagation from Omega_m^D0 or n1 into T21 is attempted, so the claim that the trough depth and position constitute a 'clear observational signature' is not supported by the stated uncertainties.
[Sec. II A - IV, Eqs. (2)-(6)] The calculation does not specify how the baryon number density nH entering Eqs. (2), (5), and (6) is defined in the inhomogeneous Buchert framework. In the two-domain model, the domain-averaged matter density <rho>_D differs from the local densities in the overdense and underdense regions, and the 21 cm optical depth depends on the local neutral hydrogen density, not the volume average. Without an explicit mapping between nH and the averaged model quantities, the T21(z) curves in Figs. 1, 3, and 4 are not well defined, and the comparison with the standard homogeneous treatment is ambiguous.
[Sec. III and Sec. V] The model's 'consistency with structure formation' is achieved by choosing the scaling exponent n1 after inspecting the N-body volume fractions; n1 is thus effectively a calibrated free parameter rather than a prediction. The four values in Table I are used to demonstrate that only n1 = -1.7 'matches consistently' with the simulation data, but no statistical measure of this match is given, and no test of how the final T21 prediction depends on the allowed range of n1 is presented. This weakens the claim that the model is jointly calibrated by supernova and structure-formation data.

minor comments (4)

[Sec. IV, paragraph after Fig. 1] The sentence 'the domination is dictated by the tau part, viz., tau ∝ e^{-H} from (Eq. 1)' is dimensionally and functionally incorrect; the intended scaling appears to be tau ∝ 1/H_D(z), as stated later in Sec. V.
[Sec. V, Table I] The table gives errors on Omega_m^D0 as superscript/subscript asymmetric values, but the text does not state whether these denote 68% or 95% credible intervals; this should be specified.
[Sec. V, Fig. 2(b)] The N-body data points for the overdense volume fraction are described only qualitatively as extracted with a block separation technique; without error bars or a precise definition of the grid and counting procedure, the visual agreement claimed for n1 = -1.7 is difficult to assess.
[Appendix A] The choice n3 = -1.8 for the ambient region is described as 'reasonable' but no motivation from simulations or observations is given; a sensitivity test over the full range -2 ≤ n3 ≤ -1.7 would strengthen the robustness claim in Fig. 4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the H0-dependence of the 21 cm trough is a derived consequence of tau proportional to 1/H_D, not a fit to 21 cm data.

full rationale

The paper's central claim is a derived prediction rather than a circular restatement of its inputs. T21 is computed from the standard expressions in Eqs. (1)-(2), with the Buchert-averaged Hubble parameter H_D(z) obtained by solving the closed system of Eqs. (28)-(35). The model parameters are calibrated to external data: Omega_m^D0 is fit to the Union 2.1 SNe sample via MCMC, and the scaling exponent n1 is chosen to match the volume fraction evolution extracted from N-body simulation data. The two Hubble constant values h=0.67 and h=0.73 are external inputs, not parameters fitted to any 21 cm measurement. The deeper trough at lower h follows algebraically from tau proportional to 1/H_D(z) in Eq. (2), so the H0-dependence is a genuine consequence of the model equations. No equation is defined in terms of its own output, and no fitted parameter is renamed as a prediction. The self-citations present in the paper (e.g., refs. [68]-[72], [77], [79]) appear in the literature survey or as forward-looking suggestions, not as load-bearing support for the main derivation; the scaling laws and N-body benchmarks come from external work. The main caveat, identified by the skeptic, is that substituting the volume-averaged H_D(z) into the homogeneous 21 cm brightness-temperature formula is an assumption that has not been derived as the actual line-of-sight averaged observable in a clumpy universe. That is a modeling and correctness concern, not a circularity, because the prediction does not reduce to its input by construction.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The model introduces no new particles, forces, or dimensions. The backreaction Q_D is a standard term within Buchert's averaging formalism. The main load-bearing inputs are the closure scaling laws and the assumption that the homogeneous T21 formula can be used with the averaged H_D.

free parameters (8)

n1 (underdense scaling exponent) = -1.7 (selected from grid -1, -1.3, -1.7, -2)
Scaling exponent for underdense backreaction and curvature. Chosen to match N-body volume fraction after SNe calibrations for other values; central to the deep trough.
n2 (overdense scaling exponent) = -2
Set to Friedmann-like curvature scaling by hand, making overdense backreaction vanish.
Omega_m^D0 (effective matter density) = 0.104 ± 0.056 for n1=-1.7
MCMC fit to Union 2.1 SNe; low value drives the deep 21 cm absorption.
lambda_M0 (present overdense volume fraction) = 0.09
Assumed from the Wiegand-Buchert two-domain model; not fit to data.
lambda_Mi (initial overdense volume fraction) = 0.5
Assumed from Gaussian early universe; used to set initial conditions for both regions.
aD0 (domain scale factor normalization) = 1000
Gauge choice; only ratios of aD enter observables, so it has no physical effect.
r (Ly-alpha injected-to-continuum intensity ratio) = 0.1
Astrophysical input for Pop-II stars; affects the thermal coupling and hence the trough depth.
n3 (ambient scaling exponent in three-domain model) = -1.8
Hand-picked in the appendix for the robustness check; not constrained by data.

assumptions (6)

domain assumption Buchert averaging applies to the real universe as a pressureless dust model with flow-orthogonal hypersurfaces, zero vorticity, and a compact domain.
Used throughout Sec. II B to derive HD, QD, and the evolution equations.
ad hoc to paper Backreaction and curvature obey scaling laws Q_F ∝ a_F^n and <R>_F ∝ a_F^n, with n=p from the integrability condition.
Imposed in Sec. III (Eqs. 28-30) to close the system; the chosen n values are not derived from first principles.
domain assumption All regions share the same foliation time with no lapse, so t_M0 = t_E0.
Assumed after Eq. (35) to eliminate Omega_RQ; ignores possible time lapses between regions.
ad hoc to paper The standard homogeneous 21 cm brightness temperature formula is valid after substituting H_D(z) for H(z).
Used to produce Fig. 3; this is the main unverified bridge between the averaged background and the observable.
domain assumption Early universe density fluctuations are Gaussian with lambda_Mi ≈ 0.5 and equal densities across regions at z ~ 1000.
Assumed after Eq. (31) to reduce the parameter space and set initial conditions.
domain assumption Union 2.1 SNe distance moduli are reliable and the likelihood used in the MCMC is adequate.
Used for calibration of Omega_m^D0; no goodness-of-fit or residual analysis is shown.

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Cite this review

Pith. "Pith review of Constraining the Hubble parameter with the 21 cm brightness temperature signal in a universe with inhomogeneities." pith.science (2026). https://pith.science/paper/SBO4JVK7

@misc{pith2026250522219,
  author       = {Pith},
  title        = {Pith review of: Constraining the Hubble parameter with the 21 cm brightness temperature signal in a universe with inhomogeneities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SBO4JVK7}},
  note         = {Machine review of arXiv:2505.22219}
}

abstract

We consider the 21\,cm brightness temperature as a probe of the Hubble tension in the framework of an inhomogeneous cosmological model. Employing Buchert's averaging formalism to study the effect of inhomogeneities on the background evolution, we consider scaling laws for the backreaction and curvature consistent with structure formation simulations. We calibrate the effective matter density using MCMC analysis using Union 2.1 Supernova Ia data. Our results show that a higher Hubble constant ($\sim73$\,km/s/Mpc) leads to a shallower absorption feature in the brightness temperature versus redshift curve. On the other hand, a lower value ($\sim67$\,km/s/Mpc) produces a remarkable dip in the brightness temperature $T_{21}$. Such a substantial difference is absent in the standard $\Lambda$CDM model. Our findings indicate that inhomogeneities could significantly affect the 21\,cm signal, and may shed further light on the different measurements of the Hubble constant.

Figures

Figures reproduced from arXiv: 2505.22219 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. 21 cm brightness temperature [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. The 21-cm brightness temperature [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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