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Constraints on the state of the IGM at $z\sim 8-10$ using redshifted 21-cm observations with LOFAR

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Pith's one-line read With 140 hours of LOFAR data at redshifts 8.3–10.1, only extreme reionization histories — sparse, bright sources carving rare large ionized or heated regions — are disfavoured, leaving ordinary reionization models untouched.

desk verdict First interpretation of the new LOFAR multi-redshift upper limits, but the headline IGM bounds are mostly prior-driven, as the paper's own Appendix A control shows. read the letter →

arxiv 2505.00373 v1 pith:GSGCAO33 submitted 2025-05-01 astro-ph.CO

classification astro-ph.CO
keywords epochofreionization21-cmpowerspectrumLOFARintergalacticmediumBayesianinferenceradiativetransferexcessradiobackgroundIGMtemperatureandionizationconstraints
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 asks what the intergalactic medium (IGM) looked like at $z \sim 8$–$10$, using LOFAR's newest 140-hour upper limits on the 21-cm power spectrum at $z = 8.3$, $9.1$ and $10.1$. The authors build a database of $10^5$ reionization simulations with the one-dimensional radiative-transfer code GRIZZLY, explore five source parameters with a Bayesian exclusion likelihood, and report which IGM states are disfavoured rather than favoured. Their central finding is that only extreme reionization histories — a sparse population of bright sources that carve out rare, large ionized or heated regions — are excluded, and that a completely neutral and unheated IGM is still consistent with the data. In a model with an excess radio background, the 95 (68) per cent disfavoured credible intervals at $z = 9.1$ correspond to ionized and heated fractions below $0.46$ ($\lesssim 0.05$), gas temperatures below $44$ ($4$) K, and heated-region sizes below $14$ ($3$) $h^{-1}\,\mathrm{Mpc}$. The upshot is that the strongest current 21-cm measurement still cannot tell ordinary reionization apart from exotic scenarios, but the same machinery is ready to turn deeper limits into genuine constraints on the first billion years.

What carries the argument

The carrying mechanism is the pairing of a simulation database with an exclusion likelihood. GRIZZLY, a one-dimensional radiative-transfer code, converts $N$-body halo catalogs into 21-cm brightness-temperature cubes; here it is run for $10^5$ combinations of five source parameters — ionization efficiency $\zeta$, minimum UV-halo mass $M_{\rm min}$, minimum X-ray-halo mass $M_{\rm min,X}$, X-ray heating efficiency $f_X$, and excess-radio-background efficiency $A_r$ — building an interpolated grid of power spectra and derived IGM quantities. A Markov chain then evaluates, at every step, the likelihood that a model is excluded by the LOFAR limits, formed from the product of error functions comparing the observed upper limits with the interpolated model spectrum at the three largest observed scales, with a conservative 30 per cent modelling error added in quadrature and a Thomson-scattering optical-depth prior capping $x_{\rm HII}$. The reported IGM parameters ($x_{\rm HII}$, $T_{\rm K}$, $\delta T_b$, $f_{\rm heat}$, $R_{\rm heat}^{\rm peak}$, $\Delta R_{\rm heat}^{\rm FWHM}$) are read off the same interpolated database, which is why the IGM constraints inherit the source-prior dependence that Appendix A exposes.

What would settle it

Re-run the $10^5$-model database with a full three-dimensional radiative-transfer code on the same five-parameter grid and repeat the MCMC: if the model power spectra shift by more than the assumed 30 per cent modelling error at $k \approx 0.076$–$0.133\,h\,\mathrm{Mpc}^{-1}$ in any region of parameter space, the disfavoured intervals move. Observationally, the paper's own Appendix C provides a sharper test: the bias-corrected upper limit of $(25\,\mathrm{mK})^2$ at $k = 0.075\,h\,\mathrm{Mpc}^{-1}$, $z = 9.1$, already lies below the power spectrum of a completely neutral, unheated IGM, so independently verifying that bias correction would immediately remove the paper's last surviving extreme-but-allowed scenario.

Watch

Extended reading notes

Core claim

The paper claims that the LOFAR upper limits of Mertens et al. (2025) at $z = 8.3$, $9.1$ and $10.1$ exclude a well-defined but narrow set of intergalactic-medium states, namely the extreme end of the model space. In the model with a free excess radio background, the disfavoured models at $z = 9.1$ have, at 68 (95) per cent credibility, volume-averaged ionized fraction $x_{\rm HII} \lesssim 0.02$ ($0.46$), neutral-gas temperature $T_{\rm K} \lesssim 4.4$ ($44$) K, heated-region volume fraction $f_{\rm heat} \lesssim 0.05$ ($0.46$), and characteristic heated-region size $R_{\rm heat}^{\rm peak} \lesssim 3$ ($14$) $h^{-1}\,\mathrm{Mpc}$. The 68 per cent interval on the disfavoured models sits at an excess-radio-background efficiency $A_r \gtrsim 4.6$, i.e. an excess background more than 100 per cent of the CMB at 1.42 GHz, while the 95 per cent interval on $A_r$ spans the entire prior range. A completely neutral and unheated IGM at $z \approx 9$ remains consistent with the data. The authors present these numbers as probabilistic statements about models disfavoured by upper limits rather than detections, and they emphasize that the intervals inherit a strong dependence on the chosen source-parameter priors.

Load-bearing premise

The entire exclusion likelihood is computed from power spectra produced by GRIZZLY's one-dimensional radiative transfer, and the paper itself notes that the code's agreement with a full three-dimensional scheme is verified only for scenarios with gas much hotter than the CMB, not for models with significant spin-temperature fluctuations, so any systematic error there would shift the quoted disfavoured IGM states.

Editorial extensions

If this is right

  • The completely neutral, unheated IGM at $z \approx 9$ remains consistent with the data; the paper estimates that 2–3 times more integration would be needed before LOFAR rules even that baseline scenario out.
  • The excluded models are extreme patchy-reionization and patchy-heating scenarios in which rare sources (minimum halo masses $\gtrsim 10^{10}\,M_\odot$) with large efficiencies create high-contrast, large ionized or heated regions; mainstream reionization histories are untouched.
  • An excess radio background of 100–207 per cent of the CMB at 1.42 GHz is disfavoured at 68 per cent credibility across $z = 8.3$–$10.1$, but the 95 per cent interval spans the whole prior range, so the radio-background question stays open.
  • Including other interferometers' upper limits — dominated by HERA at $z \approx 8$ and $10$ — substantially enlarges the disfavoured region, with HERA alone excluding a neutral, unheated IGM at $z \approx 8$ at $2\sigma$; LOFAR still dominates at $z \approx 9$.
  • The joint-redshift analysis gives a consistent redshift evolution of the disfavoured states ($x_{\rm HII} < 0.85$, $0.46$, $0.17$ at $z = 8.3$, $9.1$, $10.1$ at 95 per cent credibility), although it assumes redshift-independent source parameters and is dominated by the strongest limit.

Reading between the lines

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

  • The paper's own Appendix A control run — replacing the LOFAR likelihood with a constant — leaves the posteriors of $x_{\rm HII}$, $f_{\rm heat}$ and $T_{\rm K}$ nearly unchanged; a fair reading is that those three headline bounds are largely inherited from the source-parameter priors, while the information actually coming from the data sits in $\delta T_b$ and the heated-region size statistics.
  • Because the 95 per cent credible interval of the radio-background efficiency spans the full prior, the current data cannot separate a CMB-only sky from an excess-background sky; the paper cites the disputed ARCADE2 and LWA1 excess as the motivation for $A_r$, so a direct measurement of the sky-averaged radio temperature at tens of MHz would break this degeneracy more cheaply than additional 21-cm
  • The sharpest discriminating scale is $k \approx 0.076$–$0.133\,h\,\mathrm{Mpc}^{-1}$; if LOFAR reaches the factor-of-two sensitivity gain that the paper says is required to test the neutral-unheated baseline, the qualitative 'extreme models only' conclusion should harden into a quantitative floor on the global ionized fraction, a prediction checkable with the next round of limits.
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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

2 major / 5 minor

Summary. This paper uses the recent LOFAR upper limits on the 21-cm power spectrum at z = 8.3, 9.1 and 10.1 (Mertens et al. 2025) together with GRIZZLY simulations and a Bayesian MCMC framework to identify reionization models that are disfavoured by the data. The authors explore two source-model scenarios, one with no excess radio background (Ar = 0) and one with a variable excess radio background (Varying Ar), using both single-redshift and joint-redshift likelihoods. They derive credible intervals for the IGM parameters of the disfavoured models, with headline numbers at z = 9.1 in the Varying Ar case: xHII ≲ 0.02 (0.46), TK ≲ 4.4 (44) K, and fheat ≲ 0.05 (0.46) at 68 (95) per cent. The paper also repeats the analysis including upper limits from other interferometers and, in an appendix, applies the framework to the stronger upper limits of Acharya et al. (2024c).

Significance. If the quoted IGM intervals were genuinely set by the LOFAR upper limits, this would be an important step in using 21-cm observations to characterize the IGM during reionization. The paper is commendably transparent about its methodology: it uses a public MCMC sampler, provides an appendix that tests the effect of removing the LOFAR likelihood, and explicitly warns that the IGM constraints may be affected by source-parameter priors. However, the no-LOFAR control in Appendix A shows that for the Ar = 0 scenario the tight low-xHII and low-TK intervals are essentially unchanged when the LOFAR likelihood is replaced by a constant, so those particular bounds are not produced by the data. Because the abstract cites numbers from the Varying Ar scenario, for which no control is given, the central quantitative claim is not fully supported. The qualitative finding that the disfavoured models are extreme, with rare and large ionized or heated regions, is more robust and is a useful step toward exploiting 21-cm upper limits.

major comments (2)
  1. [Abstract; Sect. 3.2; Appendix A] The no-LOFAR control in Appendix A shows that for the Ar = 0 scenario the 68% (95%) disfavoured intervals for xHII and TK are xHII ≲ 0.024 (0.56) and TK ≲ 7.2 (618) K, whereas the LOFAR-based results in Table 5 are xHII ≲ 0.13 (0.55) and TK ≲ 7.3 (21) K. Thus the tight low-xHII and low-TK bounds are essentially the prior distribution of source parameters mapped through GRIZZLY, not a result of the LOFAR upper limits. The abstract's headline numbers, however, are taken from the Varying Ar scenario (Table 7), for which no corresponding control is presented. Given the Ar = 0 control, it is likely that the xHII and TK intervals in the Varying Ar case are also prior-dominated. The authors should either present the no-LOFAR control for the Varying Ar scenario, or explicitly state in the abstract and conclusions that the quantitative xHII, fheat and TK intervals are set by the source-parameter priors and are not direct LOFAR constraints.
  2. [Sect. 2.2.5, note 15] The authors state that "we do not have robust accuracy estimates of GRIZZLY for scenarios with TS fluctuations." The exclusion likelihood in Eqs. (3) and (4) is evaluated entirely from GRIZZLY power spectra, and the Varying Ar scenario is precisely one in which spin-temperature fluctuations can be significant because Tγ,eff can approach or exceed TS. A systematic error in GRIZZLY at the k-bins used in the analysis (0.076–0.133 h/Mpc) would directly shift the inferred disfavoured IGM parameter regions. The adopted 30% modelling error is added in quadrature per k-bin and does not account for possible correlated, shape-dependent errors across k-bins. I recommend a robustness test in which the modelling error is increased (e.g., to 50%) or a comparison with a 3D radiative-transfer scheme is carried out for a few representative Varying Ar models, in order to demonstrate that the qualitative and quantitative conclusions are stable.
minor comments (5)
  1. [Appendix A; Abstract] The abstract cites the IGM bounds from the Varying Ar case without referencing the prior-dominated nature established in Appendix A for the Ar = 0 case; a sentence in the abstract acknowledging that these numbers are strongly prior-dependent would be more accurate.
  2. [Figs. B.1–B.4] The axis labels in Figures B.1–B.4 use "Rpeak (cMpc)" and "RFWHM (cMpc)", while Table 3 and the main text define Rheat_peak and ΔRheat_FWHM in units of h⁻¹ Mpc; these units should be made consistent.
  3. [Fig. 3] The axis label "log10(Rheat_FHWM)" contains a typo; it should read "Rheat_FWHM".
  4. [Sect. 4] The text contains a typo: "databse" should be "database".
  5. [General] The term "disfavoured credible intervals" is non-standard because credible intervals usually describe a posterior distribution of the parameters. Since the posterior here is a prior-weighted distribution of models with high exclusion likelihood, the paper should more explicitly define this quantity at first use to avoid confusion with standard Bayesian constraints on the IGM.

Circularity Check

1 steps flagged · score 6.0 of 10

IGM-state bounds in the abstract are largely a prior pushforward: Appendix A's constant-likelihood control reproduces the xHII and TK credible intervals, so key 'LOFAR constraints' reduce to the chosen source priors.

  1. other [Appendix A (no-LOFAR control) vs Sec. 3.1 Table 5, Sec. 3.2 Table 7, and Abstract; see also Sec. 2.2.5 and Sec. 4.]
    "We fix to 0.5 the exclusion likelihood shown in Eqs. (3) and (4), independently from the LOFAR upper limits, but still use the priors on the maximum value of xHII, and we denote this as the no-LOFAR scenario. ... the posterior distribution of the disfavoured IGM parameter values is entirely determined by the chosen priors on the source parameters. ... The disfavoured models have xHII ≲ 0.024 (0.56), T K ≲ 7.2 (618) K, no constraints on fheat ... at 68 (95) per cent credible intervals for the Single-z no-LOFAR analysis case."

    The control sets Lex=0.5, removing all LOFAR information; the IGM posterior is then by construction the pushforward of the uniform source priors through GRIZZLY. For Ar=0 Single-z z=9.1 this no-LOFAR run gives xHII ≲ 0.024 (0.56) and TK ≲ 7.2 K at 68 (95)%, while the LOFAR-based Table 5 gives xHII ≲ 0.13 (0.55) and TK 68% ≲ 7.3 K. The 95% xHII and 68% TK limits are identical to the prior-only values; the paper itself says 'the constraints on xHII, fheat and T K are significantly affected by the chosen priors on the source parameters.' The abstract's Varying Ar headline xHII/TK intervals lie in the same prior-dominated regime (no Varying Ar control is shown), so these reported 'IGM constraints from LOFAR' reduce to the source-prior input.

full rationale

The paper's numerical IGM-state constraints are obtained by drawing source parameters from uniform priors and mapping them through GRIZZLY to derived IGM quantities. Appendix A demonstrates that with the LOFAR likelihood set to Lex=0.5, the xHII and TK credible intervals are nearly unchanged from the LOFAR-based values for the Ar=0 Single-z case; hence those specific constraints reduce by construction to the source-prior distribution and not to the 21-cm observations. The paper is transparent about this ('the constraints on xHII, fheat and T K are significantly affected by the chosen priors'), and the qualitative finding that only extreme models with rare, large regions are disfavoured may survive, so this is a partial circularity rather than a fabricated result. The self-citations to GRIZZLY and the earlier framework are not independently load-bearing here: GRIZZLY is the simulation engine, its validation limitations are disclosed, and the LOFAR data come from an observational analysis; those are correctness/robustness risks, not circularity. The central quantitative 'IGM constraints' in the abstract and Tables 5/7, however, are substantially a re-labelling of the source priors for the parameters the paper highlights, which merits a score of 6.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central inference is conditional on the GRIZZLY source model and the chosen uniform prior ranges. The paper demonstrates in Appendix A that the constraints on xHII, fheat and TK are almost entirely prior-driven. No new physical entities are introduced; the excess radio background Ar is adopted from prior literature (Fialkov & Barkana 2019; Mondal et al. 2020) and motivated by LWA1 and ARCADE2 measurements.

free parameters (6)
  • log10(ionization efficiency zeta) prior range = [-3, 3]
    Uniform prior in log space; the disfavoured constraints on zeta and derived IGM limits depend on this range.
  • log10(X-ray efficiency fX) prior range = [-3, 3]
    Uniform prior; affects heating and the 21-cm power spectrum amplitude.
  • log10(Mmin) prior range = [9, 12] in log10(M/M_sun)
    Minimum mass of UV-emitting halos; the prior strongly controls whether models have rare, large bubbles.
  • log10(Mmin,X) prior range = [9, 12] in log10(M/M_sun)
    Minimum mass of X-ray-emitting halos; controls the rarity and size of heated regions.
  • Radio background efficiency Ar prior range = [0, 416]
    Excess radio background amplitude; the 68% disfavoured lower bound Ar > 4.6 is the basis for the claim of an excess above 100% of the CMB, while the 95% interval spans the full prior.
  • Modelling error fraction = 0.3 (30%)
    Ad hoc choice for Delta^2_m,err added in quadrature to the observational error; affects the width of the exclusion likelihood.
assumptions (7)
  • domain assumption GRIZZLY 1D radiative transfer accurately predicts 21-cm power spectra across the explored parameter space
    Section 2.2.5 note 15 states the comparison with C2RAY is limited to TS >> T_gamma scenarios and that robust accuracy estimates are lacking for TS-fluctuation scenarios. The exclusion likelihood depends directly on these simulated spectra.
  • domain assumption Spin temperature equals gas kinetic temperature (TS = TK)
    Section 2.2.3 assumes strong Ly-alpha coupling at z=8-10; if the coupling is weaker, the conversion between power spectrum amplitude and gas temperature is invalid.
  • domain assumption Source parameters are redshift independent
    Required for the Joint-z likelihood (Eq. 4); the paper cautions that the joint analysis should be interpreted with care (Sect. 2.2.5).
  • domain assumption Uniform excess radio background
    Section 2.2.2 states 'we adopt a uniform radio background, but the presence of inhomogeneities could enhance the amplitude of the 21-cm signal power spectrum above our estimates'.
  • ad hoc to paper Stepwise reionization history for the Thomson optical depth prior
    Section 2.2.5 assumes neutral until z_i, partially ionized at xHII_max until z=6, then fully ionized; yields xHII_max = 1, 0.79, 0.57 at z=8.3, 9.1, 10.1.
  • domain assumption Stellar mass fraction f_star = 0.02
    Section 2.2.2, fixed based on literature; the ionizing emissivity scales with stellar mass.
  • standard math WMAP cosmology (Omega_m=0.27, Omega_L=0.73, Omega_B=0.044, h=0.7)
    Used in Eq. (2) and in the N-body simulation; consistent with the simulation inputs.

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Cite this review

Pith. "Pith review of Constraints on the state of the IGM at $z\sim 8-10$ using redshifted 21-cm observations with LOFAR." pith.science (2026). https://pith.science/paper/GSGCAO33

@misc{pith2026250500373,
  author       = {Pith},
  title        = {Pith review of: Constraints on the state of the IGM at $z\sim 8-10$ using redshifted 21-cm observations with LOFAR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GSGCAO33}},
  note         = {Machine review of arXiv:2505.00373}
}
abstract

The power spectra of the redshifted 21-cm signal from the Epoch of Reionization (EoR) contain information about the ionization and thermal states of the intergalactic medium (IGM), and depend on the properties of the EoR sources. Recently, Mertens et al 2025 has analysed 10 nights of LOFAR high-band data and estimated upper limits on the 21-cm power spectrum at redshifts 8.3, 9.1 and 10.1. Here we use these upper limit results to constrain the properties of the IGM at those redshifts. We focus on the properties of the ionized and heated regions where the temperature is larger than that of the CMB. We model the 21-cm power spectrum with the code GRIZZLY, and use a Bayesian inference framework to explore the source parameters for uniform priors on their ranges. The framework also provides information about the IGM properties in the form of derived parameters. In a model which includes a radio background in excess of the CMB, the 95 (68) per cent credible intervals of disfavoured models at redshift 9.1 for the chosen priors correspond to IGM states with averaged ionization and heated fraction below 0.46 ($\lesssim 0.05$), an average gas temperature below 44 K (4 K), and a characteristic size of the heated region $\lesssim 14 ~h^{-1} ~\mathrm{Mpc}$ ($\lesssim 3 ~h^{-1} ~\mathrm{Mpc}$). The 68 per cent credible interval suggests an excess radio background which is more than 100 per cent of the CMB at 1.42 GHz, while the 95 per cent credible interval of the radio background efficiency parameter spans the entire prior range. The behaviour of the credible intervals is similar at all redshifts. The models disfavoured by the LOFAR upper limits are extreme ones, as they are mainly driven by rare and large ionized or heated regions.

Figures

Figures reproduced from arXiv: 2505.00373 by the authors.

Figure 1
Figure 1. A set of 1000 power spectra randomly chosen out of the 105 sim￾ulated ones. Panels from top to bottom refer to z = 8.3, 9.1 and 10.1, re￾spectively. These power spectra correspond to the scenario with no addi￾tional radio background other than the CMB (Ar = 0). The down arrow points refer to the recent LOFAR 1σ upper limit (∆ 2 21(k,z)±∆ 2 21,err (k,z)) from Mertens et al. (2025). As a reference, the dashed line cor… view at source ↗
Figure 2
Figure 2. Histogram of the disfavoured values of the source parameters for the scenario with Ar = 0. These correspond to the models with power spectra values larger than ∆ 2 21(k,z)−∆ 2 21,err (k,z) in at least one k−bin, i.e. which have a high probability of being ruled out by the recent LOFAR 1σ upper limits (∆ 2 21(k,z) ± ∆ 2 21,err (k,z)) from Mertens et al. (2025). heated IGM, obtained assuming constant values of xHI = 1… view at source ↗
Figure 3
Figure 3. Histogram of the disfavoured values of the IGM parameters for the scenario with Ar = 0. These correspond to the same set of models considered in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Posterior distribution of the source parameters of the models which are disfavoured by the LOFAR upper limits from Mertens et al. (2025) at z = 9.1 for the Ar = 0 scenario. Red refers to the case when each redshift is considered separately (Single-z) at z =9.1, while b…
Figure 6
Figure 6. Figure 6: Posterior distribution of the IGM parameters of the models which are disfavoured by the LOFAR upper limits from Mertens et al. (2025) at z = 9.1 for the Ar = 0 scenario. Red refers to the case when z = 9.1 is considered separately (Single-z), while blue represents the …
Figure 7
Figure 7. Figure 7: shows the posterior distribution of the five param￾eters for the disfavoured models at z = 9.1. For the Single-z approach, the 68 per cent disfavoured credible intervals limits are ζ ≲ 2.4, Mmin ≳ 1.7 × 1010 M⊙, Mmin,X ≳ 1.6 × 1010 M⊙, fX ≲ 5.2 and Ar ≳ 4.6 (see [PITH…
Figure 8
Figure 8. Figure 8: Posterior distribution of the IGM parameters of the models which are disfavoured by the LOFAR upper limits from Mertens et al. (2025) at z = 9.1 for the Varying Ar scenario. Red refers to the case when each redshift is considered separately (Single-z), while blue repre…
Figure 10
Figure 10. Figure 10: Marginalized probability distribution of the disfavoured mod￾els’ source parameters at z ∼ 8, 9 and 10 when including upper limits from different radio interferometric observations. The results are for the Ar = 0 scenario. The black, red and green curves refer to the …
Figure 9
Figure 9. Figure 9: A set of 1000 power spectra randomly chosen out of the 105 simulated ones for the scenario with no additional radio background other than the CMB (Ar = 0). Panels from top to bottom refer to z = 8.3, 9.1 and 10.1, respectively. The dashed lines correspond to the power …
Figure 11
Figure 11. Figure 11: Marginalized probability distribution of the IGM parameters of the disfavoured models at z ∼ 9 when including upper limits from different radio interferometric observations. The results are for the Ar = 0 scenario. The red and blue curves refer to the Single-z and Joi…

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