Percent-level timing of reionization: self-consistent, implicit-likelihood inference from XQR-30+ Ly$\alpha$ forest data

Andrei Mesinger; David Prelogovi\'c; Emma Ryan-Weber; Fabian Walter; Frederick B. Davies; George Becker; Laura Keating; Manuela Bischetti; Martin G. Haehnelt; Prakash Gaikwad

REVIEW 3 major objections 5 minor 4 cited by

Percent-level timing of reionization: self-consistent, implicit-likelihood inference from XQR-30+ Ly$\alpha$ forest data

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A self-consistent Bayesian fit to quasar Lyα forest data pins reionization's end at z = 5.44 ± 0.02 and removes the need for a sharp emissivity drop.

desk verdict A careful, state-of-the-art inference that makes a strong case for a late reionization driven by faint galaxies, but the percent-level timing is conditional on a calibration whose systematics are not yet quantified. read the letter →

arxiv 2412.00799 v1 pith:BPK6WQMP submitted 2024-12-01 astro-ph.CO astro-ph.GA

Yuxiang Qin , Andrei Mesinger , David Prelogović , George Becker , Manuela Bischetti , Sarah E. I. Bosman , Frederick B. Davies , Valentina D'Odorico

show 8 more authors

Prakash Gaikwad Martin G. Haehnelt Laura Keating Samuel Lai Emma Ryan-Weber Sindhu Satyavolu Fabian Walter Yongda Zhu

This is my paper · ORCID

classification astro-ph.COastro-ph.GA

keywords EpochofreionizationLyαforestBayesianinferenceimplicitlikelihoodescapefractionintergalacticmediumXQR-30timing

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 claims that the combination of Lyα forest data from XQR-30+, galaxy UV luminosity functions, and CMB optical depth can determine the end of reionization to percent-level precision: $z = 5.44 \pm 0.02$, with midpoint $z = 7.7 \pm 0.1$. The central claim is that this is achievable without invoking an ad-hoc rapid drop in ionizing emissivity, provided unresolved small-scale structure and recombinations are modeled with sub-grid physics. If correct, the late Epoch of Reionization is known to high precision, and a prominent tension between large-scale simulations and forest observations disappears.

What carries the argument

The central object is the conditional probability distribution $p(\tau_{\rm eff} | \tau_{\rm eff,GP}; z, x_{\rm HI})$, which maps effective optical depths computed with the low-resolution Fluctuating Gunn-Peterson Approximation (FGPA) to true values calibrated against the Sherwood suite of high-resolution hydrodynamic simulations. This kernel-density-estimated conditional, combined with a sub-grid analytic model for inhomogeneous recombinations, corrects for missing small-scale structure and lets the lightcone reproduce the observed opacity fluctuations without tuning effective parameters or calibrating out the mean transmission.

What would settle it

Compute the conditional distribution $p(\tau_{\rm eff}|\tau_{\rm eff,GP}; z, x_{\rm HI})$ directly from additional high-resolution hydrodynamic snapshots at $z = 5.5$ to $6.1$; if the offset from the $z = 5$ relation exceeds roughly $0.5$ in $\tau_{\rm eff}$, the inferred end-of-reionization redshift would move by more than the quoted $0.02$ uncertainty.

Watch

Extended reading notes

Core claim

Using a forward model that connects physically-motivated galaxy scaling relations to large-scale lightcones of the intergalactic medium, the authors perform implicit-likelihood Bayesian inference over seven astrophysical parameters. The fiducial model, which allows the ionizing escape fraction to evolve with both halo mass and redshift, reproduces the observed effective optical depth distributions from $z = 5.3$ to $6.1$ and yields reionization ending at $z = 5.44 \pm 0.02$ with midpoint $z = 7.7 \pm 0.1$. The inference also implies that more than half of the ionizing photons come from galaxies fainter than $M_{\rm UV} \sim -12$, below current direct detection limits, and that the escape fraction increases toward fainter galaxies.

Load-bearing premise

The calibration assumes that the relation between true and approximate optical depths measured at a single $z = 5$ snapshot stays self-similar across $z = 5.1$ to $6.1$, a shift the paper tests only crudely.

Editorial extensions

If this is right

If correct, reionization ended at $z \approx 5.44$, so the Universe was still substantially neutral at $z \sim 6$, matching the large opacity fluctuations seen in the forest.
The required ionizing emissivity evolves smoothly from $z \sim 7$ to $5.5$, eliminating the need for a rapid factor-of-two drop over about 100 Myr that other simulations introduced.
Galaxies fainter than $M_{\rm UV} \sim -12$ contribute over half of the ionizing photon budget, meaning JWST-visible galaxies are minor players in reionization according to this model.
The model predicts a CMB optical depth of $\tau_e = 0.0589 \pm 0.001$, considerably tighter than the current Planck measurement, providing a sharp target for future CMB experiments.

Reading between the lines

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

Beyond the paper: the percent-level timing claim rests on a single-snapshot calibration, so re-running the calibration with snapshots at several redshifts would directly test whether the quoted uncertainties are realistic.
Beyond the paper: if the late reionization history is this well constrained, upcoming 21-cm experiments should see a late, rapid reionization with sizable neutral fraction at $z \sim 6$, an independent cross-check of the forest-based inference.
Beyond the paper: the implicit-likelihood forward-modelling approach could be applied to other summaries such as the dark pixel fraction without double-counting forest information, tightening joint constraints on the EoR history.

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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,

Read Qin et al. on percent-level timing of reionization from XQR-30+ Lyα forest. Bottom line: this is one of the more careful forward-modeling papers in this space, and the qualitative claims—late end of reionization, faint galaxies dominate the ionizing budget, no rapid emissivity drop—hold up better than I expected. But the headline precision (z_end = 5.44 ± 0.02) is not robust to the calibration's own acknowledged systematics. I'd send it to review, but I'd insist the authors either quantify those systematics or soften the percent-level language.

What's genuinely new: they replace the nuisance hyperparameters of Qin et al. 2021 with a conditional KDE calibrated to the Sherwood hydrodynamic simulation, embed it in an implicit likelihood, and then compare two galaxy models with Bayesian evidence. The resulting posterior reproduces the observed τ_eff CDFs across z=5.3–6.1 without tuning the mean transmission by hand, and it does so without the sharp emissivity drop that other models need. That is a real step forward. They are also unusually transparent about what they did not include—no dark pixel constraint, no z≤5.2 data, no covariance between segments—and they show the constant-fesc alternative gives qualitatively similar EoR history, which is a fair robustness check.

The soft spots are real but not fatal. The calibration relies on a single z=5 Sherwood snapshot and assumes the conditional p(τeff|τeff,GP; z, xHI) is self-similar across z=5.1–6.1. Their own crude test—rescaling the density field—gives a τeff shift ≲0.5, which is comparable to the scatter in the observed CDFs and much larger than the quoted 0.02 redshift precision. They also model partial neutral fractions by sprinkling log-normal spherical patches, and the segment-independence assumption is unquantified. These are systematics, not refutations; the central qualitative picture is consistent with independent probes shown in Fig. 4. But the percent-level timing claim is conditioned on these approximations, and the error bars do not include them.

Who is this for: anyone working on the late EoR or on simulation-based inference for the Lyα forest. It deserves a serious referee—not a desk reject—and I'd push for a major revision asking for a propagation of these calibration uncertainties, or at minimum a clear statement that the precision is internal to the model. I'd cite it for the methodology, and I'd bring it to reading group.

Referee Report

3 major / 5 minor

Summary. The paper presents a Bayesian inference framework combining 21cmFAST lightcone simulations with a stochastic calibration to the Sherwood hydrodynamic simulation to interpret XQR-30+ Lyα forest effective optical depth distributions at z ≈ 5.3–6.1, together with UV luminosity functions and the CMB optical depth. The fiducial Evolving_fesc model, with seven free galaxy parameters, is reported to constrain the reionization history at percent-level precision, yielding an end of reionization at z = 5.44 ± 0.02, a midpoint at z = 7.7 ± 0.1, and a smooth ionizing emissivity without the rapid drop inferred in some earlier works. An alternative Constant_fesc model is strongly disfavored by Bayesian evidence but gives qualitatively similar IGM histories. The authors caution that conclusions about the early EoR and the source population are more model-dependent.

Significance. If the central claims hold, this is a substantial advance: it demonstrates that a physically motivated forward model can reproduce the observed Lyα opacity fluctuations without ad hoc emissivity tuning, and it sharpens the late-EoR timeline to a level that challenges other probes. The framework is clearly specified and reproducible, building on public codes (21cmFAST, conditional_kde), and the paper is unusually transparent about its approximations, including explicit statements of the single-snapshot calibration and the segment-independence assumption. The falsifiable predictions—percent-level z_end, MFP evolution, and a smooth emissivity—are concrete and testable with upcoming data. However, the headline precision is statistical only, and the paper's own calibration test reveals a systematic shift of τeff ≲ 0.5 that is not propagated into the posteriors; this currently limits the strength of the percent-level claim.

major comments (3)

[§3.3] The single-snapshot FGPA calibration is the load-bearing step for the likelihood. The paper's own footnote states that the self-similarity assumption was crudely tested by rescaling the density field and found 'only a τeff ≲ 0.5 shift' in the conditionals. This shift is comparable to the width of the observed τeff distributions at z ≈ 5.5–6 and is about an order of magnitude larger than the quoted 0.02 uncertainty on z_end. Because the likelihood in §3.4 is built directly from p(τeff | τeff,GP; z, xHI), this systematic must be propagated into the posterior, for example by marginalizing over a calibration offset or by re-doing the calibration with multiple snapshots and evolving temperature and velocity fields. Without this, the percent-level precision claim is not supported.
[§3.4] The forest likelihood assumes that each Δz = 0.1 segment is an independent sample of p(τeff; z). The paper acknowledges this and states that the covariance 'should have only a minor impact,' but no quantitative test is provided. Given that the segments are ~40 cMpc long and that the observed opacity fluctuations are coherent on comparable scales, the effective number of independent segments is likely smaller than the number of quasar-sightline bins used in the product. This can narrow the likelihood and bias the posterior; I ask for a demonstration (e.g., a mock-based covariance estimate or a comparison with a Gaussian-process likelihood) that the independence assumption does not affect the inferred z_end and its uncertainty.
[§3.3 / Fig. 2] The treatment of partial neutral fractions by randomly placing spherical neutral patches with a log-normal radius distribution (mean 4 cMpc) is an uncontrolled approximation that directly affects the z ≈ 5.5–6.1 bins where xHI is non-negligible. The resulting conditional distributions at xHI > 0 are not validated against simulations with realistic reionization morphology. Since the high-redshift τeff CDFs are the most constraining for the late EoR, the sensitivity of the posteriors to the patch-size distribution and placement algorithm should be tested, e.g., by varying the mean radius or by using a simulation-calibrated morphology.

minor comments (5)

[§3.3/§4.2] The 'τeff ≲ 0.5 shift' from the self-similarity test is reported only in a footnote; it should be moved to the main text with a quantitative statement of how this offset translates into changes in the inferred EoR parameters, since it is essential context for evaluating the claimed precision.
[§4.2] The sentence 'most of the history constrained to better than ∆z ∼ 0.1 at the 68% C.I.' is vague; given the inset posteriors, please specify the redshift range over which this holds and define the criterion used.
[Eq. (5)] Equation (5) appears garbled in the typeset text: the superscripts on Tγ and the structure of the exponential term are unclear. Please check the equation formatting.
[Fig. 3] The caption says 'red enclosing the 95% C.I.' but the figure shows shaded red regions; please clarify whether the red region is the 95% confidence interval of the model CDF or of the sampled sightline CDFs.
[§5 / Fig. 11] The notation 'Muv ≳ –14' and 'Muv ≳ –12' in the text and caption is confusing because larger MUV means brighter galaxies; please rephrase to 'galaxies fainter than MUV = –14' or similar.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the percent-level EoR timing is an emergent posterior over independent external data, not a fitted quantity recycled as a prediction.

full rationale

The inference chain is self-contained and non-circular. The forward model maps sampled galaxy parameters to IGM lightcones with 21cmFAST; unresolved Ly-alpha opacity structure is then stochastically calibrated using conditional distributions p(tau_eff | tau_eff,GP; z, x_HI) built from the external Sherwood hydrodynamic simulation (Bolton et al. 2017). The XQR-30+ forest data, Hubble UV luminosity functions, and Planck CMB optical depth enter only as final likelihood terms against these forward predictions. The headline quantities (reionization end z = 5.44 +/- 0.02, midpoint z = 7.7 +/- 0.1) emerge from the posterior over these fixed external inputs; no fitted parameter is renamed as a prediction, and the inferred EoR history is not fed back into the calibration. The paper's own caveats in Sec. 3.3 and 3.4 (single z = 5 snapshot, self-similarity assumption, a quoted tau_eff shift of order 0.5, and unquantified segment independence) are honest robustness limitations rather than circular steps, since the Sherwood calibration and the XQR-30+ spectra are independent inputs. Same-author citations such as Q21, Park et al. 2019, Sobacchi and Mesinger 2014, and Mesinger et al. 2011 supply independent model ingredients with external grounding; they are not invoked as a uniqueness theorem and do not determine the predicted timing by construction. The claimed circularity score is therefore 0.

Assumptions & free parameters 9 free parameters · 6 assumptions · 1 invented entities

The central inference rests on seven sampled astrophysical parameters plus fixed modeling choices (RMFP,LLS, neutral patch size, Sherwood calibration). The calibration anchors small-scale opacity to an external simulation, so the circularity burden is low; the main fragility is the single-snapshot self-similarity assumption and the treatment of the likelihood as independent segments.

free parameters (9)

log10 f*10 = -1.51 ± 0.03
Normalization of the stellar-to-halo mass fraction at Mvir = 10^10 Msun; sampled in MCMC (Table 1).
alpha_* = 0.48 ± 0.05
Power-law index of stellar fraction versus halo mass; sampled in MCMC (Table 1).
log10 fesc10 = -1.52 +0.12/-0.10
Normalization of the ionizing escape fraction at Mvir = 10^10 Msun; sampled in MCMC (Table 1).
alpha_esc = -0.94 +0.09/-0.04
Power-law slope of escape fraction versus halo mass; sampled in MCMC (Table 1).
beta_esc = -1.61 +0.27/-0.21
Redshift evolution index of escape fraction in the Evolving_fesc model; sampled in MCMC (Table 1).
tau_* = 0.27 +0.02/-0.01
Star formation timescale in units of Hubble time; sampled in MCMC (Table 1).
log10(Mturn/Msun) = 8.10 +0.16/-0.07
Halo mass scale below which star formation is exponentially suppressed; sampled in MCMC (Table 1).
RMFP,LLS = 66 cMpc (z<=6), 42 cMpc (z>6)
Assumed homogeneous mean free path through the ionized IGM, fixed by hand from post-EoR measurements (Sec. 3.2). Not sampled, but directly sets the post-reionization MFP asymptote.
Neutral patch radius (mean of log-normal) = 4 cMpc
Peak of the log-normal radius distribution used to create partially neutral calibration boxes (Sec. 3.3). Motivated by Xu, Yue, and Chen (2017), not sampled.

assumptions (6)

domain assumption The Sherwood hydrodynamic simulation correctly captures small-scale Lyα opacity structure and matches observed forest at z <= 5.
Used as ground truth for calibrating FGPA in Sec. 3.3; if wrong, the conditional p(τeff | τeff,GP) is biased.
ad hoc to paper Single z = 5 snapshot calibration is self-similar across the redshift range 5.1 to 6.1.
Sec. 3.3 explicitly states snapshots at other redshifts were unavailable; crude test gave τeff shift ~ 0.5.
domain assumption Each Δz = 0.1 segment is an independent sample of the τeff distribution.
Sec. 3.4 ignores covariance between segments from the same quasar; could understate uncertainties and inflate the Bayesian evidence.
domain assumption Uniform, fixed RMFP,LLS is adequate during the EoR.
Sec. 3.2; they argue RMFP,LLS > RMFP,EoR during reionization, but it sets the post-EoR MFP asymptote and affects late-EoR behavior.
ad hoc to paper The galaxy-halo power-law scaling relations (Eqs. 1 and 2) with a duty-cycle cutoff describe high-z galaxies.
Parametrization from Park+19 with an added βesc term; the functional form is assumed, not derived from first principles.
standard math Standard cosmological simulation machinery: 2LPT, excursion-set reionization, FGPA.
Used throughout Secs. 3.1 to 3.2; well-established but approximate.

invented entities (1)

Spherical neutral patches placed in the Sherwood box
purpose: To approximate partially neutral IGM (xHI > 0) for calibrating p(τeff | τeff,GP; z, xHI)
These patches are a calibration device, not a physical claim. The radius distribution is motivated by an external simulation study (Xu, Yue, and Chen 2017), but the patches themselves have no falsifiable handle outside this paper's calibration procedure.

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Cite this review

Pith. "Pith review of Percent-level timing of reionization: self-consistent, implicit-likelihood inference from XQR-30+ Ly$\alpha$ forest data." pith.science (2026). https://pith.science/paper/BPK6WQMP

@misc{pith2026241200799,
  author       = {Pith},
  title        = {Pith review of: Percent-level timing of reionization: self-consistent, implicit-likelihood inference from XQR-30+ Ly$\alpha$ forest data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPK6WQMP}},
  note         = {Machine review of arXiv:2412.00799}
}

abstract

The Lyman alpha (Lya) forest in the spectra of z>5 quasars provides a powerful probe of the late stages of the Epoch of Reionization (EoR). With the recent advent of exquisite datasets such as XQR-30, many models have struggled to reproduce the observed large-scale fluctuations in the Lya opacity. Here we introduce a Bayesian analysis framework that forward-models large-scale lightcones of IGM properties, and accounts for unresolved sub-structure in the Lya opacity by calibrating to higher-resolution hydrodynamic simulations. Our models directly connect physically-intuitive galaxy properties with the corresponding IGM evolution, without having to tune "effective" parameters or calibrate out the mean transmission. The forest data, in combination with UV luminosity functions and the CMB optical depth, are able to constrain global IGM properties at percent level precision in our fiducial model. Unlike many other works, we recover the forest observations without evoking a rapid drop in the ionizing emissivity from z~7 to 5.5, which we attribute to our sub-grid model for recombinations. In this fiducial model, reionization ends at $z=5.44\pm0.02$ and the EoR mid-point is at $z=7.7\pm0.1$. The ionizing escape fraction increases towards faint galaxies, showing a mild redshift evolution at fixed UV magnitude, Muv. Half of the ionizing photons are provided by galaxies fainter than Muv~-12, well below direct detection limits of optical/NIR instruments including JWST. We also show results from an alternative galaxy model that does not allow for a redshift evolution in the ionizing escape fraction. Despite being decisively disfavored by the Bayesian evidence, the posterior of this model is in qualitative agreement with that from our fiducial model. We caution however that our conclusions regarding the early stages of the EoR and which sources reionized the Universe are more model-dependent.

Figures

Figures reproduced from arXiv: 2412.00799 by the authors.

**Figure 1.** A flow chart showing the steps involved in computing the likelihood for a single sample of astrophysical parameters. See text for more details. forest, which is averaged over segments of width ∆z = 0.1 (roughly corresponding to ∼ 40 cMpc at these redshifts). Nondetections (2σ) were assigned lower limits on τeff corresponding to twice the mean flux noise in the corresponding segment. The full XQR-30+ sample has at … view at source ↗

**Figure 2.** Lower sub-panels: comparisons of the Lyα effective optical depth calculated using the full integral over the Lyα cross-section at the highest available resolution (τeff ), to those calculated assuming the FGPA (τeff,GP). Both calculations use the Sherwood hydrodynamic simulation, with the latter obtained by down-sampling to the same low resolution adopted in our IGM forward-models and ignoring peculiar velocities. T… view at source ↗

**Figure 3.** Inferred τeff CDFs from our fiducial model (red) from z =5.3 to 6.1. To account for cosmic variance, we randomly select from each model in the posterior the same number of sightlines as in the XQR-30+ observational dataset. The red regions indicate the 95% C.I. For comparison, the XQR-30+ observations are shown in grey with non-detections denoted with the shaded regions spanning the flux range between zero and doubl… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: The inferred EoR history using our fiducial model. The blue shaded region uses only UV LFs and CMB τe data (a likelihood of LLF × LCMB), while the red additionally includes the Lyα forest τeff distributions (likelihood of Lforest × LLF × LCMB). In both cases the dark (…

**Figure 6.** Figure 6: The inferred UV ionizing emissivity. On the left axis we denote the number of ionizing photons per time per comoving volume, while on the right axis we show the number of ionizing photons per time per baryon. As in the previous figure, the dark (light) shaded red regio…

**Figure 5.** Figure 5: The posterior of our fiducial model in the space of the mean photoionization rate (top panel) and proper mean free path (bottom panel). As in the previous figure, the dark (light) shaded region corresponds to 68% (95%) C.I. In the lower panel, we additionally show the…

**Figure 7.** Figure 7: Comparison of the MAP model with and without recombinations. Clockwise from the upper left panel, we show the mean EoR history, photoionization rate, proper MFP and ionizing emissivity. In the bottom left panel we also show the emissivity rescaled by the ratio of Γion …

**Figure 8.** Figure 8: Top panel: Evolution of the HII filling factor from the MAP model (red curve), together with analytic estimates using equation (9) assuming the same mean ionizing emissivity as the MAP but taking a constant “clumping factor”. Curves corresponding to Ceff =1, 3, 10 are …

**Figure 9.** Figure 9: The inferred galaxy UV luminosity function. As in the previous figure, the dark (light) shaded region corresponds to 68% (95%) C.I. Observed luminosity functions are grouped into pre-JWST (light grey; Bouwens, Illingworth, Oesch, Trenti, et al. 2015; Bouwens et al. 201…

**Figure 10.** Figure 10: The inferred (68% C.I.) ionizing contribution of galaxies as a function of their UV mgnitudes at z = 6, 8 and 11. The top panel shows the normalized cumulative number of ionizing photons while the bottom panel shows the escape fraction. Our results imply reionization…

**Figure 11.** Figure 11: Upper panels: lightcones of MAP models from Evolving_fesc and Constant_fesc. From top to bottom, the panels correspond to the overdensity (∆), neutral hydrogen fraction (xHI), locally-averaged UVB (Γion), temperature (Tg), residual neutral fraction within the ionized …

**Figure 3.** Figure 3: Both models suggest a qualitatively similar conclusion – reionization finishes at z < 5.5 with the process primarily driven by ionizing photons emitted by faint galaxies. The end (corresponding to xHI = 0.01) and midpoint of reionization are at z = 5.33 ± 0.03 and z = …

**Figure 12.** Figure 12: Marginalized 1D and 2D posterior distributions of model parameters from the fiducial model Evolving_fesc (red), and this model without XQR-30+ (blue) as well as Constant_fesc (purple). Regions inside the curves or indicated in shades represent the 95th percentiles [P…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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