Reionisation time field reconstruction from 21-cm Maps: Investigating predictor coherence in WDM cosmology

Technical Summary: Reionisation Time Field Reconstruction from 21-cm Maps in WDM Cosmology

Problem Statement
The epoch of reionisation (EoR) is a critical phase in cosmic evolution, yet its detailed history remains elusive. While the reionisation time field, $t_{\text{reion}}(\vec{r})$, offers a comprehensive view of when and where the Universe became ionised, it is not directly observable. Previous work (Hiegel et al. 2023) demonstrated that Convolutional Neural Networks (CNNs) could infer $t_{\text{reion}}(\vec{r})$ from 21-cm maps at a fixed redshift. However, a significant limitation exists: these predictors are trained on specific reionisation models (e.g., Cold Dark Matter, CDM). When applied to observational data that may stem from different physical conditions (e.g., Warm Dark Matter, WDM), the reliability of these predictors is uncertain. There is a need to evaluate the "coherence" of these predictors—specifically, whether a predictor trained on one model yields consistent results when applied to data generated by a different model, and if such inconsistencies can be used to validate or exclude underlying cosmological models.

Methodology
The authors propose a method to evaluate predictor coherence by treating the CDM model as a reference input. The study utilizes mock 21-cm maps generated using the 21cmFAST simulation code under a $\Lambda$CDM cosmology, incorporating four WDM scenarios with particle masses ($m_X$) of 2, 3, 5, and 7 keV.

1. Predictor Training: Five distinct CNN predictors (based on a U-net architecture) were trained, each on 21-cm maps corresponding to a specific dark matter model (CDM, 2 keV, 3 keV, 5 keV, 7 keV) at specific redshifts ($z=8$ and $z=11$).
2. Cross-Model Application: The core of the methodology involves feeding CDM-generated 21-cm maps (the "observation") into all five predictors.
3. Coherence Evaluation: The authors compare the reconstructed $t_{\text{reion}}(\vec{r})$ fields obtained from the same predictor at two different redshifts ($z=11$ and $z=8$). The underlying hypothesis is that for a consistent predictor-model pair, the statistical properties of the reconstruction should be invariant across redshifts.
4. Metrics: Four statistical metrics were computed on the reconstructed fields to quantify self-consistency:
   • Neutral hydrogen volume fraction ($Q_{HI}$).
   • Power spectrum of the reionisation time field ($P_k$).
   • Total isocontour length (representing the interface between ionised and neutral regions).
   • Number of minima in the $t_{\text{reion}}$ field (representing reionisation sources).
5. Quantifiers: Three specific estimators were defined to compare WDM predictors against the CDM reference:
   • RMSE: Root-mean-square error of the difference between redshift reconstructions ($S_{11} - S_8$).
   • Coherence Fraction ($q$): The ratio of the CDM RMSE to the WDM RMSE.
   • Coefficient of Determination ($R^2_\Delta$): Measures the distance of a WDM reconstruction's consistency relative to the CDM reconstruction's consistency. A negative $R^2_\Delta$ indicates the WDM predictor is less consistent with the input than the CDM predictor is with itself.
6. Uncertainty Analysis: Epistemic uncertainty was estimated using Monte Carlo dropout to ensure that reconstruction noise did not dominate the statistical comparisons.

Key Results
The study finds that the ability to distinguish models depends heavily on the mass of the WDM particle and the specific metric used:

• High-Mass Models (5 and 7 keV): Predictors trained on 5 and 7 keV models exhibit high-level self-consistency when applied to CDM inputs, showing behavior statistically similar to the CDM predictor across all metrics. The $R^2_\Delta$ values are positive and high (e.g., 0.83 for $Q_{HI}$ at 7 keV), indicating that these models are difficult to distinguish from CDM using this method.
• Low-Mass Models (2 keV): The 2 keV predictor displays significant deviations across all metrics. It yields large RMSE values and negative $R^2_\Delta$ values (e.g., -4.8 for isocontour length), indicating a fundamental incompatibility with the CDM input data. The 2 keV model is easily excluded.
• Intermediate Model (3 keV): The 3 keV predictor shows intermediate behavior. While it is generally ruled out, it presents a more complex case. For some metrics (like the number of minima), it shows a high coherence fraction ($q \approx 0.97$), yet it still yields a lower $R^2_\Delta$ compared to the 5 and 7 keV models. This suggests that relying on a single metric is insufficient; multi-metric analysis is required to identify inconsistencies.
• Uncertainty: The epistemic uncertainty ($\Delta\sigma$) introduced by the CNN reconstruction remains subdominant ($\Delta\sigma < 1$) across all observables, confirming that the observed deviations are due to model mismatch rather than reconstruction noise.

Significance and Claims
The paper claims to provide a framework for validating machine learning predictors before their application to real observational data (such as from the Square Kilometre Array, SKA). The primary significance lies in demonstrating that CNN predictors retain sensitivity to the underlying physics of the training data.

• Model Selection: The method allows for the exclusion of reionisation models that are inconsistent with observational data. For instance, the 2 keV WDM model is effectively excluded based on its failure to maintain coherence across redshifts when applied to CDM-like data.
• Limitations: The authors modestly note that the method struggles to distinguish between models that are physically similar (e.g., 5 keV and 7 keV WDM vs. CDM). They emphasize that the goal is not to constrain cosmological parameters directly but to assess the robustness of predictors against model mismatches.
• Future Work: The paper concludes that while current results are promising, future applications to SKA-like observations will require improved CNN architectures (e.g., attention mechanisms) and better handling of instrumental noise and foreground contamination. The study underscores the necessity of validating predictors against their input models to ensure trustworthy applications in the study of cosmic reionisation.