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Towards end-to-end Bayesian forward models in global 21-cm cosmology: surrogate modelling and marginalisation of beam uncertainty

This paper presents an accelerated, differentiable Bayesian framework that utilizes a two-stage surrogate model and analytical marginalisation to efficiently incorporate chromatic beam uncertainty into global 21-cm cosmology forward models, thereby enabling robust parameter recovery with minimal electromagnetic simulations while avoiding the biases of fixed-beam assumptions.

Original authors: Jacob L. Tutt, Dominic J. Anstey, John Cumner, Harry T. J. Bevins, Jean Cavillot, Eloy de Lera Acedo

Published 2026-08-20

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CC BY 4.0

Original authors: Jacob L. Tutt, Dominic J. Anstey, John Cumner, Harry T. J. Bevins, Jean Cavillot, Eloy de Lera Acedo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe has a history written in light, but much of that story is hidden in a dark chapter that occurred long before the first stars ignited. Astronomers call this the Cosmic Dark Ages, a time when the cosmos was filled with neutral hydrogen gas but lacked the bright beacons of galaxies to illuminate it. To read this history, scientists are listening for a faint, specific whisper from that era: a radio signal emitted by hydrogen atoms as they cooled and settled. This signal, known as the 21-cm line, is the only direct way to map the universe during those first billion years, offering a window into how the first stars and galaxies eventually turned on and changed the nature of space itself. However, catching this whisper is incredibly difficult because it is drowned out by the roar of our own galaxy and the static of the instruments used to listen. The signal is billions of times weaker than the foreground noise, and any tiny error in how the listening device itself behaves can distort the data so severely that the true cosmic story is lost or, worse, replaced by a false one.

A team of researchers has now developed a new way to listen that accounts for these imperfections without getting overwhelmed by them. In a study focused on the REACH experiment, which uses a specialized antenna in the Karoo desert of South Africa, the scientists tackled the problem of "beam uncertainty." Every radio antenna has a unique shape and sensitivity pattern, much like a flashlight has a specific beam shape, and this pattern changes slightly depending on the frequency of the radio waves it receives. In the past, scientists had to assume they knew the exact shape of this beam to interpret their data. If their assumption was even slightly wrong, the resulting analysis would produce a distorted picture of the universe, leading to incorrect conclusions about the early cosmos. The new method presented in this paper moves away from guessing a single, perfect beam shape. Instead, it treats the antenna's behavior as a variable that is uncertain, learning from thousands of computer simulations of how the antenna might behave under different real-world conditions, such as slight manufacturing differences or changes in the soil moisture beneath it.

The researchers built a sophisticated system that simulates the antenna's response to a vast range of possible physical variations. They generated hundreds of different scenarios, altering the height of the antenna blades, their rotation, and the electrical properties of the ground beneath them. By running these simulations, they created a library of how the antenna's "flashlight" pattern might shift and warp. Rather than trying to analyze every single possibility individually, which would take too long, they used a mathematical technique to compress this massive library into a much smaller, manageable set of rules. This compression allowed them to describe the complex, shifting behavior of the antenna using just a few key numbers instead of thousands. Crucially, they designed their analysis software to "marginalize" over these numbers. In simple terms, this means the computer does not try to pick one specific beam shape as the truth; instead, it averages over all the plausible shapes allowed by their library, effectively letting the uncertainty wash out of the final result rather than contaminating it.

When the team tested this new approach, they found that the old method of assuming a fixed, perfect beam was dangerously fragile. In their simulations, when they forced the analysis to use a beam shape that was slightly different from the one actually generating the data, the results were catastrophic. The analysis failed to find the faint cosmological signal and instead produced errors thousands of times larger than the signal itself, leading to completely wrong conclusions about the early universe. In stark contrast, when they used the new method that accounted for the beam's uncertainty, the system successfully recovered the true signal. Even when the antenna in the simulation behaved in ways the researchers had never seen before, the method was able to filter out the instrumental noise and reveal the underlying cosmic signal with high precision. The recovered data matched the true input almost perfectly, with errors no larger than the natural thermal noise of the instrument itself.

The study also revealed that this powerful new approach does not require an impossibly large amount of computing power. While the researchers started with a library of five hundred different antenna simulations to build their model, they discovered that a much smaller set of just one hundred simulations was sufficient to construct an effective model that worked just as well. This finding suggests that future experiments can achieve robust, high-precision results without needing to run millions of expensive computer simulations. By treating the antenna's imperfections as a known quantity to be averaged over, rather than a fixed assumption to be guessed, the researchers have created a more reliable path to detecting the faint echo of the universe's first stars. This work provides a scalable and statistically rigorous foundation for future global 21-cm cosmology, ensuring that when the faint signal of the Cosmic Dark Ages is finally detected, it will be a true reflection of the cosmos and not an artifact of the instrument listening for it.

Technical Summary: Towards End-to-End Bayesian Forward Models in Global 21-cm Cosmology

Problem Statement
Robust statistical inference in global 21-cm cosmology requires the joint handling of the highly degenerate cosmological signal, bright foreground emission, and instrumental response. A critical bottleneck in current analysis frameworks is the treatment of the antenna beam. While electromagnetic (CEM) simulations can capture physical antenna properties, their multi-hour runtimes make direct integration into likelihood-based sampling frameworks infeasible. Consequently, most existing approaches assume a single, precomputed beam. The authors demonstrate that this is a fragile assumption: realistic mismatches between the assumed beam and the true physical response can induce chromatic distortions at the tens-of-kelvin level. These distortions are orders of magnitude larger than the expected cosmological signal (millikelvin scale) and can severely bias the recovery of both cosmological and foreground parameters, leading to catastrophic inference failures.

Methodology
To address the computational intractability of sampling high-dimensional beam uncertainties, the authors present an accelerated, differentiable Bayesian framework that incorporates an informed surrogate representation of chromatic beam uncertainty directly into a forward-modelling pipeline. The methodology proceeds in three stages:

Beam Simulation and Ensemble Generation:
The authors generate an ensemble of 500 distinct antenna configurations using a Method-of-Moments (MoM) solver. These configurations sample a seven-dimensional physical parameter space representing geometric perturbations (dipole blade height, ground-plane height, rotation, and lateral displacements) and environmental variations (soil permittivity). This ensemble serves as a conservative upper limit on instrumental uncertainty.
Two-Stage Surrogate Parameterisation:
To compress the high-dimensional beam space (angular and spectral) into a tractable parameter set, a two-stage decomposition is applied:
- Stage 1 (SVD): A Singular Value Decomposition (SVD) is performed on the flattened directivity patterns of the 400-beam training set. This yields 400 orthogonal angular basis functions ( $B_j$ ) and their corresponding frequency-dependent weights ( $w_{b,j}$ ). This stage captures the dominant angular structure and chromatic evolution.
- Stage 2 (PCA on Weights): Recognizing that a direct parameterisation of the 400 weights across all frequency channels would still result in ~34,000 parameters, a second decomposition is performed on the weight vectors themselves. A Principal Component Analysis (PCA) is applied to the residual variations of the SVD weights across the beam ensemble. This yields a compact set of correlated correction modes ( $e_m$ ) and coefficients ( $a_{b,m}$ ).
- Result: This hierarchical approach compresses the beam parameterisation by two orders of magnitude. The authors determine that retaining 400 SVD modes and 100 PCA modes is sufficient to reconstruct the beam such that the resulting antenna temperature error remains below the instrument's thermal noise level (25 mK).
Analytical Marginalisation:
The forward model is extended to include the beam coefficients as nuisance parameters. Crucially, because the beam model enters the antenna temperature calculation linearly with respect to the PCA coefficients, the authors exploit this linearity to perform analytical marginalisation. Instead of sampling the beam coefficients directly (which would be computationally prohibitive for Nested Sampling), they integrate them out analytically. This inflates the instrumental noise covariance matrix to form an effective data covariance ( $C_{eff}$ ), allowing the continuous instrumental uncertainty to be propagated into the final posteriors and Bayesian evidence without explicitly sampling the high-dimensional beam space.

Key Contributions

First Joint Inference Framework: This work presents the first joint inference framework for global 21-cm cosmology that propagates continuous instrumental beam uncertainty, learned from CEM simulations, alongside foreground and cosmological uncertainties within a physically motivated Bayesian forward model.
Efficient Surrogate Modelling: The introduction of a two-stage SVD-PCA decomposition that reduces the beam parameterisation from ~34,000 dimensions to just 100 coefficients while maintaining reconstruction accuracy at the thermal noise level.
Analytical Marginalisation: A rigorous method to propagate beam uncertainty into cosmological constraints without sampling the beam space, significantly accelerating inference while avoiding the bias associated with fixed-beam assumptions.

Results
The framework was tested against a suite of unseen beams (validation set) and various cosmological signal injections:

Fixed-Beam Failure: When conditioned on an incorrect beam (even one with a small geometric mismatch), the fixed-beam approach resulted in residuals at the tens-of-kelvin level and strongly biased recovery of the 21-cm signal and foreground spectral indices.
Marginalised Success: Using the marginalised framework, the authors recovered the true cosmological inputs at approximately the instrumental-noise level (RMSE ~25 mK) for the vast majority of cases across 11 unseen beam realisations and varying signal profiles.
Robustness to Training Set Size: Reducing the training set from 400 to 100 beams (the minimum required for the PCA truncation) did not degrade signal recovery, suggesting that the method is not limited by the sheer number of simulations but by the number of retained modes.
Extrapolation Limits: The method demonstrated the ability to extrapolate to beams slightly outside the training support, though performance degraded for the most extreme realisations lying far outside the training prior volume.

Significance
The paper claims that this framework provides a scalable, statistically rigorous route towards hardware-accelerated uncertainty quantification in global 21-cm cosmology. By moving away from the assumption of a single, fixed beam template and instead treating physical antenna properties as nuisance quantities to be marginalised, the authors establish a practical foundation for robust, evidence-based detection of the sky-averaged 21-cm signal. This approach ensures that instrumental uncertainties are rigorously propagated into the final cosmological constraints, preventing artificially narrow posteriors and biased signal recovery that have plagued previous analyses. The work suggests that future analyses need not rely on prohibitively large simulation suites, provided the training data spans the physically relevant directions of instrumental variation.

Technical Summary: Towards End-to-End Bayesian Forward Models in Global 21-cm Cosmology

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