🔭 astrophysics

Implicit inference of the reionization history with higher-order statistics of the 21-cm signal

This paper demonstrates that combining higher-order statistics, particularly Betti numbers, with cylindrical power spectra significantly enhances the constraining power on the Universe's reionization history using future Square Kilometre Array 21-cm observations, outperforming traditional Gaussian statistics alone.

Original authors: Nicolas Cerardi, Sambit K. Giri, Michele Bianco, Davide Piras, Emmanuel de Salis, Massimo De Santis, Merve Selcuk-Simsek, Philipp Denzel, Kelley M. Hess, M. Carmen Toribio, Franz Kirsten, Hatem Ghorbe

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicolas Cerardi, Sambit K. Giri, Michele Bianco, Davide Piras, Emmanuel de Salis, Massimo De Santis, Merve Selcuk-Simsek, Philipp Denzel, Kelley M. Hess, M. Carmen Toribio, Franz Kirsten, Hatem Ghorbel

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 Big Picture: Listening to the Cosmic "Dark Ages"

Imagine the universe as a giant, dark room. For a long time after the Big Bang, this room was filled with a thick, invisible fog made of neutral hydrogen gas. Nothing was lit up yet; this was the "Cosmic Dark Ages."

Eventually, the first stars and galaxies turned on like lightbulbs. Their light started to burn holes in the fog, creating bubbles of clear air. This process is called Reionization.

Scientists want to know exactly how this happened. Did the lightbulbs turn on all at once? Did they flicker slowly? Did they burn out quickly? To find out, we need to "see" the fog. We can't see it with regular eyes, but we can listen to it using a special radio signal called the 21-cm signal.

The Problem: A Noisy Room

The Square Kilometre Array (SKA), a massive new radio telescope, is about to listen to this signal. But there are two big problems:

The Signal is Faint: The radio signal from the fog is incredibly weak, like trying to hear a whisper in a stadium full of cheering fans.
The Signal is Weird: The fog doesn't just fade away evenly. It breaks apart into complex, lumpy shapes (bubbles). This makes the data "non-Gaussian," which is a fancy way of saying the patterns are messy and don't follow simple, smooth curves.

The Solution: A New Way to "Read" the Data

The authors of this paper built a smart computer system (using a method called Implicit Inference) to figure out the history of the fog.

Think of it like this: Imagine you are trying to guess the shape of a hidden object inside a box by shaking it and listening to the sound.

Old Method (Power Spectrum): This is like listening to the average volume of the shake. It tells you how much stuff is in the box, but it misses the details of the shape.
New Method (Higher-Order Statistics): This is like listening to the rhythm and the topology of the shake.
- Betti Numbers: These count the "holes" and "loops" in the fog. It's like counting how many tunnels or islands are in the fog.
- Bispectrum: This looks at how three different parts of the signal interact with each other, like checking if three specific notes in a song always play together.

What They Did

The researchers created thousands of fake universes on a computer. They simulated the fog turning into bubbles at different speeds (Early, Middle, and Late reionization). They then added "static" (noise) to these simulations to mimic what the real telescope will see.

They trained their AI on these fake universes to learn: "If the data looks like this, the fog was probably clearing up at this specific rate."

The Results: The Power of Teamwork

Here is what they found, using simple comparisons:

The "Lumpy" Stats are Better: The "Betti numbers" (counting the holes in the fog) were actually better at guessing the history than the traditional "Power Spectrum" (average volume) on their own. It's like realizing that counting the number of islands in a lake tells you more about the landscape than just measuring the water's average depth.
The "Bispectrum" is a Bit Picky: The "Bispectrum" (the three-way interaction) didn't do much on its own. It was like a specialist who is great at one specific thing but not very helpful in general.
The Winning Combo: The magic happened when they combined the tools.
- Power Spectrum + Betti Numbers: This was the best team. It improved the accuracy by about 33%. It's like having a map (Power Spectrum) and a compass (Betti Numbers) together; you get a much clearer picture of where you are.
- Adding the Bispectrum: Adding the Bispectrum helped a little bit more, but only in certain situations.

A Crucial Twist: It Depends on the Fog

The most interesting discovery is that the "best" tool changes depending on how much fog is left.

When the fog is mostly gone (low neutral fraction), the "Betti numbers" are super helpful.
When the fog is still thick (high neutral fraction), the "Bispectrum" suddenly becomes useful.
The Lesson: You can't just use one tool for the whole job. You have to know what stage of the process you are looking at to pick the right combination of tools.

Conclusion

This paper proves that to understand the history of the universe's first light, we shouldn't just listen to the "average" sound. We need to listen to the complex, lumpy, and topological patterns of the signal. By combining different types of data analysis (like counting holes and measuring interactions), the SKA telescope will be able to reconstruct the history of the universe's "dark ages" much more accurately than previously thought.

In short: To solve the mystery of the cosmic fog, don't just count the volume; count the holes, check the connections, and mix your tools depending on how thick the fog still is.

Technical Summary: Implicit Inference of the Reionization History with Higher-Order Statistics of the 21-cm Signal

Problem Statement
The Epoch of Reionization (EoR) represents a critical phase transition in cosmic history, yet reconstructing its timeline remains challenging. Traditional probes, such as the Lyman- $\alpha$ forest and CMB optical depth, offer indirect views or integrated constraints. The 21-cm signal from neutral hydrogen (HI) offers a direct, three-dimensional tomographic view of the intergalactic medium (IGM). However, extracting the global neutral hydrogen fraction ( $\bar{x}_{HI}$ ) and the reionization history from 21-cm data is hindered by the signal's extreme faintness, bright foregrounds, and, crucially, its highly non-Gaussian nature. The standard summary statistic, the power spectrum (PS), is a two-point statistic that fully describes Gaussian fields but fails to capture the complex morphology and topology of ionized bubbles. Furthermore, the likelihood function connecting summary statistics to astrophysical parameters is often intractable, particularly for higher-order statistics or when complex instrumental effects are modeled. Traditional Markov Chain Monte Carlo (MCMC) methods struggle with the computational cost of evaluating millions of simulations and the intractability of the likelihood.

Methodology
The authors employ a state-of-the-art Implicit Inference framework, specifically Neural Posterior Estimation (NPE), to bypass the need for an explicit analytical likelihood function. The methodology proceeds as follows:

Simulation and Data Generation:
- Forward Model: The semi-numerical code 21cmFAST is used to generate a large suite of mock 21-cm lightcones. The model varies six key astrophysical parameters governing galaxy UV properties and star formation ( $f_{*,10}, \alpha_*, f_{esc,10}, \alpha_{esc}, M_{turn}, t_*$ ) while fixing X-ray parameters.
- Sampling Strategy: To avoid the bimodal distribution of reionization histories resulting from uniform priors on astrophysical parameters, a custom rejection sampling strategy using the 21cmEMU emulator was implemented. This ensured a balanced distribution of $\bar{x}_{HI}$ across the redshift bins of interest ( $z=8.0, 7.2, 6.5$ ).
- Instrumental Effects: Mock observations were generated for the SKA-Low AA* configuration (307 stations). The pipeline incorporates Gaussian noise realizations for 100-hour and 1000-hour integration times, uv-coverage convolution, resolution filtering (removing modes beyond the core baseline), and the subtraction of the spatial mean to mimic the lack of zero-spacing baselines.
Summary Statistics:
The study evaluates the information content of several statistics:
- Gaussian (2-point): Spherically averaged power spectrum (PS1D) and cylindrically averaged power spectrum (PS2D).
- Non-Gaussian (Higher-order):
  - Bispectrum: Reduced bispectrum in Equilateral and Squeezed-limit configurations.
  - Betti Numbers: Topological invariants ( $\beta_0, \beta_1, \beta_2$ ) calculated as a function of signal intensity thresholds, capturing the connectivity and voids of ionized regions.
Inference Framework:
- Neural Architecture: The framework uses Variational Mutual Information Maximization (VMIM) to compress the high-dimensional summary statistics into a latent representation, followed by NPE to learn the posterior distribution $p(\bar{x}_{HI} | t)$ .
- Training: The network was trained on $\sim330,000$ pairs of ( $\bar{x}_{HI}$ , statistic) derived from $\sim15,900$ unique simulations with multiple noise realizations.
- Calibration: The posteriors were validated using the Test of Accuracy with Random Points (TARP) to ensure unbiased and well-calibrated inference.

Key Contributions and Results
The study systematically compares the constraining power of these statistics using the Figure of Merit (FoM), defined as the inverse of the posterior volume (scaled by dimension).

Superiority of Non-Gaussian Statistics: Betti numbers alone were found to be, on average, more informative than the power spectra (PS1D and PS2D). The bispectrum alone provided limited constraints, likely due to the loss of large-scale information from angular cuts and the cancellation of 2-point information in the reduced bispectrum.
Synergy of Combined Statistics: The most significant finding is the complementary nature of the statistics. Combining the cylindrical power spectrum (PS2D) with Betti numbers improved the mean FoM by $\sim0.25$ dex (a $\sim33\%$ reduction in the uncertainty $\sigma(\bar{x}_{HI})$ ). The full combination of PS2D + Betti + Bispectrum yielded the tightest constraints overall.
State-Dependence: The constraining power of the statistics is not uniform; it depends on the reionization state ( $\bar{x}_{HI}$ $\overset{x}{ˉ}_{H I}$ ).
- Betti numbers provided significant complementary information across all stages of reionization.
- The bispectrum's utility varied: it improved constraints in low-to-mid neutral fraction scenarios ( $\bar{x}_{HI} \lesssim 0.6$ ) but could degrade constraints in highly neutral regimes (e.g., the "Late" model), suggesting that the optimal choice of statistics is epoch-dependent.
Noise Robustness: The benefits of combining higher-order statistics were evident in both 100-hour and 1000-hour noise scenarios, with deeper observations naturally yielding tighter constraints.

Significance and Claims
The authors claim that this work demonstrates the necessity of moving beyond the power spectrum to fully exploit the information content of future 21-cm observations. By utilizing implicit inference, the study successfully bypasses the intractability of likelihoods for complex, non-Gaussian statistics. The results indicate that a hybrid approach, combining the power spectrum with topological descriptors (Betti numbers) and, where appropriate, the bispectrum, can significantly maximize the scientific return of the SKA-Low. This approach allows for the direct inference of the reionization history ( $\bar{x}_{HI}$ ) without relying on intermediate astrophysical parameter mapping, thereby offering a more robust and efficient pathway to charting the end of the cosmic dark ages. The paper acknowledges limitations regarding foreground residuals and model misspecification (training and testing on the same simulator), identifying these as critical areas for future investigation.