Field-level multi-tracers simulation-based inference of cosmological parameters from 3D maps

This paper presents a proof-of-concept Simulation-Based Inference pipeline that utilizes neural emulators trained on hydrodynamical simulations to extract cosmological parameters from 3D galaxy and neutral hydrogen maps, demonstrating that field-level multi-tracer analysis significantly outperforms traditional summary statistics by improving constraints by factors of 2 to 7 while robustly marginalizing over astrophysical uncertainties.

The Gist

The Big Picture: Reading the Universe's "Fingerprint"

Imagine the universe as a giant, complex 3D puzzle. For decades, cosmologists have tried to solve this puzzle by looking at the "summary statistics"—basically, they took the puzzle, flattened it out, and counted how many pieces of a certain color were next to each other. This is like trying to understand a symphony by only listening to the average volume of the music, ignoring the melody, the instruments, and the rhythm.

This paper proposes a new way to listen. Instead of just counting notes, the authors built a system that listens to the entire symphony (the full 3D map of the universe) to figure out the rules of the universe (cosmological parameters like how much matter exists and how clumpy it is).

The Problem: The Universe is Too Complicated to Simulate

To understand the universe, scientists use supercomputer simulations. However, simulating the universe with all its details (gas, stars, black holes) is like trying to simulate a hurricane in a bathtub; it takes millions of hours of computer time. You can't run enough of these perfect simulations to test every possible version of the universe.

Usually, scientists use "approximate" simulations (like a rough sketch) and then try to guess what the "perfect" version would look like. But this guesswork often throws away valuable information, especially the messy, non-linear details that happen on small scales.

The Solution: The "AI Translator" (Emulators)

The authors created a clever workaround using Artificial Intelligence (AI).

1. The Sketch (Fast Simulations): They first run fast, rough simulations of dark matter (the invisible skeleton of the universe). These are cheap and quick to make.
2. The Translator (The Emulator): They trained a neural network (an AI) on a limited set of perfect, high-detail simulations. This AI learned how to "translate" the rough dark matter sketch into a detailed map of Galaxies and Neutral Hydrogen (HI).
   • Analogy: Think of the AI as a master chef who has tasted a few perfect dishes. Now, if you give them a list of basic ingredients (the rough sketch), they can instantly cook up a perfect meal without needing to start from scratch every time.

The Experiment: Two Ways to Listen

The team tested two different ways to use this AI to learn about the universe:

• Method A: The Summary Sheet (Power Spectrum)
  They took the detailed maps the AI generated and compressed them into a simple summary statistic called a "Power Spectrum." This is like taking the symphony and turning it into a single graph showing the average volume at different frequencies.
• Method B: The Full Recording (Field-Level Inference)
  They fed the entire 3D map directly into a new AI system. This system looked at the full, un-compressed data, preserving all the complex shapes, clumps, and structures.
  • Analogy: Method A is reading a book report. Method B is reading the actual book, word for word, including the footnotes and the messy handwriting in the margins.

They also tested using two different "tracers" (Galaxies and Hydrogen gas) together, rather than just one.

• Analogy: Trying to solve a mystery by only looking at footprints (Galaxies) is hard because the ground is uneven. But if you also look at tire tracks (Hydrogen gas) and see how they overlap, you get a much clearer picture of what happened.

The Results: Why "Full 3D" Wins

The results were clear and surprising:

1. The Full 3D Map is King: The method that looked at the full 3D maps (Method B) was 3 times better at pinning down the universe's secrets than the method that used the summary sheet (Method A).
   • Why? The summary sheet throws away the "messy" details. The full 3D map keeps the non-linear structures (the complex clumps) that hold the most valuable clues about the universe's history.
2. Two Tracers are Better Than One: Combining Galaxy maps and Hydrogen maps improved the precision by a factor of 2 to 7 compared to using just one.
   • Why? Galaxies are "spotty" and noisy (like a sparse crowd of people), while Hydrogen gas is a smooth, continuous fog. When you combine them, the smooth fog fills in the gaps of the spotty crowd, canceling out the noise.
3. Robustness: Even when the authors told the AI, "We don't know exactly how stars form or how black holes behave" (marginalizing over astrophysical parameters), the 3D method still worked well. The summary method failed miserably in this scenario, producing very vague answers.

The Catch: It's Expensive

There is a trade-off. While the "Full 3D" method is much more accurate, it is also much more computationally expensive.

• Analogy: Reading the whole book (3D method) takes longer and requires more brainpower than reading the book report (Summary method), but you get a much deeper understanding of the story.

Conclusion

The paper demonstrates that to get the most out of future telescopes (like the ones that will map the entire sky), we need to stop compressing the data into simple summaries. Instead, we should use AI to analyze the full, raw 3D structure of the universe. By combining different types of cosmic "tracers" and looking at the full picture, we can unlock a much deeper understanding of the universe's composition and history.

Note: The authors emphasize that this is a "proof-of-concept." They used idealized simulations without real-world noise (like telescope errors or atmospheric interference). While the results are promising, they acknowledge that applying this to real-world data will require further work to handle those messy real-life factors.

Technical Summary

Technical Summary: Field-level multi-tracers simulation-based inference of cosmological parameters from 3D maps

Problem Statement
Current and upcoming Large-Scale Structure (LSS) surveys (e.g., Euclid, SKAO, DESI) will generate data volumes where statistical uncertainties are subdominant to systematic effects and modeling accuracy. Extracting maximum cosmological information requires moving beyond compressed summary statistics, such as the power spectrum, which discard non-Gaussian information encoded in the full matter and tracer fields. Traditional likelihood-based inference (e.g., MCMC) is computationally prohibitive when applied to full-field data from high-fidelity hydrodynamical simulations, as these simulations are too expensive to run across the necessary high-dimensional parameter space. Furthermore, standard approaches often rely on Gaussian approximations that fail to capture the complex, non-linear gravitational dynamics and baryonic processes governing structure formation.

Methodology
The authors develop a proof-of-concept Simulation-Based Inference (SBI) pipeline designed to infer cosmological parameters $\{\Omega_m, \sigma_8\}$ from galaxy number counts and neutral hydrogen (HI) intensity mapping. The pipeline integrates three core components:

1. Fast Approximate Simulations: The pipeline utilizes FastPM to generate approximate dark matter (DM) maps at low computational cost. These maps serve as the input for the subsequent emulation step.
2. Neural Emulators (HGL): A modified U-Net architecture, termed HGL (HALOgen-like), is trained on the CAMELS (Cosmology and Astrophysics with MachinE Learning Simulations) Latin Hypercube suite. This emulator learns the mapping from approximate DM fields to full 3D galaxy and HI fields. Crucially, the emulator is conditioned on both cosmological parameters ($\Omega_m, \sigma_8$) and astrophysical feedback parameters (Supernovae and AGN feedback strengths), allowing it to generate realistic tracer fields without running full hydrodynamical simulations for every inference step.
3. Simulation-Based Inference (SBI): The authors employ Neural Posterior Estimation (NPE) using a Masked Autoregressive Flow (MAF) to learn the posterior distribution $p(\theta | d)$ directly from simulated parameter-data pairs. They compare two distinct data representations:
   • Case P (Power Spectrum): Inference performed on the isotropic power spectrum $P(k)$ extracted from the emulated maps.
   • Case M (Field-Level): Inference performed on latent representations of the full 3D maps. A Convolutional Neural Network (CNN) compresses the full field into a 64-dimensional latent space, which is then fed to the normalizing flow. This is tested in both 2D (projected) and full 3D configurations.

The study investigates single-tracer (galaxy-only, HI-only) and multi-tracer (combined) scenarios, with analyses performed both with fixed astrophysical parameters and with astrophysical parameters marginalized over.

Key Contributions

• Pipeline Development: The authors present a scalable SBI framework that combines fast DM simulations, neural emulators for baryonic observables, and neural density estimators. This approach bypasses the need for expensive hydrodynamical simulations during the inference phase.
• Multi-Tracer Analysis: The work explicitly demonstrates the integration of galaxy and HI fields within an SBI framework, assessing the information gain from cross-correlations.
• Field-Level vs. Summary Statistics: The paper provides a systematic comparison between inference based on the power spectrum and inference based on full-field latent representations, specifically isolating the impact of 3D structural information versus 2D projections.

Results

• Precision Gains: Moving from summary statistics (power spectrum) to field-level inference yields a consistent gain in constraining power of approximately a factor of 3. The most precise and well-calibrated posteriors are obtained using full 3D maps (Case 3DM).
• Multi-Tracer Synergy: Combining galaxy and HI fields improves constraints on $\{\Omega_m, \sigma_8\}$ by a factor of 2 to 7 compared to single-tracer cases, depending on the configuration. The multi-tracer approach effectively breaks degeneracies and reduces the impact of tracer-specific noise (e.g., shot noise in galaxy maps).
• Robustness to Astrophysics: When marginalizing over astrophysical parameters, power-spectrum-based inference degrades significantly, often resulting in uninformative posteriors (particularly for $\sigma_8$). In contrast, field-level inference retains substantial cosmological information, suggesting that non-Gaussian features encoded in the full field help disentangle cosmology from astrophysical uncertainties.
• Dimensionality: Flattening 3D maps to 2D projections (2DM) does not yield the same gains as full 3D analysis, indicating that line-of-sight structures contain significant cosmological information that is lost in projection.
• Bias and Calibration: The 3DM configurations generally exhibit the smallest bias (typically $<1.5\%$) and show good calibration across the parameter space, even when marginalizing over astrophysics.

Significance and Claims
The paper positions itself as a proof-of-concept exploration. The authors claim that their results demonstrate the feasibility and potential of field-level SBI for multi-tracer LSS analysis. They assert that:

1. Information Retention: Field-level inference captures information irreversibly lost when reducing data to the power spectrum, particularly non-Gaussian features sensitive to $\sigma_8$.
2. Astrophysical Marginalization: The approach allows for the marginalization over complex astrophysical effects without a catastrophic loss of cosmological constraints, a capability where traditional summary-statistic methods struggle.
3. Future Applicability: While the current analysis is idealized (neglecting instrumental noise, survey systematics, and foregrounds), the framework establishes a baseline for applying these methods to upcoming surveys.

The authors explicitly note limitations: the study relies on emulated maps generated within a single simulation framework (CAMELS) and does not yet address out-of-distribution (OOD) shifts between different simulation codes or real observational data. They emphasize that incorporating realistic survey effects and improving emulator faithfulness are necessary next steps before applying this pipeline to real data.