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Field-Level Inference of Primordial Non-Gaussianity with the Quijote Simulation Suite

This paper demonstrates, for the first time using the Quijote simulation suite, that a Bayesian field-level inference algorithm (BORG) significantly outperforms traditional power spectrum and bispectrum estimators in constraining local primordial non-Gaussianity ( $f_{\rm NL}^{\rm local}$ ) by jointly inferring initial conditions and nuisance parameters from realistic halo catalogues.

Original authors: Adam Andrews, Jens Jasche, Guilhem Lavaux, William Coulton, Francisco Villaescusa-Navarro, Marco Baldi, Drew Jamieson, Gabriel Jung, Dionysios Karagiannis, Florent Leclercq, Michele Liguori, Marco Mar

Published 2026-03-24

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CC BY 4.0

Original authors: Adam Andrews, Jens Jasche, Guilhem Lavaux, William Coulton, Francisco Villaescusa-Navarro, Marco Baldi, Drew Jamieson, Gabriel Jung, Dionysios Karagiannis, Florent Leclercq, Michele Liguori, Marco Marinucci, Benjamin Wandelt

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

Imagine the universe as a giant, cosmic loaf of bread rising in an oven. The "dough" is the dark matter and gas that eventually forms galaxies, stars, and us. But before the oven even turned on, the dough had to be mixed. The way that dough was mixed in the very first split second of the universe (a period called Cosmic Inflation) determines how the bread rises today.

Scientists want to know: Was the mixing perfectly smooth and uniform? Or were there tiny, random lumps and swirls? These "lumps" are called Primordial Non-Gaussianity. If we can find them, it tells us exactly how many different "ingredients" (fields) were swirling around during the Big Bang.

This paper is about a new, super-smart way to find those lumps.

The Old Way: Counting Crumbs

Traditionally, cosmologists have tried to understand the universe by taking a big picture, chopping it up into tiny pieces, and counting the crumbs. They measure the "Power Spectrum" (how much clumping there is on average) and the "Bispectrum" (how those clumps relate to each other).

Think of this like trying to understand a complex symphony by only counting how many times the violin plays a note, or how often the drums hit. You get some information, but you miss the melody, the harmony, and the emotion. You are throwing away a massive amount of the music's detail.

The New Way: Listening to the Whole Song (Field-Level Inference)

The authors of this paper are using a method called Field-Level Inference (FLI). Instead of chopping up the data, they treat the entire universe like a single, continuous song.

Imagine you are a detective trying to figure out what a crime scene looked like before the police arrived.

The Old Way: You look at the scattered evidence (a broken vase, a muddy footprint) and guess what happened based on statistics.
The New Way (FLI): You have a time machine. You start with a blank slate (the initial conditions) and run a simulation forward, second by second, to see if you can recreate the crime scene exactly as you see it now. If your simulation matches the real scene perfectly, you know exactly what happened at the start.

In this paper, the authors use a sophisticated computer program (called BORG) to do exactly this. They take the map of galaxies we see today and run the physics "backwards" to reconstruct the universe's initial state.

The "Quijote" Test Kitchen

To prove their method works, they didn't just look at real data yet (which is messy and incomplete). Instead, they went into a "test kitchen" called the Quijote Simulation Suite.

Imagine they baked 30 different loaves of cosmic bread in a computer.

Some loaves were baked with a "perfectly smooth" mix (no lumps).
Some had a "huge lump" added to the mix.
Some had a "negative lump" (a weird dip).

They then hid the recipe (the initial conditions) and the amount of "lump" (the parameter $f_{NL}$ ) from themselves. They fed the finished bread (the simulated galaxy maps) into their BORG algorithm and asked: "Can you figure out how much lump was in the original mix?"

The Results: A New Level of Clarity

The results were impressive:

Better Precision: Their new method was about 30% more precise than the old "counting crumbs" methods. It's like going from guessing the weight of a person by looking at their shadow, to actually weighing them on a scale.
Reconstructing the Past: Not only did they guess the "lump" amount correctly, but they also successfully reconstructed the entire initial map of the universe. They could see the "ghost" of the early universe's structure, which is usually impossible to see directly.
Resolution Matters: They found that the more detail they looked at (higher resolution), the better their guesses became. It's like looking at a painting: the more pixels you have, the clearer the picture.

Why This Matters

We are about to launch massive new telescopes (like the Euclid mission) that will map millions of galaxies. This is a treasure trove of data.

If we use the old methods, we are leaving a lot of that treasure on the table. This paper shows that by using Field-Level Inference, we can unlock the full potential of these new telescopes. It allows us to peer deeper into the first fraction of a second of the universe's existence, potentially revealing the number of fields that drove the Big Bang and helping us understand the fundamental laws of physics.

In short: The authors built a time machine that doesn't just guess the past; it reconstructs it in high definition, proving that we can finally hear the full symphony of the universe, not just count the notes.

1. Problem Statement

The physical mechanism generating the initial conditions of the Universe remains a central open question in cosmology. While cosmic inflation is the leading theory, distinguishing between single-field and multi-field inflation models relies heavily on detecting Local Primordial Non-Gaussianity (PNG), parameterized as $f_{\text{NL}}^{\text{local}}$ .

Current Limitations: Current constraints from the Cosmic Microwave Background (CMB) have reached the cosmic-variance limit on large scales. Next-generation galaxy redshift surveys (e.g., Euclid, LSST) offer vast volumes but traditional analysis methods (power spectrum and bispectrum estimators) compress data into summary statistics. This compression discards higher-order correlations and non-Gaussian features essential for constraining $f_{\text{NL}}$ with the required precision (targeting $\sigma(f_{\text{NL}}) \sim 1$ ).
The Challenge: There is a need for analysis techniques that exploit the full statistical content of galaxy fields without information loss, while simultaneously reconstructing initial conditions and marginalizing over nuisance parameters (bias, initial phases) in realistic, non-linear environments.

2. Methodology: Field-Level Inference (FLI)

The authors employ the BORG (Bayesian Origin Reconstruction from Galaxies) algorithm, a Bayesian hierarchical forward-modelling framework. Unlike traditional estimators, FLI fits a physical model directly to the observational data field.

Forward Model:
- Initial Conditions: Generates a Gaussian white-noise field ( $\epsilon$ ) convolved with a transfer function.
- Primordial Non-Gaussianity: Introduces local PNG via the Bardeen potential: $\Phi_{\text{NG}} = \phi_G + f_{\text{NL}}(\phi_G^2 - \langle\phi_G^2\rangle)$ .
- Structure Formation: Evolves initial conditions to redshift $z=0.5$ using Second-Order Lagrangian Perturbation Theory (2LPT).
- Bias Model: Maps the dark matter field to a halo field using a bias expansion that includes linear ( $b_1$ ), quadratic ( $b_2$ ), tidal ( $b_K$ ), and Laplacian ( $b_\nabla$ ) terms. Crucially, it includes scale-dependent bias terms ( $b_\phi, b_{\phi\delta}$ ) arising from PNG coupling, modeled via the universal mass function.
- Likelihood: Assumes a Gaussian likelihood for halo counts in voxels, comparing observed data ( $\rho_h^{\text{data}}$ ) with model predictions ( $\rho_h^{\text{pred}}$ ).
Sampling Framework:
- Uses Hamiltonian Monte Carlo (HMC) to sample the high-dimensional initial conditions (white noise field).
- Uses Slice Sampling and Gibbs sampling for nuisance parameters ( $f_{\text{NL}}$ , bias parameters, noise level $\sigma_G$ ).
- Inference Strategy: The study performs joint inference of $f_{\text{NL}}$ , initial conditions, and bias parameters, marginalizing over the latter to avoid biases.
Data Source:
- The Quijote-PNG simulation suite: 30 dark-matter-only $N$ -body simulations in a $1 (\text{Gpc}/h)^3$ volume.
- Simulations cover three ground-truth values: $f_{\text{NL}} = 0, +100, -100$ .
- Analysis performed at redshift $z=0.5$ with halos above $M_{\text{min}} = 10^{13.5} h^{-1} M_\odot$ .

3. Key Contributions

First Joint Inference in Realistic Simulations: This is the first demonstration of joint field-level inference of $f_{\text{NL}}$ , initial conditions, and bias parameters applied to realistic halo catalogues generated from full $N$ -body simulations (rather than self-consistent mock data).
Validation of Forward Model: The authors validate the 2LPT forward model and bias parameterization against $N$ -body truth, quantifying the model mismatch noise ( $\sigma_G$ ).
Resolution and Model Sensitivity: The study systematically tests how inference constraints improve with increased voxel resolution and different structure formation models (1LPT, 2LPT, tCOLA).
Benchmarking: Provides a direct comparison between Field-Level Inference (BORG) and traditional Fisher forecasts based on power spectrum + bispectrum estimators.

4. Key Results

Constraint Improvement:
- Compared to a combined power spectrum and bispectrum estimator (Fisher forecast), the BORG field-level inference achieves a $\sim 1.3\times$ improvement in the uncertainty of $f_{\text{NL}}$ .
- Fisher Forecast: $\sigma(f_{\text{NL}}) \approx 27$ (using $P+B$ ).
- BORG FLI: $\sigma(f_{\text{NL}}) \approx 21$ (marginalizing over initial conditions and bias).
- This demonstrates that FLI captures information beyond 2-point and 3-point statistics.
Recovery of Ground Truth:
- Across 30 simulations with varying $f_{\text{NL}}$ (0, $\pm 100$ ), the method successfully recovers the ground truth values within the $1\sigma$ confidence interval in $\sim 68\%$ of cases.
- The method correctly marginalizes over the initial conditions and bias parameters, despite degeneracies (e.g., between $f_{\text{NL}}$ and linear bias $b_1$ ).
Field Reconstruction:
- Large Scales: High cross-correlation ( $r \approx 0.9$ ) between inferred and ground-truth fields for the gravitational potential, matter density, and halo distribution.
- Small Scales: Correlation drops at smaller scales ( $k > 0.1 h \text{Mpc}^{-1}$ ) due to the limitations of the 2LPT approximation in capturing full non-linear $N$ -body dynamics.
Resolution and Model Dependence:
- Resolution: Increasing the resolution (decreasing voxel size from $125$ to $31.25 h^{-1} \text{Mpc}$ ) significantly tightens constraints on $f_{\text{NL}}$ , confirming that small-scale information is valuable.
- Structure Formation Model: At the tested resolution ( $k_{\text{max}} \approx 0.1 h \text{Mpc}^{-1}$ ), switching from 1LPT to 2LPT or tCOLA yields only marginal improvements in $f_{\text{NL}}$ constraints. This suggests that at these scales, the bias model and resolution are the limiting factors, not the fidelity of the structure formation solver.

5. Significance and Future Outlook

Transformative Potential: The results underscore that Field-Level Inference is a viable and superior alternative to summary statistics for upcoming surveys like Euclid. It leverages the full statistical power of the data to constrain inflationary physics.
Readiness for Data: By validating the method on realistic $N$ -body simulations rather than idealized mocks, the authors demonstrate the robustness of the pipeline for application to real observational data.
Future Work:
- Refining the bias model for $b_\phi$ and $b_{\phi\delta}$ (currently fixed by the universal mass function) to reduce systematic errors.
- Extending the framework to infer other primordial parameters ( $n_s, A_s$ ) and non-Gaussian shapes ( $f_{\text{NL}}^{\text{equi}}, f_{\text{NL}}^{\text{orth}}$ ).
- Pushing to higher resolutions to capture non-linear scales, which requires improved forward models and noise modeling.

In conclusion, this paper establishes that field-level inference can robustly constrain primordial non-Gaussianity in realistic halo catalogues, outperforming traditional estimators and paving the way for high-precision cosmology with next-generation galaxy surveys.