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Primordial non-Gaussianity -- Fast simulations and persistent summary statistics

This paper introduces a large suite of fast primordial non-Gaussianity simulations (PNG-pmwd) to demonstrate that topological persistent statistics, particularly when applied to large-mass halos, offer superior constraints on equilateral non-Gaussianity and maintain robust transferability across different simulation suites compared to traditional power spectrum and bispectrum analyses.

Original authors: Juan Calles, Gabriella Contardo, Jorge Noreña, Jacky H. T. Yip, Gary Shiu

Published 2026-08-21

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CC BY 4.0

Original authors: Juan Calles, Gabriella Contardo, Jorge Noreña, Jacky H. T. Yip, Gary Shiu

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 began as a nearly uniform soup of energy, but tiny, random fluctuations in that early state eventually grew into the vast cosmic web we see today: a sprawling network of galaxies, filaments, and empty voids. For decades, cosmologists have assumed these initial fluctuations were perfectly random, following a simple bell-curve pattern known as a Gaussian distribution. However, many theories about the universe's first moments suggest that this randomness was not perfect, but contained subtle, non-random deviations called primordial non-Gaussianity. Detecting these deviations is like finding a fingerprint left by the physics of the Big Bang itself, offering a way to distinguish between different theories of how the universe started. While the cosmic microwave background—the afterglow of the Big Bang—has provided tight constraints on this idea, scientists are now turning their attention to the large-scale structure of the universe, mapping the distribution of matter across billions of light-years to find these hidden signatures.

To do this, researchers need to understand how the invisible scaffolding of the universe changes when those initial conditions vary. In a new study, a team of scientists introduced a massive, custom-built collection of computer simulations designed specifically to test how different statistical tools can spot these subtle deviations. They created over 22,000 distinct catalogs of dark matter halos—clumps of invisible matter that act as the gravitational seeds for galaxies. These simulations were not just random variations; they were carefully constructed to explore how the universe would look if the initial randomness were tweaked in specific ways, known as "local" and "equilateral" shapes, while also changing other fundamental properties like the total amount of matter and how clumpy the universe is. The goal was to see which methods could best read the map of the cosmos to reveal these hidden initial conditions.

The researchers tested two main approaches to reading these maps. The first was a traditional method that measures how matter clusters at different distances, looking at the strength of connections between points. The second was a newer, more geometric approach called topological data analysis. Imagine looking at a cloud of points and asking not just how close they are, but how they connect to form shapes: do they form isolated islands, long chains, or closed loops? This method tracks the birth and death of these shapes as you zoom in and out, creating a summary of the universe's connectivity. The team fed these maps into powerful computer programs, essentially training artificial intelligence to look at the data and guess the specific type of initial randomness used to create it. They tested these tools on halos of different sizes, from small, barely resolved clumps to massive, well-defined structures.

The results revealed a clear winner in the race to detect these cosmic fingerprints. The most effective tool turned out to be a surprisingly simple set of descriptive statistics derived from the topological shapes. By calculating basic measures like the average size and count of the loops and voids in the cosmic web, the researchers could constrain the strength of the initial non-Gaussianity more accurately than with complex, high-powered neural networks or traditional clustering measurements. This finding suggests that the most useful information is not hidden in the most complex, long-lasting structures, but rather in the simpler, short-lived features that appear and disappear quickly in the data. Furthermore, the study showed that the size of the objects being measured matters immensely. The most massive halos provided the clearest signal, while including smaller, less resolved halos actually introduced noise that confused the results and made the measurements less precise.

A critical part of the research involved testing whether models trained on these fast, approximate simulations could be trusted to analyze data from more accurate, slower simulations. This is a vital question for the future, as running the most precise simulations for every possible scenario is too computationally expensive. The team found that the models could indeed transfer their knowledge from the fast simulations to the more accurate ones, but only under strict conditions. If the analysis included the smallest scales or the smallest halos, the predictions became biased and unreliable. The fast simulations, while good at capturing large-scale trends, tend to blur the details of the smallest structures, causing the artificial intelligence to misinterpret the data. However, when the researchers focused only on the most massive halos, the models performed robustly, maintaining their accuracy across the different simulation types.

Ultimately, this work provides a roadmap for future cosmological surveys. It demonstrates that we do not need the most complex algorithms to find the earliest signatures of the universe; sometimes, a straightforward look at the shape and connectivity of the cosmic web is more powerful. It also highlights a practical lesson for the field: to get the most accurate results, we must be careful about which parts of the universe we include in our analysis. By focusing on the largest, most reliable structures and using simple, interpretable summaries of their topology, scientists can better distinguish between different theories of the Big Bang. As upcoming telescopes prepare to map the universe with unprecedented detail, these findings suggest that the key to unlocking the secrets of our origins may lie in understanding the simple, enduring shapes of the cosmic web.

Technical Summary: Primordial Non-Gaussianity – Fast simulations and persistent summary statistics

Problem Statement
The statistical nature of primordial curvature perturbations remains a central open question in cosmology. While the Cosmic Microwave Background (CMB) provides stringent constraints on primordial non-Gaussianity (PNG), its information content is approaching the cosmic-variance limit. Consequently, Large Scale Structure (LSS) surveys (e.g., Euclid, LSST) are poised to serve as complementary probes. However, traditional LSS analyses relying on two-point statistics (the power spectrum) compress data and discard structural richness at small scales. Higher-order statistics like the bispectrum encode crucial PNG information but are challenging to extract and interpret due to non-linear gravitational evolution and galaxy biasing. Furthermore, the choice of tracers (halos) introduces a "topological bias," where different mass populations delineate the cosmic web differently. There is a need for robust, interpretable statistical frameworks capable of extracting higher-order information from LSS at small scales, particularly those that can generalize across different simulation suites and numerical approximations.

Methodology
The authors employ a likelihood-free inference framework using neural networks to map data vectors directly to the PNG amplitude parameters ( $f_{\text{NL}}^{\text{loc}}$ and $f_{\text{NL}}^{\text{equil}}$ ). The study utilizes two primary simulation suites:

PNG-pmwd Suite: A new dataset comprising 22,410 dark matter halo catalogs generated using the particle-mesh with derivatives (pmwd) solver, a fast N-body code implemented in JAX. This suite varies local and equilateral PNG shapes, as well as cosmological parameters ( $\Omega_m, \sigma_8$ ), using Latin hypercube designs.
QuijotePNG Suite: A set of full N-body simulations using the Gadget-III TreePM code, serving as a benchmark for transferability tests.

The methodology involves a systematic comparison of various summary statistics derived from these halo catalogs:

Topological Descriptors: Using Persistent Homology (PH) via $\alpha$ -DTM- $\ell$ filtration to generate persistence diagrams. These are vectorized using several methods: Persistence Images (PI), Histograms of Counts, Betti Curves, Persistent Betti Curves, Persistence Statistics (PD-statistics), Persistence Landscapes, and Persistence Silhouettes.
Traditional Clustering Statistics: The power spectrum (monopole, quadrupole, hexadecapole) and bispectrum monopole (combined as PSBS).
Mass Binning: Analyses are performed across seven distinct halo mass bins (from low-mass $H_{\text{Low}}$ to high-mass $H_{\text{High}}$ ) to assess the dependence of constraining power on tracer mass.
Architectures: Various neural architectures are tested, including Multilayer Perceptrons (MLP), Convolutional Neural Networks (CNN), DeepSets, and PersLay, alongside Gradient-Boosted Decision Trees (XGB) as a non-neural baseline.

Key Contributions

PNG-pmwd Simulation Suite: The release of a large-scale, open-access suite of 22,410 halo catalogs specifically designed for machine learning analyses of PNG, varying both local and equilateral shapes and cosmological parameters.
Systematic Benchmarking: A comprehensive evaluation of topological descriptors against traditional clustering statistics (PSBS) for inferring $f_{\text{NL}}$ , highlighting the performance of simple descriptive statistics (PD-statistics) over complex neural vectorizations.
Mass Dependence Analysis: A detailed investigation into how halo mass selection influences inference, revealing that small-mass halos often degrade performance by introducing noise, while large-mass halos provide the primary constraining power.
Transferability Assessment: An evaluation of whether models trained on "fast" (pmwd) simulations can robustly infer parameters on "full" (Quijote) simulations, identifying specific regimes (mass and scale) where transferability holds or fails.

Results

Optimal Summary Statistics: PD-statistics (simple descriptive statistics of birth/death/persistence values) consistently delivered the best performance for constraining both local and equilateral $f_{\text{NL}}$ , particularly for the equilateral template which is typically harder to detect. Combining PD-statistics with PSBS offered only marginal improvements, suggesting limited complementary information.
Mass Dependence: Constraining power is dominated by high-mass halos. Including small-mass halos (e.g., $H_{\text{Low}}$ ) degrades model performance by adding noise. For equilateral PNG, the best performance comes from the $H_{\text{Mid}}$ bin (mid- to high-mass), while excluding high-mass halos significantly weakens constraints.
Vectorization Efficiency: Complex neural vectorizations (DeepSets, PersLay) and Persistence Images (CNN) generally underperformed compared to simpler, handcrafted summaries like PD-statistics and Betti curves. DeepSets and PersLay faced challenges with convergence, computational cost, and input dimensionality.
Feature Importance: Contrary to the intuition that long-lived (high-persistence) features are most informative, the results suggest that short-lived, early-birth features (particularly in 0-cycles) carry significant cosmological information. Weighting schemes that amplify short-lived features ( $p < 0.5$ ) performed better than those emphasizing persistence.
Transferability: Models trained on PNG-pmwd simulations can transfer to QuijotePNG simulations, but this robustness is conditional. Transferability degrades when small scales or small-mass halos are included. The bias observed in transfer (e.g., upward offsets in predictions) is attributed to solver systematics where fast solvers may diffuse particles, leading to halo fragmentation and altered small-scale clustering. Restricting analysis to high-mass halos ( $H_{\text{High}}$ ) or excluding small-scale modes significantly improves transfer robustness.
Parameter Degeneracies: When $\Omega_m$ , $\sigma_8$ , and $f_{\text{NL}}$ are varied simultaneously, constraints on $f_{\text{NL}}$ weaken significantly due to degeneracies in halo abundance. Fixing cosmological parameters substantially improves performance.

Significance and Claims
The paper claims that simple topological summary statistics, specifically PD-statistics, offer a highly effective and computationally efficient alternative to complex neural architectures for inferring primordial non-Gaussianity. The study demonstrates that the information content regarding PNG is heavily concentrated in the large-mass end of the halo population, and that including poorly resolved small-mass halos introduces noise that obscures the signal.

Regarding the use of fast simulations for training, the authors claim that while transferability is possible, it requires careful handling of small scales and low-mass tracers to avoid systematic biases introduced by numerical approximations in fast solvers. The findings suggest that for likelihood-free inference to be robust across different simulation suites (and potentially to real data), one must either filter out unreliable small-scale features or focus on high-mass tracers that are less sensitive to solver-specific systematics. The work underscores the importance of understanding the specific systematics of simulation suites when applying machine learning models trained on one dataset to another.

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