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Differentiable Halo Mass Prediction and the Cosmology-Dependence of Halo Mass Functions

This paper introduces a differentiable 3D U-Net model trained on initial density fields to predict dark matter halo abundance and accurately capture its cosmological dependence, demonstrating superior performance in gradient-based parameter estimation and model extrapolation compared to traditional parametric halo mass functions.

Original authors: Jim Buisman, Florian List, Oliver Hahn

Published 2026-08-25

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CC BY 4.0

Original authors: Jim Buisman, Florian List, Oliver Hahn

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 is not a static stage but a vast, evolving tapestry woven from invisible threads of dark matter. These invisible clumps, known as haloes, act as the gravitational scaffolding for everything we see. Without them, galaxies would never have formed, and the stars and planets that make up our world would not exist. For decades, astronomers have tried to predict exactly how many of these dark matter clumps form, and how massive they become, based on the fundamental rules of the universe. This prediction, called the halo mass function, is a sensitive test of our cosmological models. If the rules of gravity or the nature of dark matter were slightly different, the number of massive galaxy clusters or tiny dwarf galaxies would change in a measurable way. However, calculating these numbers has traditionally been a slow, rigid process. Scientists usually run massive computer simulations that take days or weeks to complete, and then they count the resulting structures. Because this counting process involves sharp, discrete steps, it is mathematically impossible to ask the simulation how the result would change if the rules were tweaked just a tiny bit. This limitation has made it difficult to use these simulations for the most precise types of statistical analysis that modern astronomy requires.

A team of researchers at the University of Vienna has now found a way to bypass this bottleneck by teaching a computer to see the future of the universe directly from its past. Instead of running a slow, step-by-step simulation to watch gravity pull matter together, they trained a sophisticated neural network to look at the initial, smooth distribution of matter and instantly predict where the clumps will form and how heavy they will be. They built this system using a type of artificial intelligence architecture known as a U-Net, which is particularly good at recognizing patterns in three-dimensional data. The researchers fed the network data from thousands of fast, simplified simulations where the fundamental rules of the universe were slightly different in each case. The network learned to map the initial, smooth density of the universe to the final, clumpy distribution of dark matter haloes. Crucially, they designed the system so that it remains mathematically smooth and continuous, meaning the computer can calculate exactly how the prediction shifts when the underlying rules of the universe are adjusted.

The results show that this new approach is remarkably accurate. When the researchers tested the network on data it had never seen before, it successfully identified the regions that would become dark matter haloes with high precision. It correctly distinguished between empty space and the dense cores of future galaxies, and it could assign a likely mass to each clump. While the system was slightly more cautious about predicting the existence of the very largest, rarest clusters, it performed exceptionally well for the vast majority of structures. The most significant achievement, however, was not just in predicting the count of these haloes, but in understanding how that count changes. Because the network is "differentiable," the researchers could ask it to calculate the rate of change for the number of haloes if the density of matter or the strength of initial fluctuations were altered. The network's answers matched the results of traditional, slow simulations and existing theoretical models with impressive fidelity. It correctly identified that increasing the density of matter leads to more haloes across the board, while changing the initial smoothness of the universe affects the formation of massive structures more than small ones.

Beyond simply counting haloes, the researchers demonstrated that this method could be used to improve existing models that are not mathematically smooth. By using the precise sensitivity data from their neural network, they could take a standard, non-smooth prediction and adjust it to account for new cosmological parameters with extreme accuracy. This allows scientists to use powerful statistical tools that were previously incompatible with these types of models. Furthermore, the team showed that the system could also reveal how the final structure of the universe depends on the specific, random fluctuations present at the very beginning. They found that stronger fluctuations on the largest scales tend to suppress the formation of small haloes while boosting the growth of medium-sized ones, a result that aligns with our physical intuition about how gravity works. This work represents a significant step toward a fully data-driven approach to cosmology, where the complex relationship between the initial conditions of the universe and its final structure can be explored instantly and with mathematical precision, opening the door to more rigorous tests of the laws that govern our cosmos.

Technical Summary: Differentiable Halo Mass Prediction and the Cosmology-Dependence of Halo Mass Functions

Problem Statement
Modern cosmological inference increasingly relies on gradient-based methods, such as Hamiltonian Monte-Carlo (HMC), to efficiently sample posterior distributions in high-dimensional parameter spaces. However, a critical bottleneck exists: the standard pipeline for deriving the Halo Mass Function (HMF) from cosmological simulations involves non-differentiable steps. These include discrete halo finding algorithms (e.g., Friends-of-Friends or Spherical Overdensity), thresholding, and clustering operations used to construct discrete halo catalogues. These operations introduce discontinuities that prevent the direct computation of derivatives of the HMF with respect to cosmological parameters or initial conditions. Consequently, the HMF cannot be directly integrated into field-level inference pipelines that require differentiable forward models.

Methodology
To address this, the authors propose a fully differentiable, field-level approach to predict the abundance of dark matter haloes and their cosmological dependence using a neural network (NN).

Data Generation: The study utilizes a dataset of 2,000 independent cosmological N-body simulations generated using the GPU-accelerated Disco-Dj code. The simulations employ a flat $\Lambda$ CDM model, varying the matter density ( $\Omega_m$ ), the primordial scalar spectral index ( $n_s$ ), and the amplitude of matter fluctuations ( $\sigma_8$ ) via Latin hypercube sampling. The simulations evolve to $z=0$ using a particle-mesh (PM) scheme with $N=256^3$ particles and a $1024^3$ grid.
Halo Labeling: Haloes are identified using the Rockstar halo finder with a Spherical Overdensity (SO) criterion. The particles belonging to these haloes are traced back to their initial Lagrangian positions. The target labels for the NN are "protohalo patches" in the initial linear density field, binned by their final virial mass.
Neural Network Architecture: A 3D U-Net (V-Net) architecture is employed to perform multiclass semantic segmentation. The input is the initial linear density field, and the output is a probability distribution over 12 classes: 10 logarithmically spaced mass bins ( $4 \times 10^{12} M_\odot/h$ $4 \times 1 0^{12} M_{⊙} / h$ to $10^{15} M_\odot/h$ $1 0^{15} M_{⊙} / h$ ), a background class, and an overflow class for massive haloes.
- The network uses periodic padding and performs down-sampling and up-sampling in Fourier space (using sharp-k low-pass filters and zero-padding) to preserve cosmological power spectrum properties.
- The loss function is the squared Earth Mover's Distance (EMD), chosen to respect the ordered nature of the mass bins, penalizing predictions based on the distance between the predicted and true mass bins.
Differentiability: The entire pipeline, from the linear density field to the predicted mass function, is implemented in Jax, enabling automatic differentiation. This allows for the direct computation of gradients of the HMF with respect to cosmological parameters and the initial white noise field.

Key Results

Protohalo Identification: The U-Net achieves a True Negative Rate (TNR) of 96% and a True Positive Rate (TPR) of 73%, resulting in an F1-score of 0.77. The model is conservative, accurately identifying background regions but tending to smooth out patch edges, leading to some under-prediction of small haloes.
HMF Accuracy: The network predicts the HMF with an average accuracy of approximately 8% across the mass range. It performs robustly for low-to-medium mass haloes but slightly underestimates the high-mass end, a limitation attributed to cosmic variance and the small simulation volume ( $100 \text{ Mpc}/h$ box).
Cosmological Dependence: The derivatives of the predicted HMF with respect to $\Omega_m$ $Ω_{m}$ , $n_s$ $n_{s}$ , and $\sigma_8$ $σ_{8}$ were computed via automatic differentiation and compared against finite-difference derivatives from simulations and analytical models (Tinker08, Mira-Titan emulator).
- The NN derivatives for $\Omega_m$ and $n_s$ agree extremely well with the Mira-Titan emulator and Tinker08 predictions.
- For $\sigma_8$ , the NN captures the general trend but shows slightly larger deviations at low masses compared to the emulator, though it remains within the 1 $\sigma$ range of the emulator's uncertainty.
Extrapolation Capability: The authors demonstrated that the differentiable gradients from a parametric model (Tinker08) can be used to extrapolate a non-differentiable emulator (Mira-Titan) to perturbed cosmologies. Using a first-order Taylor expansion with Tinker08 gradients, the Mira-Titan HMF for perturbed parameters was reproduced at the sub-percent level, despite the base Tinker08 model having a ~10% deviation at low masses.
Initial Condition Sensitivity: The framework successfully computed the sensitivity of the HMF to perturbations in the initial white noise field. The results confirmed that large-scale modes in the initial conditions primarily drive the abundance of intermediate-to-high mass haloes, while small-scale modes have a negligible effect, consistent with theoretical expectations.

Significance and Claims
The paper claims to present a novel, differentiable method for predicting halo abundances that bridges the gap between field-level simulations and gradient-based cosmological inference. The primary significance lies in the ability to:

Enable Gradient-Based Inference: By providing a differentiable mapping from initial conditions to the HMF, the method allows for the use of HMC and other gradient-based samplers in large-scale structure cosmology, potentially offering tighter constraints on cosmological parameters than summary-statistic-based analyses.
Quantify Model Dependence: The approach allows for the direct investigation of how HMFs respond to changes in cosmological parameters and specific realizations of the initial density field, offering a tool to study cosmic variance and model systematics.
Enhance Existing Models: The study demonstrates that differentiable models can serve as "gradient correctors," enabling the extrapolation of complex, non-differentiable emulators (like Mira-Titan) to new parameter spaces with high precision, even when the base parametric model used for gradients is imperfect.

The authors characterize this work as a "proof-of-concept" that achieves accuracy at the few-percent level. They acknowledge limitations, such as the conservative nature of the classification and the difficulty in capturing the high-mass end due to volume constraints, and suggest future work involving continuous property prediction and larger volume training.

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