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Approximating the peculiar velocity distribution of dark matter halos with Tsallis statistics

By combining N-body simulations with non-extensive statistical mechanics and superstatistics theory, this paper demonstrates that a concise two-parameter Tsallis distribution accurately models the non-Gaussian peculiar velocities of dark matter halos across redshifts 0–2, offering a promising tool for probing cosmological parameters.

Original authors: Jun Pan, Ming Li

Published 2026-08-28

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CC BY 4.0

Original authors: Jun Pan, Ming Li

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 dynamic web of invisible scaffolding, where vast clouds of dark matter gather to form halos that cradle galaxies and galaxy clusters. These structures are not sitting still; they drift through space with their own unique speeds and directions, known as peculiar velocities. For decades, astronomers have tried to understand how these invisible halos move, hoping that their motion would reveal the hidden rules of gravity and the history of the cosmos. A common assumption in the past was that these velocities followed a simple, bell-shaped pattern, much like the predictable spread of speeds in a calm gas. However, the universe is far more chaotic than a calm gas. Dark matter halos are massive, self-gravitating objects that interact strongly over long distances, creating a complex, turbulent environment where simple rules often break down. Understanding the true shape of their speed distribution is crucial because it helps scientists interpret the distortions seen in galaxy maps and measure the expansion of the universe with greater precision.

In a recent study, researchers Jun Pan and Ming Li set out to find a better way to describe this chaotic motion. They turned to a mathematical framework originally developed for systems that are far from equilibrium, known as Tsallis statistics. This approach is designed to handle complex systems where particles interact in ways that standard physics cannot easily capture. The team did not observe the universe directly for this work; instead, they used a massive collection of computer simulations that modeled the evolution of dark matter over billions of years. These simulations, generated by the Quijote and Mira-Titan projects, contained millions of dark matter halos, allowing the researchers to track the speeds of these invisible structures across different epochs of cosmic history.

The researchers took the velocity data from these simulations and tested how well the Tsallis model could fit the actual distribution of speeds. They found that for the vast majority of halos moving at speeds below 1,000 kilometers per second, the model provided an exceptionally accurate description, matching the simulation data with an error margin of less than 5 percent. This level of precision held true across a wide range of cosmic time, from the present day back to when the universe was only a fraction of its current age. The study revealed that as the universe ages and gravity continues to pull matter together, the distribution of velocities becomes increasingly complex and deviates further from the simple bell curve. The model successfully captured this growing complexity, showing that the universe is moving further away from a state of simple equilibrium over time.

One of the most significant findings was how the model behaved across different sizes of dark matter halos. While the researchers expected the fit to vary significantly between small and large halos, they discovered that the Tsallis model worked remarkably well for all of them. The parameters that defined the shape of the distribution changed only slightly with mass, suggesting a universal behavior in how these structures move. However, the study did note that the model was less accurate for the fastest-moving halos, particularly those traveling at speeds greater than 1,000 kilometers per second, where the simulation data showed a higher number of extreme outliers than the model predicted. Despite this limitation at the very high end of the speed spectrum, the researchers concluded that the single mathematical function they used was sufficient for most practical applications in cosmology.

The work also clarified how the motion of these halos relates to the underlying physics. The researchers demonstrated that the complex, non-standard distribution of speeds could be understood as a combination of many simpler, standard distributions, each with slightly different speeds. This insight connects the chaotic motion of the cosmos to a broader statistical theory, suggesting that the irregularities we see are the result of averaging over many different local environments. By confirming that this approach works across different cosmological models and simulation setups, the study provides a robust tool for future astronomers. It offers a ready-made, accurate description of how dark matter moves, which can be used to refine measurements of the universe's expansion and the nature of gravity itself, without needing to rely on overly complex or untested assumptions.

Technical Summary: Approximating the peculiar velocity distribution of dark matter halos with Tsallis statistics

Problem Statement
Dark matter halos, the hosts of galaxies and clusters, possess peculiar velocities that deviate significantly from thermal equilibrium. While the spatial clustering of halos is well-described by standard statistics (e.g., correlation functions), the statistical description of their velocity fields remains challenging. Traditional approaches often assume that peculiar velocities evolve linearly from Gaussian initial conditions or rely on the "halo model," which integrates Gaussian conditional velocity distributions over local density contrasts. However, the halo model suffers from empirical parameters lacking solid physical justification (e.g., the smoothing scale and the variance-density relation) and requires extensive calibration. Furthermore, simple linear extrapolations fail to match N-body simulation results, particularly regarding the non-Gaussian features and heavy tails of the velocity distribution. There is a need for a concise, accurate, and physically motivated model to characterize these nonlinear, non-Gaussian velocity distributions for cosmological applications such as redshift-space distortions and the kinematic Sunyaev–Zel'dovich effect.

Methodology
The authors investigate the efficacy of Tsallis statistics, derived from non-extensive statistical mechanics, in modeling the one-point distribution function of halo peculiar velocities. The study employs two primary datasets:

Quijote N-body Simulations: 100 high-resolution realizations of a fiducial $\Lambda$ CDM model, providing halo catalogs at redshifts $z=0, 1, 2$ .
Mira-Titan Universe Suite: Three realizations with distinct cosmological parameters to test the robustness of the model across different cosmologies.

The methodology involves:

Data Extraction: Constructing halo catalogs using the friends-of-friends algorithm with specific linking lengths and particle mass thresholds.
Statistical Fitting: Applying Maximum Likelihood Estimation (MLE) to fit the Tsallis distribution (Eq. 10 in $\kappa$ -notation) to the simulated velocity distributions. Both continuous likelihoods and coarse-grained (binned) likelihoods are utilized.
Theoretical Framework: Reformulating the halo model within the superstatistics framework (Beck & Cohen 2003). The authors posit that the peculiar velocity distribution is a superposition of Gaussians with fluctuating variances. By expanding the distribution of the inverse variance ( $s = 1/\sigma_G^2$ ) using a generalized Gram–Charlier expansion based on the gamma distribution, they theoretically derive the halo velocity distribution as a series of Tsallis functions.

Key Contributions and Results

Accuracy of the Tsallis Model: The study demonstrates that the Tsallis distribution provides an excellent fit to the peculiar velocity distributions of dark matter halos for velocities $|v| \lesssim 1000$ km s $^{-1}$ across the redshift range $0 \le z \le 2$ . The model achieves an accuracy of approximately 5% for $z \le 2$ , with performance improving at lower redshifts.
Non-Gaussian Evolution: The results indicate that halo velocity distributions become increasingly non-Gaussian and depart further from thermodynamic equilibrium over time (as $z$ decreases). This is quantified by the invariant index $\kappa_0$ , which decreases as $z \to 0$ .
Mass Dependence: The best-fit parameters show a weak but discernible dependence on halo mass. Low-mass halos generally exhibit stronger non-Gaussianity (smaller $\kappa_0$ ) and larger velocity dispersions ( $\sigma_{\kappa_0}$ ) compared to massive halos, though this trend varies slightly at high redshifts.
Correlation Structure: The Tsallis model correctly predicts that while linear correlations between velocity components vanish (due to isotropy), the components are not statistically independent. Specifically, the model predicts nontrivial quadratic correlations (e.g., $\langle v_x^2 v_y^2 \rangle / \langle v_x^2 \rangle \langle v_y^2 \rangle = \kappa_0 / (\kappa_0 - 1)$ ), which align excellently with simulation data.
Robustness Across Cosmologies: Validation using the Mira-Titan suite confirms that the Tsallis model remains effective across different cosmological parameters. While the velocity dispersion $\sigma_{\kappa_0}$ varies significantly with cosmology, the non-extensive index $\kappa_0$ shows less variation, suggesting $\sigma_{\kappa_0}$ is a more sensitive probe for cosmological constraints.
Theoretical Interpretation: The authors establish that the halo velocity distribution can be theoretically expressed as a superposition of Tsallis distributions. The simulation results imply that the underlying distribution of the inverse variance ( $s$ ) is close to a gamma distribution, making the zeroth-order approximation (a single Tsallis function) sufficient for practical accuracy.

Significance and Claims
The paper claims that the one-point statistics of halo peculiar velocities at low redshifts can be effectively characterized by the Tsallis distribution. This offers a "concise and accurate" alternative to the cumbersome, parameter-heavy halo model. The authors emphasize that while the Tsallis model does not replace pairwise velocity statistics (which are crucial for kinematic Sunyaev–Zel'dovich and redshift-space distortion analyses), it provides a necessary theoretical prior and building block for these applications. By accurately capturing the non-Gaussianity of the one-point distribution, the model prevents the propagation of errors into higher-order statistics.

The study concludes that the Tsallis distribution serves as an effective, ready-to-use non-Gaussian prior for halo peculiar velocities, particularly at low redshifts. The authors modestly note that while the single-function approximation is sufficient for halos, future work is required to determine if this framework extends to dark matter particles and galaxies, which exhibit more complex internal motions. The paper does not claim to solve the full cosmological parameter estimation problem but identifies the fitted parameters ( $\kappa_0, \sigma_{\kappa_0}$ ) as promising candidates for such constraints in future studies.

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