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Cosmological evolution of collisionless relativistic gases as dark matter

This paper proposes a dark matter model consisting of a collisionless relativistic kinetic gas and demonstrates, through Planck 2018 CMB data analysis, that it is observationally consistent with the standard $\Lambda$ CDM model provided the particles' present-day velocity is sufficiently small.

Original authors: Francisco X. Linares Cedeño, Ulises Nucamendi, Olivier Sarbach

Published 2026-02-10

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CC BY 4.0

Original authors: Francisco X. Linares Cedeño, Ulises Nucamendi, Olivier Sarbach

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 Cosmic "Speed Limit" of Dark Matter: A Simple Guide

Imagine you are watching a massive, slow-moving parade of heavy tanks rolling through a city. This is how we usually think of Dark Matter: a heavy, silent, and "cold" substance that provides the gravitational glue holding galaxies together. It moves slowly, it doesn't bump into things, and it just sits there, building the foundations of the universe.

But what if, in the very beginning, those tanks weren't tanks at all? What if they were actually high-speed racing cars that eventually slowed down and turned into tanks?

This paper explores a theory called the Relativistic Kinetic Gas (RKG) model. It suggests that Dark Matter might have had a "wild youth"—a period in the early universe where it was moving at near-light speeds before cooling down into the slow, heavy stuff we see today.

1. The Metaphor: From Race Cars to Heavy Tanks

To understand this paper, think about a crowd of people in a massive stadium:

The Standard View (Cold Dark Matter): Imagine everyone in the stadium is sitting perfectly still in their seats. They are heavy, they don't move, and they just provide a steady "weight" to the stadium. This is the standard model scientists use to explain the universe.
The New Theory (RKG Model): Imagine that at the start of the game, everyone is sprinting around the stadium at full speed. Because they are moving so fast, they are bumping into the "idea" of gravity differently—they create a kind of "pressure" that makes it hard for small groups to form. But as the game goes on, everyone gets tired, slows down, and eventually sits back down in their seats.

The researchers are trying to figure out: How fast were those people sprinting at the start, and how much does that "early sprinting" change the way the stadium (the universe) looks today?

2. The "Speed Parameter" ( $\beta$ )

The scientists use a single magic number called $\beta$ (Beta).

If $\beta$ is zero, the dark matter was always a heavy tank (the standard model).
If $\beta$ is large, the dark matter was a high-speed race car for a long time.

The paper's goal was to use data from the Planck Satellite (which took a "baby picture" of the universe via the Cosmic Microwave Background) to see if we can detect any evidence of this "early sprinting."

3. The "Small-Scale" Problem

One of the biggest reasons scientists look at these models is because the standard "Heavy Tank" model has a tiny flaw. It predicts that the universe should be filled with thousands of tiny, little "mini-galaxies" (sub-halos). However, when we look through our telescopes, we don't see as many of them as we expected.

The RKG model solves this beautifully. If dark matter was "sprinting" (moving fast) in the early universe, that speed would have acted like a cosmic broom, sweeping away small clumps of matter and preventing those tiny mini-galaxies from forming. It "smooths out" the small stuff while leaving the big stuff (like our Milky Way) alone.

4. The Verdict: How fast can they go?

After running complex computer simulations (using a tool called CLASS) and comparing them to the actual maps of the universe, the researchers found something important:

The "sprinting" couldn't have been too wild.

If the dark matter was moving too fast for too long, the "baby picture" of the universe would look totally different from what the Planck satellite actually saw. The researchers set a "speed limit." They found that the dark matter must be moving very, very slowly today—specifically, less than about 61 meters per second (that's roughly 136 mph).

While that sounds fast to us, in the cosmic scale of light-speed, it is incredibly slow.

Summary: Why does this matter?

This paper tells us that while Dark Matter could have had a high-energy, relativistic "youth," it must have settled down into its "cold" adult phase very early on.

By setting these limits, the scientists are narrowing down the search. They are telling future astronomers: "If you want to find the truth about Dark Matter, look for a substance that was once a racer but is now a slow-moving giant, and make sure its 'speeding ticket' doesn't break the laws of the early universe."

Technical Summary: Cosmological Evolution of Collisionless Relativistic Gases as Dark Matter

1. Problem Statement

The standard $\Lambda$ CDM model assumes dark matter (DM) is perfectly cold, pressureless, and collisionless. While highly successful at large scales, $\Lambda$ CDM faces "small-scale challenges," such as discrepancies in the abundance and internal structure of low-mass halos. These tensions suggest that dark matter might not be perfectly "cold" at all times. Specifically, the authors investigate whether a dark matter component that undergoes a transient relativistic phase during early cosmological evolution can alleviate these issues without violating large-scale observations.

2. Methodology

The authors propose a phenomenological model termed Relativistic Kinetic Gas (RKG) dark matter.

Kinetic Framework: Instead of treating DM as a perfect fluid, they use a covariant relativistic kinetic theory approach. They model the DM as a mono-energetic collisionless gas described by a single-stream distribution function (DF) that is sharply peaked at a fixed conserved momentum $C$ .
Mathematical Formulation: By applying the Vlasov (collisionless Boltzmann) equation in a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, they derive closed analytic expressions for the energy density ( $\rho$ ), pressure ( $P$ ), and equation of state ( $w = P/\rho$ ).
Key Parameter: The model is fully specified by a single dimensionless parameter, $\beta$ , which represents the present-day particle velocity in units of the speed of light ( $c=1$ ). This parameter controls the transition from a radiation-like regime ( $w \simeq 1/3$ ) at early times to a cold dark matter regime ( $w \simeq 0$ ) at late times.
Numerical Implementation: The authors implemented this RKG model into a modified version of the CLASS (Cosmic Linear Anisotropy Solving System) Boltzmann code to compute background evolution and linear cosmological perturbations.
Statistical Analysis: They performed a Bayesian parameter inference using Planck 2018 CMB data (temperature, polarization, and lensing) combined with Baryon Acoustic Oscillation (BAO) measurements from BOSS DR12.

3. Key Contributions

Unified Kinetic Model: They provide a rigorous derivation showing how a collisionless gas naturally interpolates between relativistic and non-relativistic behaviors based on its momentum distribution.
Analytic Predictivity: The paper provides exact expressions for the adiabatic sound speed $c_s^2$ , which is critical for understanding how the "effective pressure" of the gas suppresses the growth of structure.
Scale-Dependent Suppression: They demonstrate that the RKG model induces a characteristic suppression in the matter power spectrum at small scales (high $k$ ), similar to Warm Dark Matter (WDM) or massive neutrinos, but driven by the kinetic velocity dispersion rather than thermal broadening.

4. Results

BBN Consistency: Big Bang Nucleosynthesis (BBN) constraints on the effective number of relativistic species ( $N_{\text{eff}}$ ) require $\beta \lesssim 4.89 \times 10^{-6}$ . This ensures the relativistic phase is confined to the very early universe.
CMB and LSS Constraints: Using Planck 2018 and BAO data, the authors found that the model is highly consistent with $\Lambda$ $Λ$ CDM for small $\beta$ $β$ . They established the following upper limits on the velocity parameter:
- 68% Credibility: $\log_{10} \beta < -6.6890$
- 95% Credibility: $\log_{10} \beta < -5.3054$
Physical Velocity Bounds: Translating $\beta$ $β$ back to physical velocity, the present-day dark matter velocity is constrained to:
- $|v_0| \lesssim 61.4 \text{ m/s}$ (68% CL)
- $|v_0| \lesssim 1,484 \text{ m/s}$ (95% CL)
Observational Agreement: The results are consistent with previous estimates of dark matter velocity dispersion ( $v_{\text{rms}} \leq 54 \text{ m/s}$ ) and show that the RKG model reproduces the successful large-scale predictions of $\Lambda$ CDM while allowing for small-scale modifications.

5. Significance

This work establishes that a collisionless relativistic gas is a mathematically consistent and observationally viable alternative to the standard CDM paradigm. It provides a controlled framework to study "non-cold" dark matter. By showing that the relativistic phase is transient, the authors demonstrate how one can modify small-scale structure formation (potentially solving sub-galactic tensions) without spoiling the high-precision agreement between $\Lambda$ CDM and the Cosmic Microwave Background. The study highlights that future large-scale structure surveys (like LSST and Euclid) will be essential to further tighten these kinetic constraints.