Cosmological evolution of collisionless relativistic gases as dark matter

We study a phenomenological dark matter model described as a collisionless relativistic kinetic gas in a spatially flat Friedmann-Lemaître-Robertson-Walker universe. After normalization to the observed present-day dark matter abundance, the model is fully specified by a single dimensionless parameter $β$, interpreted as the present particle velocity in units of the speed of light. The resulting energy density, pressure, and sound speed admit closed analytic expressions, interpolating between a radiation-like regime at early times and cold dark matter at late times. We implement the model in a modified version of the Boltzmann code CLASS and confront it with Planck 2018 CMB data. We find that sufficiently small values of $β$ are observationally indistinguishable from $Λ$CDM, while larger values inducing relativistic effects at early times are constrained. These results establish the consistency of the relativistic kinetic gas scenario with current cosmological observations.

💡 Research Summary

The paper presents a phenomenological dark‑matter model in which the dark sector is described as a collision‑less relativistic kinetic gas evolving in a spatially flat Friedmann‑Lemaître‑Robertson‑Walker (FLRW) universe. By normalising the model to the observed present‑day dark‑matter density, the authors show that the entire cosmology is characterised by a single dimensionless parameter, β, which can be interpreted as the present‑day particle velocity in units of the speed of light.

The theoretical framework starts from the covariant kinetic theory of gases. The one‑particle distribution function f(x,p) is defined on the tangent bundle of spacetime and must be invariant under the Killing vectors associated with spatial translations and rotations of the FLRW background. This invariance forces f to depend only on cosmic time and the magnitude of the comoving momentum. For massive particles the mass‑shell condition eliminates the temporal component of momentum, leaving a conserved quantity C = a²|p| (with a(t) the scale factor). The Vlasov (collisionless Boltzmann) equation then implies that f is independent of time and can be written as f = F(C).

To obtain a tractable model the authors adopt a mono‑energetic ansatz: all particles share the same value C₀, i.e. the distribution is sharply peaked at a fixed conserved momentum. Physically this corresponds to an ensemble of identical particles with isotropic momentum directions but a common kinetic energy. This idealisation is distinct from thermal distributions (Maxwell‑Boltzmann, Fermi‑Dirac, Bose‑Einstein) and allows fully analytic expressions for the macroscopic quantities.

From the mono‑energetic distribution the energy density ρ(a) and pressure p(a) are derived as
ρ(a) = ρ₀ / √