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-Lema\^itre-Robertson-Walker universe. After normalization to the observed present-day dark matter abundance, the model is fully specified by a single dimensionless parameter $\beta$, 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 $\beta$ are observationally indistinguishable from $\Lambda$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.