Cosmological evolution of collisionless relativistic gases as dark matter

TL;DR

This work develops and tests a relativistic kinetic gas model for dark matter in a flat FLRW universe, parameterized by a single velocity-like parameter $β$ that governs a smooth transition from radiation-like to cold dark matter behavior. By solving the covariant Vlasov framework for a mono-energetic distribution, the authors obtain closed-form expressions for the background density $ ho(a)$, pressure $P(a)$, and the adiabatic sound speed $c_s^2(a)$, and implement the model in CLASS to compare with Planck 2018 and BAO data. They derive tight BBN-based constraints on $β$ (via $ ext{Δ}N_{ ext{eff}}$) and, from CMB and Lyman-$α$ observations, obtain robust upper limits on $ ext{log}eta$ that favor CDM-like behavior at late times while allowing transient relativistic effects at early times. The results demonstrate that a collisionless relativistic kinetic gas can be fully consistent with current cosmological observations, while offering a framework to systematically explore departures from perfect coldness in the dark sector and future probes could further sharpen these bounds.

Abstract

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.

Figures (11)

• Figure 1: Cosmological evolution of the energy density $\rho_{DM}(a)$ for several values of the relativistic parameter $\beta$. The blue and red curves represent, respectively, the CDM and radiation components, while the colored lines show the kinetic--gas dark matter model for increasing values of $\beta$. All curves are normalized to the present-day abundances.
• Figure 2: Evolution of the ratio $\rho_{DM}(a)/\rho_r(a)$ for several values of the relativistic velocity parameter $\beta$. The blue line shows the standard cold matter scaling $\rho_m/\rho_r \propto a$, while the red line corresponds to pure radiation behavior $\rho_r/\rho_r = 1$.
• Figure 3: Cosmological evolution of the dark matter density parameter $\Omega_{DM}(z)$ for several values of $\beta$. The standard CDM evolution (black curve) is shown for comparison.
• Figure 4: Relative deviation of the dark matter density parameter with respect to CDM, $\Delta\Omega_{DM}$, for the same set of $\beta$ values shown in Fig. \ref{['Omegas']}.
• Figure 5: Cosmological evolution of the density parameters of all relevant components. The standard CDM case is shown as the black reference curve, while the RKG model is represented by the dashed cyan curve.