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Large-scale structure formation is studied in a kinetic theory approach,extending the standard perfect pressureless fluid description for dark matterby including the velocity dispersion tensor as a dynamical degree of freedom.The evolution of power spectra for density, velocity and velocity dispersiondegrees of freedom is investigated in a non-perturbative approximation schemebased on the Dyson$\unicode{x2013}$Schwinger equation. In particular, thegeneration of vorticity and velocity dispersion is studied and predictions forthe corresponding power spectra are made, which qualitatively agree well withresults obtained from $N$-body simulations. It is found that velocitydispersion grows strongly due to non-linear effects and at late times its meanvalue seems to be largely independent of the initial conditions. By taking thisinto account, a rather realistic picture of non-linear large-scale structureformation can be obtained, albeit the numerical treatment remains challenging,especially for very cold dark matter models.

Dark matter vorticity and velocity dispersion from truncated Dyson$\unicode{x2013}$Schwinger equations