  Cosmological perturbation theory for the late Universe dominated by dark
matter is extended beyond the perfect fluid approximation by taking the dark
matter velocity dispersion tensor as an additional field into account. A proper
tensor decomposition of the latter leads to two additional scalar fields, as
well as a vector and a tensor field. Most importantly, the trace of the
velocity dispersion tensor can have a spatially homogeneous and isotropic
expectation value. While it decays at early times, we show that a back-reaction
effect quadratic in perturbations makes it grow strongly at late times. We
compare sterile neutrinos as a candidate for comparatively warm dark matter to
weakly interacting massive particles as a rather cold dark matter candidate and
show that the late time growth of velocity dispersion is stronger for the
latter. Another feature of a non-vanishing velocity dispersion expectation
value is that it destroys the apparent self-consistency of the single-stream
approximation and allows thereby to treat times and scales beyond
shell-crossing.
