  We compute the dark matter velocity dispersion tensor up to third order in
perturbation theory using the Lagrangian formalism, revealing growing solutions
at the third and higher orders. Our results are general and can be used for any
other perturbative formalism. As an application, corrections to the matter
power spectrum are calculated, and we find that some of them have the same
structure as those in the effective field theory of large-scale structure, with
"EFT-like" coefficients that grow quadratically with the linear growth function
and are further suppressed by powers of the logarithmic linear growth factor
$f$; other corrections present additional $k$ dependence. Due to the velocity
dispersions, there exists a free-streaming scale that suppresses the whole
1-loop power spectrum. Furthermore, we find that as a consequence of the
nonlinear evolution, the free-streaming length is shifted towards larger
scales, wiping out more structure than that expected in linear theory.
Therefore, we argue that the formalism developed here is better suited for a
perturbation treatment of warm dark matter or neutrino clustering, where the
velocity dispersion effects are well known to be important. We discuss
implications related to the nature of dark matter.
