How to generate a significant effective temperature for cold dark
  matter, from first principles
  I show how to reintroduce velocity dispersion into perturbation theory (PT)
calculations of structure in the Universe, i.e., how to go beyond the
pressureless fluid approximation, starting from first principles. This
addresses a possible deficiency in uses of PT to compute clustering on the
weakly non-linear scales that will be critical for probing dark energy.
Specifically, I show how to derive a non-negligible value for the (initially
tiny) velocity dispersion of dark matter particles, <\delta v^2>, where \delta
v is the deviation of particle velocities from the local bulk flow. The
calculation is essentially a renormalization of the homogeneous (zero order)
dispersion by fluctuations 1st order in the initial power spectrum. For power
law power spectra with n>-3, the small-scale fluctuations diverge and
significant dispersion can be generated from an arbitrarily small starting
value -- the dispersion level is set by an equilibrium between fluctuations
generating more dispersion and dispersion suppressing fluctuations. For an
n=-1.4 power law normalized to match the present non-linear scale, the
dispersion would be ~100 km/s. This n corresponds roughly to the slope on the
non-linear scale in the real \LambdaCDM Universe, but \LambdaCDM contains much
less initial small-scale power -- not enough to bootstrap the small starting
dispersion up to a significant value within linear theory (viewed very broadly,
structure formation has actually taken place rather suddenly and recently, in
spite of the usual "hierarchical" description). The next order PT calculation,
which I carry out only at an order of magnitude level, should drive the
dispersion up into balance with the growing structure, accounting for small
dispersion effects seen recently in simulations.
