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, , 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 A 100 km s(-1). This n corresponds roughly to the slope on the non-linear scale in the real ACDM Universe, but ACDM 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 drives the ACDM dispersion up into balance with growing structure, and should eventually account for dispersion effects seen recently in simulations (I have not yet carried out that calculation).