Orodruin said:
This is not an accurate description of thermal dark matter. Dark matter is produced shortly after the Big Bang in any viable dark matter model and it does not have to be thermally produced. Axions are a prime example of this. In addition, I strongly doubt a dark matter candidate with only gravitational interactions would be thermally produced anywhere below the reheating scale.
I have used the terms "thermal relic" and "relic" interchangeably, which is sloppy. A paper suggesting ways to distinguish between thermal and non-thermal DM with experimental constraints is here:
https://arxiv.org/pdf/1311.5297.pdf
A rather comprehensive recent review of axion dark matter that explains and/or refers to papers that explain how axions could be non-relativistic despite being produced in the early universe can be found at
http://iopscience.iop.org/article/10.1088/1367-2630/11/10/105008 although to be honest, it is not the most readable presentation. A more readable discussion is here:
http://web.mit.edu/redingtn/www/netadv/specr/345/node3.html
I had thought, perhaps mistakenly, that axions could also be an example of dark matter that is not just produced shortly after the Big Bang, because it has to be constantly produces in day to day QCD interactions to keep the CP violation constant of the strong force theta, naturally or near zero, which would also allow it to have non-relativistic velocities (as any viable dark matter candidate must) despite having a mass less than that of hot dark matter neutrinos. But, I am too tired to run down a reference for that at the moment.
While I'm at it, a generalized and somewhat outdated case for the WIMP miracle involving thermal relic WIMPs can be found at
http://web.mit.edu/redingtn/www/netadv/specr/345/node2.html
FWIW, I am highly unimpressed by the "natural" motivations for both the axions (on the theory that the zero or nearly zero CP violation of the strong force is unnatural) and for SUSY which would naturally solve the "hierarchy problem", both of which have at their foundations a scientists presumption about what physical constants Nature should have that really have no meaningful scientific basis and are mere guesswork, neither of which have born any fruit to date.
Reference please. Dividing the density by the mass gives you the number density, not the velocity.
Many papers do the analysis in a model dependent manner specific to WIMPs but a more general model-independent analysis can be found, for example, in the following 2014 paper.
http://arxiv.org/pdf/1309.6971.pdf (Note that I am not citing this paper in support of the claim that everything in it is true, merely because it is an example of many that lays out the basic equations involved in the mass-velocity relationship relevant to dark matter, which it does preliminarily to reach its further conclusions.)
Another fairly general model-independent analysis that focuses on the free streaming length of DM which phenomenologically has the same sort of impacts that could be inferred from velocity is here: http://chalonge.obspm.fr/Dark_Matter.pdf
The relationship between free streaming length and velocity is spelled out here:
http://www.thphys.uni-heidelberg.de/~smp/view/Delta09/Slides_rubakov.pdf