What does it mean for CDM to "decouple from baryons" I am currently working on my Final Year Project as an undergraduate, mostly intended to be a review of $\Lambda$CDM model and some of its modern challenges.
When talking about Cold Dark Matter I stated it must be cold in order to clump in the galactic volumes, and my tutor wrote me to "Improve the definition. these are the velocities the particles must have when decoupling from baryons."
While it feels intuitively easy for me to understand, for example, photon decoupling in this context (CMB is a pretty "visual" thing), I'm not sure what it would mean in the case of DM. My guess is that CDM is expected to interact significantly with matter at some specific energies, and once the energies are below certain point it becomes the nowadays "weakly interacting stuff" we all know and love, but I'm not sure what role does speed play here. And why should CDM be more interacting at higher energies? What evidence supports this idea?
 A: You are entirely correct in stating that the "Cold" in "Cold Dark Matter" refers to the fact that dark matter needs to be non-relativistic in order to explain the strong clumping we observe in structure formation. This is a statement about the speed of particles at times at and after the emission of the Cosmic Microwave Background.
A separate issue is the nature of dark matter. One particularly popular candidate is the WIMP, which in turn is a special case of a Thermal Relic Particle. For Thermal Relic Particles see this discussion, or in short for what is important here: In the very early universe, normal particles and these hypothetical thermal relic particles are in thermal equilibrium, which is to say that they all are constantly produced and annihilate into each other, and thus each particle species has the same temperature distribution. They are "coupled". As the universe cools, at some point lighter standard model particles may no longer have enough energy to produce heavier dark matter particles. So while heavier dark matter particles can still annihilate into standard model particles, the reverse is no longer possible. Their number density drops, and their temperature develops differently than the temperature of the standard model particles. At that point, they are said to have decoupled from the standard model (or baryons, as it may). Their decoupling happens at a temperature (corresponding to some median speed) characteristic of the coupling and mass of these particles.
There is no evidence to support this idea other than saying "the model works". Of course, speed is energy, and at particle accelerators the same idea holds where accelerating particles to faster and faster speeds may, in collisions, produce new types of (heavy) elementary particles.
In the context of e.g. the Weak (with capital W) interaction, one can also see that when the temperature of the universe (aka the temperature of the primordial plasma, thus the energy of particles, as indicated by their speed) is much higher than the mass of the Weak interaction bosons (W, Z), then the mass of these bosons can be neglected. The Weak force at that time thus isn't weak at all, since the interaction bosons are readily produced. Only once the temperature drops below the mass of these bosons does it become difficult to create them (even as virtual particles), and thus the Weak force becomes weak. At that point too one can say that e.g. Weakly interacting particles decouple from the plasma.
However, please note that there are many other dark matter candidates, and this story may not apply to those at all. As a specific example, consider Axions. These are never in thermal equilibrium with baryons. Axions are never coupled to baryons and therefore also never decouple. You may want to keep these aspects clearly separate in your project to avoid confusion about what is known about dark matter (it is nonrelativistic, i.e. cold) and what is hypothesized in some models (it decoupled at some time in the past).
Finally, if you are looking for a formal definition of "decoupling", one could in this context use the condition that the interaction rate of your dark matter and baryons becomes slow compared to the expansion rate of the universe.
