# Dark matter in the time window between freeze-out and kinetic decoupling

Background

After the freeze-out, when all annihilations have stopped, the abundance ($$Y=\frac{n}{s}$$) of thermal dark matter species no longer changes with time. However, it is still kept in kinetic equilibrium for some amount of time after freeze-out via scatterings with ordinary particles in the thermal bath. It's only after kinetic decoupling it completely departed from equilibrium.

Now, in the window of temperature (or time) between freeze-out ($$T_{\rm fo}$$) and kinetic decoupling $$(T_{\rm d})$$, during which the kinetic equilibrium was maintained (though not chemical equilibrium), how do we expect the number density $$n$$ to change?

Question

The confusion is about the temperature (or time) interval between freeze-out and kinetic decoupling.

On one hand, since all annihilations have stopped, one would expect that any reduction in the number density $$n$$ must solely be an effect of the expansion, and hence, $$n$$ must dilute as $$n\sim a^{-3}\tag{1}$$ with the expansion.

However, we can also think a bit differently. Since in this interval, it is in kinetic equilibrium, one might expect $$n$$ is equal to the equilibrium number density, and therefore decreases as $$n=n_{\rm eq}\sim T^{3/2}e^{-M/T}.\tag{2}$$

At least one reasoning must be incorrect. I think that the second thought is less reliable but I would like to have some expert opinion.