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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.

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The first reasoning is correct, the second is not.

Kinetic equilibrium does not mean full chemical equilibrium as described by your equation (2) which only holds before freeze out. The time window you are referring to just means that the average thermal energy of the dark matter follows that of the ordinary ones, but precisely they are not in chemical equilibrium.

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