# Where are particles dissappearing in thermal equilibrium?

I am incredibly puzzled over dark matter relic abundance plots like this (taken randomly from google), where it shows number density of some particle species in a comoving volume

In thermal equilibrium x=m/T < 10, the particle density decays exponentially, this is because of Boltzmann suppression factor sitting in the particle density: $$n=\int \frac{d^3p}{(2\pi)^3}e^{-E/T}$$

But, what does it mean physically? Say I am keeping track of number density of particle species $S$ and it is in thermal equilibrium (say x=1), as I slowly decrease the temperature (while still being in thermal equilibrium say x~10) the number density of this particle (in a comoving volume) will decrease! So where the heck these particle are going to?

In thermal equilibrium particles should be annihilating back and forth how can they disappear, we have not approached the freezeout so there is no reason for number density to decrease? And expansion of the universe has no impact, because we are studying comoving number density.

I am incredibly puzzled, any thoughts are much appreciated!

Good question!

You are completely right that in thermal equilibrium the creation and annihilation rates of any given particles are identical. I think your confusion comes from the fact that the universe is not actually in complete thermal equilibrium.

As the universe expands, the temperature changes, but if the interaction rates are high, then the particles will follow thermal distributions at the current temperature of the universe. So you can roughly think of the universe as a thermal system with a temperature that changes so slowly that the particles have time to constantly equilibrate to the new temperature.

If the interaction rate becomes too slow, i.e. $\Gamma \sim H$, then the particles do not have time to equilibrate befor the universe expands further and go out of thermal equilibrium. This is what happens to dark matter in your figure.

You ask also where the dark matter particles go to, and the answer is that they annihilate to other particles. We get the exponential suppression in the number density for low temperatures, since few of the non-dark matter particles have the energy to create DM particles (since $T \ll m_{DM}$), so the number density of dark matter has to be really low for the annihilation rate to equal the creation rate.

I hope this was clear.

• But this plot comes from Boltzmann equation-theoretical prediction you assume thermal equilibrium in the beginning, it doesn't matter what real universe is like. In last section you are talking about $T<<m_{DM}$ (decoupling) it does not have exponential suppression anymore. What I am asking about is $m/T\in(1,10)$ range (where energy is fine enough to produce DM), why does number density decreases exponentially and what happens to these particles? – Wint Jul 20 '17 at 6:33
• If you read my answer, you see that I say that we are never really in thermal equilibrium. In the last paragraph I (try to) explain what happens to the equilibrium number density (nothing about decoupling!) at low temperatures. What actually happens to these particles is that they annihilate to other particles in order to keep up the equilibrium number density as the temperature changes. – Ihle Jul 20 '17 at 6:45
• If you're still confused, try to understand what we actually mean by thermal equilibrium in the universe (which is basically what I tried to explain). Also, do you understand why we get the exponential suppression of the number density at low temperatures (T<<m) in actual thermal equilibrium (with a constant unchanging temperature?) – Ihle Jul 20 '17 at 6:50