Standard texts say that whenever the rate of annihilation goes below the rate of expansion "decoupling" takes place and when the abundance of the decoupled entity does not change with time we say "freeze-out" has taken place. First of all, why an entity decouples when the temperature of the plasma (earlier with which it was in thermal equilibrium) goes below the mass of the entity concerned? In an annihilation process A+B->C+D, after A and B are decoupled, what happens to chemical potential ?
In thermal equilibrium (temperature is much higher than mass thresholds of A,B,C,D) when A+B-->C+D is taking place, the inverse particle scattering process is also taking place that is C+D-->A+B. But if the temperature falls below the mass threshold of total mass of A and B, then inverse scattering process is blocked. The physics is that C and D does not have sufficient kinetic energy to produce A+B. The abundance of C and D remains fixed.
Decoupling occurs when any one of the following two reasons. (i) Lack of sufficient energy to maintain equilibrium (ii) Expansion of the universe.
In cosmological models, when the reaction rate becomes less than the expansion rate of the universe, the particle of mass Mx can no longer maintain it's equilibrium abundance (which falls with increasing Mx/T) and the value of abundance becomes a constant. Note that Mx/T increases with decreasing of temperature that is cooling.
In early universe photons were in thermal equilibrium with electrons. The interaction between electron and photon is just Compton scattering. The rate of this process became less than the expansion rate of the universe at the point of photon decoupling. Similarly when weak interaction rate becomes less than the expansion rate of universe, then neutrino decoupling takes place.This explains why cosmic microwave background and cosmic neutrino background have different temperatures.
Incidentally if dark matter interacts with the visible sector weakly, that is if the interaction between dark matter and visible matter is of the weak interaction strength and if mass of the dark matter particle is of 100 GeV to few TeV range then it has also decoupled from the rest of the universe quite early and therefore it's abundance has become a constant. This value of abundance matches well with those measured by the satellite experiments such as COBE, WMAP, PLANCK.
If mass of the dark matter particle is too small (eV range) then after decoupling it will move freely with relativistic speeds. Being massive it will also carry out energy with it. This causes some problems to the formation of structures. For this reason the theory of hot dark matter (massive neutrinos) was ruled out after COBE results were announced in early nineties.
Ref: The Early Universe by E.W. Kolb and M.S. Turner
Sorry I don't have much time to give you an answer right now.
You will find some informations in this pdf section 3.2.1 Equilibrium thermodynamics
http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter3.pdf
In thermal equilibrium (temperature is much higher than mass thresholds of A,B,C,D) when A+B-->C+D is taking place, the inverse particle scattering process is also taking place that is C+D-->A+B. But if the temperature falls below the mass threshold of total mass of A and B, then inverse scattering process is blocked. The physics is that C and D does not have sufficient kinetic energy to produce A+B. The abundance of C and D remains fixed.
Decoupling occurs when any one of the following two reasons. (i) Lack of sufficient energy to maintain equilibrium (ii) Expansion of the universe.
In cosmological models, when the reaction rate becomes less than the expansion rate of the universe, the particle of mass Mx can no longer maintain it's equilibrium abundance (which falls with increasing Mx/T) and the value of abundance becomes a constant. Note that Mx/T increases with decreasing of temperature that is cooling.
In early universe photons were in thermal equilibrium with electrons. The interaction between electron and photon is just Compton scattering. The rate of this process became less than the expansion rate of the universe at the point of photon decoupling. Similarly when weak interaction rate becomes less than the expansion rate of universe, then neutrino decoupling takes place.This explains why cosmic microwave background and cosmic neutrino background have different temperatures.
Incidentally if dark matter interacts with the visible sector weakly, that is if the interaction between dark matter and visible matter is of the weak interaction strength and if mass of the dark matter particle is of 100 GeV to few TeV range then it has also decoupled from the rest of the universe quite early and therefore it's abundance has become a constant. This value of abundance matches well with those measured by the satellite experiments such as COBE, WMAP, PLANCK.
If mass of the dark matter particle is too small (eV range) then after decoupling it will move freely with relativistic speeds. Being massive it will also carry out energy with it. This causes some problems to the formation of structures. For this reason the theory of hot dark matter (massive neutrinos) was ruled out after COBE results were announced in early nineties.
Ref: The Early Universe by E.W. Kolb and M.S. Turner
^Thank you so much for the explanation, but I still have some confusions (which might be pretty naive). The point is, if I am not wrong, apart from "scattering" there are also "annihilation" processes going on. What happens to those annihilation processes after decoupling? I mean the same explanation goes to annihilation cases as well?
Your question is not naive. Essentially it means if there are more than one process involved, how to compare expansion rate of the universe with rate of these processes. It is quite natural for a particle to undergo more than one process simultaneously in thermal equilibrium.
In this case one has to solve Boltzmann transport equations (relativistic form). This is a first order differential equation in time. One solves Boltzmann equations to get number density of the particle when there are more than one process. Abundance is roughly m n/rho_c. Here m is mass of the particle n is the number density and rho_c is the critical density. I am attaching a typical plot of the number density, which I have downloaded from net (https://ned.ipac.caltech.edu/level5/Kolb/Kolb5_1.html). Here the abundance becomes constant after a value of Mx / T is reached. This means that the particle number density no longer follows the equilibrium trajectory, or the particle has decoupled from the rest of the universe.
You had also asked about the role of chemical potential. Note that in the Boltzmann equation there are distinct contributions such as,
dn/dt = contribution from expansion + contribution from scattering + contribution from annihilation + contribution from decay +...
The expansion term is always present. But depending on the case you are considering, you have to include thermally averaged cross sections and decay rates. In doing so, that is in doing the thermal averaging, one has to integrate (over momenta) |M|2 folded with a statistical distribution function of the form e-(E+μ)/T for particles and e-(E-μ)/T for antiparticles. Here M is the matrix element and μ is the chemical potential. You can see the paper by Kolb and Wolfram Nucl. Phys. B172, 224 (1980).
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