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A particle (say neutrino) decouples from the plasma in the early universe when its interaction rate $\Gamma$, with the plasma, is slower than the expansion rate $=H = \frac{\dot{a}}{a}$, i.e. $\Gamma < H$.

One rough explanation is that: when the universe expands, the two particles never get to interact with each other, as the distance between them keeps growing. On the other hand, if the interaction $\Gamma$ is much faster than $H$, then the universe can be assumed to be pseudo-static for the interaction purpose.

But this picture does not seem very compelling and definitely not rigorous. In an S-matrix approach (from QFT), the input and output particle wavefunctions are taken at asymptotically infinite time ($+\infty$ for outgoing and $-\infty$ for incoming). So, the S-matrix in QFT doesn't actually deal with the above situation in Question.

So, the question is: Is there a somewhat rigorous rationale for saying that $\Gamma < H$ for decoupling? Is there any explanation based on QFT in curved spacetime?

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You're absolutely right. In the case where $\Gamma < H$, then we are out of equilibrium!

But I guess we should have said that in the case $H<\Gamma$ we have a thermal state, so need thermal quantum field theory.

Small departures from equilibrium can be dealt with when linearised in the imaginary time formalism, but, to consistently deal with the case $\Gamma<H$, then one would need non-equilibrium QFT and use the Keldysh in-in formalism.

But decoupling doesn't have any interesting physical phenomena, so we don't normally worry about these details. I'm not 100% sure but it's more important for baryogenesis & leptogenesis (and reheating in the inflationary era).

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