# Particle decoupling, universe expansion and QFT

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?

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, then one would need non-equilibrium QFT and use the Keldysh in-in formalism.