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I have a question about cosmology. At popular level people explain the time of decoupling of matter and radiation as the moment when temperature falls enough for nuclei and electrons to recombine into atoms. People say that the Universe became “transparent”. The photon cross section by an electro-neutral system is smaller than Thomson's one, i.e., technically it means that the mean free path became very large or even infinite.

However, there is another mechanism for the mean free path to become infinite. It is because during its expansion the universe becomes less dense. Let me explain it by example. Imagine that you are in the forest where the diameter of tree is $a$, and mean distance between trees is $b$. What is the mean diameter of the observed area? It is proportional to $b^{2}/a$, I suppose. Now let's imagine that our forest is in the expanding universe, i.e., $b$ grows (linearly, for example) with time while $a$ remains constant. Then at some moment “the diameter of the observed area” starts to grow faster than the speed of light, i.e., becomes infinite.

It implies that the recombination is not necessary for the decoupling. Did the recombination start earlier than the moment I described above? Or both mechanisms (recombination and density fall) are equally important for the decoupling?

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You are right in that you don't need recombination to decouple and that Hubble expansion is an important part of decoupling. Decoupling basically happens when the interaction rate drops to the Hubble expansion:

$$\Gamma = H.$$

In terms of the cross section:

$$\Gamma = n_X \langle\sigma\,v\rangle,$$

where $n_X$ is the number density, $\sigma$ is the interaction cross section, and $v$ is the speed of the particle. So, if there is expansion the cross section need not vanish for decoupling. Also, even if the free electron number density didn't drop precipitously due to recombination, the photons would decouple eventually. See Modern Cosmology by Dodelson p. 73 for example (available in Google books).

Dodelson states the free electron fraction drops to about 2% to get decoupling, so in some sense recombination had to proceed partially (though not to completion) to get decoupling.

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I'm not a cosmologist, but I find you scaling argument a little off in that a) you scale the finite distance between trees, but want to keep the finite radii of your trees constant and b) mean free paths are usually inversely proportional to the cross sections, which scale with a^2 as well. Scaling both a and b with the same factor gives me a constant mean free path.

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For the sake of simplicity, I used 2D analogy, thus the cross section is "a". It is also well known that cosmological expansion hasn't an influence on the size of proton/electron/atom/etc at the late stages of the expansion. Certainly, there is a some sort of tidal forces but at the time of radiation-matter decoupling their influence was absolutely negligible. – Grisha Kirilin Nov 10 '10 at 12:36

The only thing I can think of for the 'diameter of the observed area' is the inter-atom distance within the IGM, and for recombination to not take place, I would assume that the IGM in this model would still be hot enough to be a plasma. A naïve answer would be that this is unlikely, considering that we're still not at infinite redshifts with respect to neighboring galaxies, much less the IGM between us and the neighboring galaxies.

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In plasma physics there is a notion of plasma transparency or so. When you heat plasma in TOKAMAK, it is first transparent to radiation and the heat losses are essential. The plasma radiates "from volume". With temperature increasing the plasma becomes non transparent and radiates only "from surface".

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