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?