How does the photon scattering rate imply the universe is transparent? In Barbara Ryden's Intro to Cosmology, she calls the 'epoch of photon decoupling' as the time when the rate at which photons scatter from electrons becomes smaller than the Hubble parameter. I am having problem with understanding why the fact that the universe is expanding faster than the rate at which photons are scattered, tells us that the universe became transparent? The rate at which photons are scattered is a function of the expansion, so it is clear why the two are connected, but how does this imply transparency?
 A: The Compton scattering rate of photons is easily related to the mean free path of photons as
$$\Gamma=\frac{c}{\lambda}$$
where $\Gamma$ is the Compton scattering rate and $\lambda$ is the mean free path.
Moreover, the Hubble parameter can be related to the approximate size of the cosmological horizon as $H^{-1}$. From that, you can see that
$$\Gamma<H\to\lambda>H^{-1}$$
That is, a scattering rate less than the Hubble rate means the mean free path of photons is greater than the approximate size of the cosmological horizon. Why does that mean the universe is transparent? Because when the mean free path is smaller than the horizon, there is a higher probability that photons will be scattered, which leads to a more opaque universe. But, when the mean free path is larger than the horizon, there is a greater probability that photons can essentially travel forever without scattering (allowing them to travel for billions of years and eventually be detected by us). They won't be able to interact with anything outside their horizon, so for all intents and purposes, we have free streaming. Obviously, this is a transparent universe.
