What was the mean distance photons travelled before colliding with electrons in the matter plasma before recombination?
I have checked other answers close to this but they only mention a mean distance but not what it actually is.
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Since the cat is out of the bag and I calculated this in another (non-duplicate) question, I'll repeat the calculation here.
The baryon density today is about $\rho_0 \sim 4\times 10^{-28}$ kg/m$^3$, then at a redshift of $z\sim 1200$ (just prior to recombination), the number density of free electrons (assuming a fully ionised hydrogen gas) is roughly $$n_e = \frac{\rho_0 }{m_u}(1+z)^3= 4\times 10^8\ {\rm m}^{-3}$$
The mean free path of a photon in the plasma is $1/(\sigma n_e) = 4\times 10^{19}$m, where $\sigma$ is the Thomson scattering cross-section. Thus the average photon can travel about 4000 light years before being scattered.
Since the "size" of the universe is $\sim ct$, where $t$ is the time since the big bang, and since $t \sim 300,000$ years at $z\sim 1200$, then the universe is effectively opaque to the radiation within it.
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$\begingroup$ Thanks Rob can you explain the last statement ie how that demonstrates the opaqueness, as I dont understand that considering photons are travelling 4000 light years. Does that mean any restriction on how far a photon can travel creates opaqueness? $\endgroup$– JonathanCommented Apr 10, 2020 at 8:08
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$\begingroup$ @Jonathan it means a photon cannot travel in a straight line across the universe. After (re)combination, the mean free path increases by several orders of magnitude and becomes bigger than $ct$. $\endgroup$– ProfRobCommented Apr 10, 2020 at 8:27
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$\begingroup$ Thanks Rob so its not how far photons can travel that has anything to do with the opaqueness its back to the fact that they are continually scattering off free electrons. $\endgroup$– JonathanCommented Apr 11, 2020 at 4:37
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$\begingroup$ @Jonathan that's a bit subtle, however I think you can see (or rather you couldn't) that any light source placed much more than 4000 light years away would not be visible. Like looking into fog. There are other absorption processes, with lower cross-section that come into play when photons are scattered many times. $\endgroup$– ProfRobCommented Apr 11, 2020 at 7:06
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$\begingroup$ Thanks Rob the fog has lifted! :) $\endgroup$– JonathanCommented Apr 13, 2020 at 6:08