Big Bang Glow

Using the ultra reliable Wikipedia I see at

that the mean free path of a photon in deep space is approx 10 billion light years. That of course is the mean and so photons will travel further or less in some linear(?) distribution.

By deduction some 50% of photons from beyond 10 billion light years won’t reach us. They will be absorbed by some atom and re-radiated in a random direction with vanishingly small probability that direction will be towards Earth

By corollary 50% of daughter photons will hit earth out of some foggy glow after being re-radiated uniformly from space all around us ( actually that figure is too low as there is more visible universe than 10 billion light years).

So we have more energy from the fog than there is from distant galaxies. This fog radiation will be pretty uniform - assuming uniform gas distribution in deep space - and I expect it will be at low thermal temperatures.

How is this fog glow any different to a ‘big bang’ glow? And is it accounted for in the various cosmological theories?

Specifically we’re talking about Thomson scattering here and the intensity of the scattered radiation will depend on the cosine of the scattering angle, so there is a bias to photons being scattered parallel or antiparallel to their initial direction.

I think the 10 billion ly figure quoted may be for comoving distance, but tbh I’m not sure.

Thomson scattering does not change the wavelength, so there is no way of distinguishing between the portion of the CMB that has been scattered and that which has not been scattered and the effect will be to increase the isotropy of the CMB over small angles. Clearly this must be accounted for more detailed comparisons between observations of the CMB and theoretical cosmological models.