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The CMB, once the Doppler shift from the Earth's motion is subtracted, is fairly uniform, which seems to imply that all the matter that emitted it moved at more or less the same velocity when it did. It seems unlikely that all the matter in the universe at that moment would be moving at the same velocity, so I think it is natural to deduce that an observer far away, with their own observable universe, would likely see their own CMB as if emitted by matter moving at a different velocity. Is this right?

And if, as I think it is with my limited knowledge, this smoothness of velocity was caused by inflation, it doesn't seem like such a smoothness could be explained by the particle picture of matter. There is no obvious (to me) reason why particles right next to each other would move at similar velocities. So, my reasoning would be, inflation stretched the universe very rapidly, along with the quantum fields that compose it. The electron field, for example, would be stretched so much that, if you were to look at a small portion of it, the electron field would have the form of a plane wave. As such, any electrons and positrons inside this portion of the universe would, at this moment, have more or less the same momentum. Is this also right?

And, assuming it is, why would all the fields (electron and quark for example) have the same average momentum, so all matter has the same average motion? Or would they have had different momentums that cancel each other out once they interact?

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    $\begingroup$ Maybe I'm not seeing the precise point of your question. Of course there's variation in the velocities of the particles in a plasma or gas. But the temperature during the era of recombination was from 4000 K to 3000 K, so the range of velocities isn't huge. $\endgroup$ – PM 2Ring Oct 28 '19 at 7:37
  • $\begingroup$ I will suggest you to look royalsocietypublishing.org/doi/full/10.1098/rsta.2011.0289 $\endgroup$ – Layla Oct 28 '19 at 21:02
  • $\begingroup$ Also search for Ehlers–Geren–Sachs theorem. $\endgroup$ – Layla Oct 28 '19 at 21:03
  • $\begingroup$ @PM2Ring Actually, now that I think of it, the Boltzmann Distribution of particle velocities in a gas has the velocities relative to the center of mass of the gas as a whole. But how would one obtain the velocity distribution of an infinite gas in equilibrium, if it has no clear center of mass? $\endgroup$ – Phineas Nicolson Oct 28 '19 at 22:44
  • $\begingroup$ @Reign I'll have a look $\endgroup$ – Phineas Nicolson Oct 28 '19 at 22:44

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