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If my understanding is correct, the Cosmic Microwave Background always come from our event horizon. By event horizon I mean the edge of the observable universe.

If the universe is finite does it mean that after some time (maybe in billions of years?) there will start being a hole in CMB. As our event horizon let us see CMB from regions further and further away, isn't there a time where it will reach the edge of the universe?

If this is the case, based on the known number of atom and mass of the universe, is it possible to calculate when this will happen?

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First, let me clear up a misunderstanding:

Particle horizon

The "edge" of the observable Universe is called the particle horizon, and lies roughly 47 Gly (billion lightyears) away. It is always receding, both because the Universe expands and because light from increasingly large distances eventually reach us. In comoving coordinates (the coordinate system that expands along with the Universe), it asymptotically reaches a finite size (some 63 Gly), but in physical coordinates it increases forever$^\dagger$.

Event horizon

The event horizon is the boundary between the part of the Universe that contains the galaxies (and other stuff) from which light emitted today will be able to reach us, and the part where it won't. Currently, it is roughly 17 Gly away, but his horizon is always shrinking because the expansion of the Universe accelerates. In other words, stuff outside the event horizon is so far away that the expansion carries it away too fast for the photon amitted today to ever reach us.

Thus, at Big Bang (BB), the event horizon was 63 comoving Gly away, because that was the maximum distance from which we would ever receive information.

The CMB will always lag behind the particle horizon

The farther away we look, the more we look back in time. If there were galaxies a second after BB, and if we had the technology to look that far, we would see them right after BB, and they would be 47 Gly away. However, galaxies weren't born until several hundred Myr (million years) after BB. And now to your question: The CMB wasn't emitted at BB, so it doesn't origin from a region that is now 47 Gly away. Instead, it was emitted 380,000 yr after BB, and it was emitted from all of the Universe, in all directions.

This emittance wasn't instantaneous, but almost (took something like 10,000 yr), so let's assume for now that it was. Let's call that time $t_\mathrm{CMB}$. In that case, because light travels at $c = 300\,000\,\mathrm{km}\,\mathrm{s}^{-1}$, if you were present one second after $t_\mathrm{CMB}$, you would see the CMB that came from all regions that were 300,000 km away. That is, the CMB would be the light from a thin shell of radius 300,000 km.

Two seconds after BB, the CMB would be the light from a shell of radius 600,000 km, and so on. However, because the Universe is expanding, it is actually at little more than 600,000 km in radius. And now, 13.8 Gyr after BB, and hence 13.7996 Gyr after the CMB was emitted, this shell is not almost 14 Gyr away, but in fact 45 Gly away which is almost, but quite, at the particle horizon.

Because of the expansion, the distance to the CMB shell always increases. But so does the distance to the particle horizon, and because the CMB was emitted after BB, the particle horizon will always "be ahead" of the CMB. As the CMB shell recedes, it becomes more and more redshifted, slowly fading into darkness but never disappearing. This is irrespective of whether the Universe is finite or infinite, except in the possible, but rather fantastic, event that the Universe is larger than the ~47 Gly we can see today, but smaller than the 63 comoving Gly that is the maximum size we will ever see.

Finite or infinite?

We don't know if the Universe is finite or infinite, but if the un-observable part of it looks like the observable Universe, then it is infinite. What we can say is that, at least in the observable Universe, the CMB was emitted everywhere in space, and there is currently no reason to believe that it was different elsewhere (although it could bee, if it expanded differently in other parts). To the best of our knowledge, space is homogeneous, meaning that (on average) there are as many baryons, photons, dark matter particles, and bicycles in one place as there is in another. In other words, to the best of our knowledge, there is no edge. Even if it's finite, it is difficult to (and most, if not all, astronomers don't) imagine an edge; rather it is thought that it would "curve back on itself", just as the surface of a ball is finite in area, but doesn't have an edge.

2D analogy

As an analogy, imagine living on a flat, inifinite Earth, populated evenly by people with loud voices. At some point in time, everbody yells "CMB". First you'll hear your neighbors yell, after 1 sec you'll hear those living 340 meters away, after 1 minute those living 20 km away, after a year those 10 million km away, and so on and so on forever.

Now imagine the same on a round Earth. After 16 hours, you would hear the folks living at your antipode, and if I understand correctly, this is when you expect to "hear a hole in the sound". But now you would start hearing the ones that are closer, but whose sounds have been traveling more than halfway around Earth. After 32 hours, you'd hear your own yell. And so on.

But, our Earth analogy is also expanding. If it were expanding at a steady rate, we would eventually hear our antipode friends. You might think that if it were expanding fast enough, their sound would never reach you, but as explained by the ant-on-a-rubber-band puzzle, that's not the case).

However, our fictional Earth is not only expanding, but in fact this expansion is accelerating. And in that case, there is a limit to how far away you can hear people.

Back to the real Universe, there is a limit to from how distant regions you may receive CMB photons (or any other information). Moreover, if the Universe were finite, and were not accelerating, you might think that you would eventually see the CMB that was emitted from our own neighborhood and had been traveling around the Universe. However, it can be shown that, in this case, the Universe will reach a maximum size, start collapsing, and end in a Big Crunch exactly at the time where you expect to see your antipode.

Only in a finite Universe with a rather special mix of gravitating matter and accelerating dark energy might it be possible to delay the collapse or the eternal expansion to a point where we eventually look back at ourselves. But that's sort of fine-tuning.


$^\dagger$Unless the Universe contains a yet unknown substance that is able to halt the acceleration. This is entirely possible, but in this answer I consider only a well-behaving Universe that acts according to current knowledge.

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  • $\begingroup$ I'm confused by the statement that the edge of the observable universe asymptotically reaches a finite size. I don't understand this. Could you enlighten me, please? $\endgroup$ – jmh Dec 9 '17 at 0:50
  • $\begingroup$ If I understand, because the expansion of the universe between us and the CMB shell is exponentially increasing and happen so fast that light from the CMB shell can never reach us, we will never have a hole in the CMB, and all we will see is more redshift in the CMB. Thank you for your answer. $\endgroup$ – user1097111 Dec 9 '17 at 23:59
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    $\begingroup$ @jmh It's slightly complicated, but hold on: If the Universe were expanding at a steady rate, there would be no limit how far we could see, in principle. We just have to wait long enough for the light to arrive. Even if a galaxy now is receding at twice the speed of light, or $10c$, or a million $c$, eventually the light would reach us. In say "in principle", because in practice the light is redshifted and diluted beyond detection. However, due to the accelerated expansion there is a limit to how far a galaxy can be and still emit light that is able to reach us [cont'd below] $\endgroup$ – pela Dec 10 '17 at 13:32
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    $\begingroup$ […] The farthest we can see at the moment is (in principle) 47 Gly. In, say, 15 Gyr from now, a point that is now 47 Gyr away will be roughly 90 Gly away, due to expansion. But since light from more distant regions will at that time have had the time to reach us, we can see farther than 90 Gly, in fact roughly 110 Gly. So, all the time points which we see now will get farther away. In comoving coordinates, however, all points lie still (because comoving coordinates are defined to expand along with the Universe). [cont'd below] $\endgroup$ – pela Dec 10 '17 at 13:33
  • $\begingroup$ […] In comoving coordinates, a point that is now 47 Gly away, will always be 47 Gly away (because comoving coordinates are defined to coincide with today's coordinates). But in 15 Gyr, will will be able to see 55 comoving Gly away. In, say, 30 Gyr, we will be able to see roughly 60 comoving Gly away, which in physical coordinates will at that time be ~250 Gly. In comoving coordinates, you will never be able to see farther away than 63 Gly, but in physical coordinates this so-called particle horizon always gets farther away. $\endgroup$ – pela Dec 10 '17 at 13:33

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