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As I understand it, even though our current measurements of the Riemann curvature of the universe overlaps with flat spacetime, this doesn't restrict the shape of the universe to being infinite. It's still possible that we live in a closed shape (such as S3). And a closed shape like S3 or S1xS1xS1 (the latter of which isn't isotropic, but I think that constraint only applies to the curvature, not shape of the universe, correct me if I'm wrong) combined with flat curvature means we live in the kind of manifold where you can look forward and see the back of your head.

In that case, is it possible, given the confines of what we know, that we live in a flat universe with a closed shape, and one of the ancient active galactic nuclei we're seeing is just the Milky Way closer to its infancy? This would be the cosmological version of us looking forward and seeing the back of our heads. I suppose this would mean that the size of the observable universe is bigger than the actual size of the universe. Is that possible given what we expect from the historic expansion rate of the universe?

Keep in mind that I'm not asking if this is the case, I'm only asking if it's possible given what we know.

EDIT: What I mainly want to know is this. If the universe is finite-shaped, does the historical expansion of the universe, as we understand and believe it happened since the time of ancient AGNs, inherently prevent a photon from ever reaching the same point in the manifold at a future time? Assuming (which I believe is a safe assumption) that the universe was matter-dominated both back at the time when AGNs dominated and is still so today, the scale factor of the universe grew as $t^{2/3}$ that whole time. Let's say, hypothetically, that the universe has an S3 shape and, at some time $t_0$, it had a radius $R_0$ and the Milky Way had an AGN. Then if we call today $t$ the radius of the universe would be $R = R_0 (t - t_0)^{2/3}$, right? Does there exist some finite $t$ such that the distance travelled by a photon since $t_0$ is equal to a full revolution (inflation-adjusted) or does the matter-dominated expansion inherently prevent this?

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    $\begingroup$ Related: physics.stackexchange.com/questions/167999 $\endgroup$ – Kyle Kanos Aug 5 '15 at 16:11
  • $\begingroup$ @KyleKanos That question seems somewhat related, but it seems to be implying that the possibility in question is the universe being a hypersurface of a larger dimensional space, and the accepted answer imposes curvature on that metric given the assumption. Mine is clearly different. I'm assuming the possibility of a flat spacetime with a closed shape. $\endgroup$ – Bridgeburners Aug 5 '15 at 16:24
  • $\begingroup$ @HDE226868 Those questions aren't really related. They're only asking if it's possible that the Milky Way used to be an AGN, which I already knew was possible. I'm asking if it's possible, given the FRW solution, that the Milky way was an AGN that's currently visible in our line of sight. $\endgroup$ – Bridgeburners Aug 5 '15 at 16:25
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    $\begingroup$ @Bridgeburners They're relevant. If the Milky Way was never a quasar, then clearly any quasars we see can not be the Milky Way. $\endgroup$ – HDE 226868 Aug 5 '15 at 16:27
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Is it possible that one of the quasars we see is an ancient Milky Way?

It's possible I suppose. I can't rule it out.

As I understand it, even though our current measurements of the Riemann curvature of the universe overlaps with flat spacetime

Yes, it would appear from WMAP and the Planck mission that the universe is "flat". And what this means is that light goes straight$^*$. We have no evidence that light somehow goes round in some great big circle.

this doesn't restrict the shape of the universe to being infinite.

Agreed. This doesn't mean the universe is infinite. We have no evidence whatsoever that the universe is infinite. Some people say the universe is flat and therefore it must be infinite, but that's a non-sequitur. It just doesn't follow, and IMHO it's totally at odds with Big Bang cosmology.

It's still possible that we live in a closed shape (such as S3). And a closed shape like S3 or S1xS1xS1 (the latter of which isn't isotropic, but I think that constraint only applies to the curvature, not shape of the universe, correct me if I'm wrong) combined with flat curvature means we live in the kind of manifold where you can look forward and see the back of your head.

Like I said, everything is possible. It's also possible that the universe is shaped like a teapot.

In that case, is it possible, given the confines of what we know, that we live in a flat universe with a closed shape, and one of the ancient active galactic nuclei we're seeing is just the Milky Way closer to its infancy? This would be the cosmological version of us looking forward and seeing the back of our heads. I suppose this would mean that the size of the observable universe is bigger than the actual size of the universe. Is that possible given what we expect from the historic expansion rate of the universe?

The Planck mission found no evidence of any intrinsic-curvature toroidal universe, see http://arxiv.org/abs/1303.5086. And we have no evidence whatsoever of any "higher dimensions". IMHO if you want to pursue this you should look into the hall of mirror universe. You don't look forward and see the back of your head. You just see your own reflection.

$^*$ Actually, light can curve somewhat due to the expansion of the universe, a bit like a slimy slug track across a spreading pool of pitch. But it doesn't end up coming round full circle, and radial slug tracks don't curve at all.

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  • $\begingroup$ Thanks. I get the spirit of the answer here; anything is possible really. But we do have an idea of how, historically, the universe expanded. Since the universe was matter-dominated back when quasars were around and is still so today, we can assume roughly that the scale factor grew as $t^{2/3}$ in that interval. I'm wondering if that form of expansion inherently prevents photons from ever reaching the same point in a finite shaped manifold. A while ago I did a napkin calculation that convinced me that it doesn't, but I don't remember what I did, nor do I know if it was valid. $\endgroup$ – Bridgeburners Aug 5 '15 at 18:58
  • $\begingroup$ I don't think it was ever matter-dominated, Bridgeburners. Not really. Yes, people say it used to be radiation-dominated, then matter-dominated, and it's now dark-energy-dominated. But I know of no breach of conservation of energy, or any ongoing method by which substantial percentages of dark energy are converted into radiation or matter. Which means that in truth the universe has always been dark-energy dominated. $\endgroup$ – John Duffield Aug 5 '15 at 19:11
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    $\begingroup$ Also, the universe being spatially flat doesn't necessarily mean that it's not a flat 3-torus. But there's no real a priori reason for expecting that, either. $\endgroup$ – Jerry Schirmer Aug 5 '15 at 19:27
  • $\begingroup$ @ John Duffield--The arxiv item you cite gives, for the shape of a flat universe (like our own), the highest confidence level of probability for the "cubic torus". Surely that's the same as the 3-torus that Bridgeburner is asking about. You're implying that its curvature's extrinsic (not observable except to beings in a higher-dimensional spacetime), but, given the Kerr metric, what he's hypothesized seems to be an unusually large and complicated version of a watch observed to read "3:00" on different days. I appreciate the neutrality of your answer, but have I misunderstood "cubic torus"? $\endgroup$ – Edouard Feb 25 at 20:37
  • $\begingroup$ The comment I posted on Yimiller's answer shows that, on more careful consideration, I've realized that the watch might show "8:00" on one day and "3:00" on another, but I'm still hoping someone will let me know whether I'm right about the "cubic torus".. $\endgroup$ – Edouard Feb 26 at 19:02
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Figure 1 in this document: http://arxiv.org/abs/astro-ph/0310808 (Davis and Lineweaver, 2003) appears to show that it is possible to see 'the back of our heads' one or more times from successively greater distances. The comoving diameter of the universe would have to be small enough to encounter it one or more times on our historical light cone. Of course an ancient version of the Milky Way would be very hard to recognize- its location and orientation would be quite unpredictable! I have often wondered whether anyone has made a serious search for additional, older versions of some nearby astronomical objects that have unique characteristics.

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  • $\begingroup$ @ yimiller--"All events that we currently observe are on our past light cone", but the converse isn't true: The light cone represents the cross-section of a conical solid, and, as LD explain later in their caption, the light reaching us now is from much further away (in GLYR) than the universe is old (in GYR), due to spatial expansion (which is not relative motion). If you look closely at the top panel of LD's figure, you'll see that there is a missing portion, between the surface of the Hubble sphere and the particle horizon, that we'd have to reconstruct by guesswork. $\endgroup$ – Edouard Feb 26 at 18:21

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