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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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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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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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