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Imagine the following situation: more and more accurate measurements of the average density of the Universe reveal that it is greater than the critical one, which corresponds to the model of a closed Universe with positive curvature. Will this necessarily mean that the entire large Universe is closed and has a positive curvature? Could it be that the curvature will simply be positive in our region of the large Cosmos, but at very large distances it will change, or is this impossible?

Qmechanic
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Arman Armenpress
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    A fundamental assumption of cosmology is that the universe is the same everywhere and in every direction and hence curvature would be the same everywhere. It's not impossible for that not to be the case however when we look into the night sky it appears that way – Toby Peterken Dec 23 '20 at 13:52
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    The spatial curvature of the universe could have some more exotic topology that isn't described by the FRW metric (which is $\mathbb{R} \times \mathcal{S}$). e.g. https://arxiv.org/abs/1903.00323 – Eletie Dec 23 '20 at 13:53

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If the universe had a positive average curvature then it would be finite. The converse is not correct - a universe with zero or positive average curvature could be either finite or infinite, depending on its overall topology.

Note that even a finite universe could still be so large that the limited size of the observable universe might mean we can only observe a very small part of it.

gandalf61
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  • Maybe on a very large scale, the Universe is inhomogeneous, there are regions with a large amount of matter (where space is curved) and with less (where it is almost flat or negative). I'm wondering if there is any reason to believe that this is not the case, other than that we proceed from the assumption of the homogeneity of the entire universe? – Arman Armenpress Dec 23 '20 at 13:57
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    @ArmanArmenpress Just to add, the answerer is correct when they talk about 'average positive curvature' as they're talking about the whole universe. A semi-local measurement indicating positive curvature doesn't conclude a finite universe though, which seems to be what you're suggesting! – Eletie Dec 23 '20 at 14:16
  • @Eletie That is, the measurement of positive curvature does not give absolute confidence that the universe is finite? – Arman Armenpress Dec 23 '20 at 14:18
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    @ArmanArmenpress I think absolute confidence is always pretty hard to ascertain, but that observation would be strong evidence for a finite universe. But one can still consider more exotic topologies and cosmological models (and people most certainly still would). – Eletie Dec 23 '20 at 14:24
  • @Eletie And what is meant by the average curvature? For example, if in our observable part the curvature turns out to be positive, and at a distance of 100 billion light years it becomes negative, and then changes again at some distance, will it be necessary to calculate the average value from all this? – Arman Armenpress Dec 23 '20 at 14:28
  • @ArmanArmenpress We talk about the average because the universe isn't perfectly homogeneous, so we need to account for small overdensities & underdensities. & the only reason I allow for these exotic possibilities is with reference to inflationary models (or models within string theory) which make specific predictions about the this type of thing - or else we shouldn't really be talking about the anything beyond our observable patch at all. We obviously can't do any calculations with data we can't observe (because it doesn't exist). But I think your question was answered well by ganadalf61. – Eletie Dec 23 '20 at 14:42
  • @Eletie And how do we know that the entire universe is homogeneous? Could it be that in our observable region the curvature will be positive, and on a much larger scale, the universe will be flat? – Arman Armenpress Dec 23 '20 at 14:49
  • We can only talk about the observable universe, so there's no real point in considering something like this (unless it's a specific prediction from a precise theory which has other observational signatures). It couldn't be as simple as the universe being flat on a larger scale because you need to consider its evolution through the field equations (i.e. it wouldn't stay this way trivially). As I said before, you'd need some more exotic topology (check the arxiv link in my comment). I'll leave the comments there to prevent this becoming too cluttered, maybe you could ask this as a new question. – Eletie Dec 23 '20 at 14:57
  • If the universe had a positive average curvature then it would be finite This is misleading statement, leading to the wrong conclusion, because the “averaging” in this sentence in not the same averaging needed to obtain Friedmann equations. – A.V.S. Dec 23 '20 at 15:26
  • It may be true that a positive global average spatial curvature implies a spatially finite universe, but this question is about the average curvature of the observable universe. I don't think this answer addresses the question at all. – benrg Dec 24 '20 at 01:51
  • I wonder if the concept of alternative topology is helpful when comparing the configuration of our observable universe with the entire universe. If might be helpful if the alternative topology preserved the assumption of homogeneity, but I am unaware of any such topology. – Buzz Dec 31 '20 at 17:43
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Yes, it's possible that the uniformity that we observe is only local.

The Lemaître–Tolman–Bondi family of exact solutions to GR generalizes the FLRW and Schwarzschild geometries, and can describe, for example, a huge uniform FLRW region with a slight positive curvature (small enough to be consistent with current data) surrounded by an infinite Schwarzschild vacuum. I think similar exact solutions can be constructed with a nonzero cosmological constant.

If you want the FLRW region to be irregular in shape instead of precisely spherical, or you want to include radiation or any matter with a peculiar velocity, then there's no hope of finding an exact solution. But that doesn't mean such configurations don't make sense. There's no special cosmological magic to the FLRW geometry; it just happens to be the gravitational field of a matter distribution that's simple enough that its gravitational field can be calculated exactly. The lack of more realistic solutions is due to the difficulty of solving coupled second-order partial differential equations and not any deeper property of GR.

In some inflationary models, inflation starts in a small region of a larger non-FLRW universe when the conditions for it are randomly met in that region. The rest of the universe is still there after the end of inflation. As far as I know, the almost-uniform almost-flat region produced in these models has a finite volume.

benrg
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  • Thanks for the answer. One clarification. Is it necessary to introduce an infinite Schwarzschild vacuum surrounding the FLRW region? – Arman Armenpress Dec 24 '20 at 12:26
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    @ArmanArmenpress No, that's just an arrangement for which an exact solution happens to exist. In principle there could be practically anything out there if the edge of the FLRW region is far enough from us. In the sort of inflation model I was talking about, the larger universe is assumed to be nonuniform since explaining the uniformity of our region is the point of inflation. – benrg Dec 24 '20 at 17:37
  • Let's say that the entire universe is many orders of magnitude larger than the observable part. And let's say, sometime in the future, more accurate measurements will show that the space in the observed part is slightly curved. Could this, among other models, correspond to the following scenario: the observable universe is a tiny portion of the cosmic region of high density that positively bends space. This region is analogous to our galaxies. – Arman Armenpress Dec 24 '20 at 17:42
  • There are many such regions. However, there are also regions (again, much larger than the observable universe) of low density in which space is either flat or negatively curved (analogous to voids). And on average, these "mega galaxies" and "mega voids" make the entire large Universe flat. Could this kind of large-scale structure exist? – Arman Armenpress Dec 24 '20 at 17:42
  • @ArmanArmenpress You could have large regions of higher and lower density in a critical-density universe. The problem is that the CMBR is incredibly uniform. If the local curvature was large enough to detect then the whole higher-density region couldn't be dramatically larger than the observable universe. I doubt there's any model that could produce that uniformity on a scale not much smaller than the size of the whole fluctuation. A few people have suggested that our region could look more uniform than it is because we happen to be at the exact center of it, but that seems pretty implausible. – benrg Dec 24 '20 at 18:37
  • George Ellis, for example, writes that positive curvature does not necessarily mean that we live in a closed universe. Maybe, he writes, we are inside a lump of high density, which is surrounded by an area of low density. How does this contradict what I wrote? Why shouldn't other high density lumps exist? – Arman Armenpress Dec 24 '20 at 19:17
  • @ArmanArmenpress It's certainly possible; I just don't see how it could come about with a detectably large positive curvature. Imagine drawing circles on a nonuniform surface. At small scales you get what seems to be a perfect circle with a circumference of $2πr$. At large scales you don't get a circle at all. There's no scale at which you'd expect to get a perfect circle with a circumference less than $2πr$. I think Ellis is one of the people I mentioned in my last comment. His model is non-FLRW even within the observable universe, so direct confirmation of it would directly overthrow FLRW. – benrg Dec 24 '20 at 19:52
  • Following the link, I tried to illustrate the idea: https://ibb.co/syJ0Tw2 It is not necessary that the curvature be large, it can be small. Regions with higher density are shown in black, and regions with lower density are shown in red. The yellow circle is our observable universe. Does this contradict any ideas? – Arman Armenpress Dec 24 '20 at 20:43
  • That is, I want to say that although the universe as a whole is not homogeneous, we can live in a small part of some homogeneous region. – Arman Armenpress Dec 24 '20 at 20:50