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I remember back in the day, I took a course in cosmology where I was taught that there are three possible types of universes based on their curvature, the flat universe which may be compact or infinite, the spherical which is compact and the hyperbolic which is infinite.

I don't understand why a hyperbolic universe must be infinite, I know many manifolds which admit a hyperbolic structure and are compact, in fact most manifolds are hyperbolic, an example is, all the surfaces of genus greater than or equal to 2.

Now I am not a physicist and don't have good physics intuition, so I want to ask why does hyperbolic universe means infinite universe? Have I misunderstood something?

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You're correct that in general a hyperbolic geometry does not imply the manifold is infinite: it could have some non-trivial topology and be compact (e.g. 3-torus). In FLRW cosmology we usually assume$^1$ the topology is simply-connected, in which case negative curvature does imply an infinite universe, but of course this isn't the only option.

e.g. see What are the allowed topologies for a FRW metric?

Edit:
$^1$I'd also add that if one picks some of these more interesting compact topologies with constant negative curvature for the FLRW metric, then there are no continuous group isometries (no global Killing vector fields) and the spacetime is not isotropic at every point. So there are physical reasons why people often make the assumption that hyperbolic implies infinite, but it should be stated explicitly.

Eletie
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  • https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric This says homogeneity is necessary for this metric, and therefore no odd topology is acceptable. – Buzz Jan 08 '21 at 15:16
  • @Buzz what I mean is, starting from the FLRW metric, without assuming simply-connectedness, you can obtain non-trivial topologies by identifying different points on the spatial 3-manifold: but it does ruin the isotropy, which is the reason for considering the metric in the first place. So I agree if you want to stick to homogeneity and isotropy globally then you do restrict the topology, but you can consider deviations in general. – Eletie Jan 08 '21 at 15:40
  • @Buzz to be more specific, the symmetries implied by assuming isotropy and homogeneity (i.e. in FLRW spacetimes) determine the geometry locally, not globally. One is free to still make these identifications to get nontrivial spacetime topologies. I don't see anything on the wiki in disagreement with this. – Eletie Jan 08 '21 at 15:55
  • Hi @Eletie. I really am unable to understand this distinction. My understanding is that homogeneity implies, for example, that for any sufficiently large sphere of a specified volume in the universe, there is a value for its property, for example say density, which is very closely approximately the same as for any other such sphere. I would guess that the locality refers to the slight differences of the value in different spheres. A different topology would create much different observable values for different spheres. – Buzz Jan 09 '21 at 18:22
  • https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#Evidence says astrophysicists now agree that the universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. – Buzz Jan 09 '21 at 18:27
  • @buzz a different global topology wouldn't necessarily produce different local observables (by local here I mean anything we could ever observe). You can have local homogeneity and isotropy but different global topologies which can be compact. For a good discussion which explains this in detail see The Large Scale Structure of Spacetime, Hawking & Ellis, section 5.2 on Robertson Walker spaces. – Eletie Jan 09 '21 at 19:38
  • @buzz also there are notion of globally isotropic and locally isotropic manifolds, which also gives more room to explore different topologies whilst still not being in conflict with observation. But to summarise, in FLRW we assume things like connectedness and choose to make no global identifications, not because they'd be wrong or in conflict, but because they're simpler or more natural. So the answer to OP's question is as I wrote in my answer, that $k<0$ doesn't necessarily imply an infinite universe. – Eletie Jan 09 '21 at 20:01
  • @Electie I have recently made a similar coment about another Q: Topology in cosmology. There is a distinction betrween (a) a mathematical specuation that cannot be verified with observation, and (b) a observational based presentation with probability ranges for various model variables. – Buzz Mar 15 '21 at 21:11
  • @Buzz I fail to see a single physical reason why a non-trivial topology is "mathematical speculation" whilst a trivial topology isn't. You cannot calculate probabilities for things that we have no evidence for or against, or that have one singular occurrence, unless you have a physical model of how they work (which we obviously don't with cosmology). Moreover, the other answers are in agreement with mine here, and in disagreement with your comments that "no odd topology is acceptable": other topologies are indeed possible even with isotropy. – Eletie Mar 15 '21 at 23:46
  • My point is that there ARE models for what you call "trivial topology" (i.e 3D hypersphere, 3D Eucliean flat and 3D hyperbola) and I have not been able find any cosmological models for any "non-trivial topology". – Buzz Mar 16 '21 at 02:07
  • @buzz I think you're still missing the point. Cosmological models generally don't depend on the global topology, only the local (observable universe) geometry. You can find many papers that consider non-trivial topologies and prove they're consistent, and this has been known since the 70's. You seem to think having a different global topology necessarily leads to a different cosmological model: it doesn't. – Eletie Mar 16 '21 at 11:15
  • Just take a look through any of the hundreds of papers from this search: https://scholar.google.com/scholar?hl=en&as_sdt=2005&sciodt=0%2C5&cites=10106687952996312853&scipsc=&q=topology+cosmology&btnG=&oq=topology+cosmolog or this recent review https://arxiv.org/abs/1903.00323 – Eletie Mar 16 '21 at 11:17
  • @buzz Lastly, this whole discussion started because you thought non-trivial topologies were ruled out by homogeneity/isotropy. I assume you at least now see that isn't the case? I've also never seen any papers claiming a simple topology is 'more likely' than a complicated one, because this is a calculation that cannot be done. On the contrary, many people will say a simple topology is 'easier to work with' (with no observational differences), but that is all. OP's question and knowledge about more interesting compact finite manifolds being consistent is correct, as said in my answer. – Eletie Mar 16 '21 at 11:36
  • "...you thought non-trivial topologies were ruled out by homogeneity/isotropy. I assume you at least now see that isn't the case? " I like the referenes you cited, I have only just started reviewing them. I have not yet seen any constancy between non-trivial topology and isotropy. The first item that mentioed this topic acknowledged they were not compatible. https://arxiv.org/abs/1903.00323 "a surprising dynamical mechanism at work within the (vacuum) Einstein `flow' that strongly suggests that many closed 3-manifolds that do not admit a locally homogeneous and isotropic metric..." – Buzz Mar 17 '21 at 16:02
  • @Buzz the continuation of that quote "... that many closed 3-manifolds that do not admit a locally homogeneous and isotropic metric at all will nevertheless evolve, under Einsteinian evolution, in such a way as to be asymptotically compatible with the observed, approximate, spatial homogeneity and isotropy of the universe". This is saying the exact opposite of what your partial quote seems to imply! I.e. as well as the many other topologies that do* admit a locally homogeneous and isotropic metric, there are still more that don't yet are still compatible with observation*. – Eletie Mar 17 '21 at 16:07
  • @Buzz I'm baffled that you can still say something like "I have not yet seen any constancy between non-trivial topology and isotropy". This has been mathematically demonstrated and claiming otherwise is just definitively incorrect. The Hawking & Ellis text I referenced way earlier shows this too, as does every other work on the topic. At this point I believe you're being disingenuous; either a fatal misunderstanding of the mathematics or purposefully ignoring what is basically consensus in the literature. So I will not go on with this discussion here or elsewhere. – Eletie Mar 17 '21 at 16:13
  • I have triedyto find a copy of the Hawking and Ellis I can download. So far the closest I have gotten is https://www.cambridge.org/core/books/the-large-scale-structure-of-space-time/1E6B961EC9878EDDBBD6AC0AF031CC93 . This gives a table of contents with a PDF link for each item. – Buzz Mar 17 '21 at 20:51
  • There are 10 chapters: 1. The role of gravity, 2. Differential Geometry, 3. General Relativity, 4. The Physical Significance of Curvature 5. Exact solotions, 6. Causal Structure, 7. The Cauchy Problem in GeneralRelativity, 8. Space-time Singularities, 9. Gravitational Collapse and Black Holes, 10.The Initial Singularity in the Universe. Regarding the topic of our disagreement, which do you suggest I look at first? – Buzz Mar 17 '21 at 20:51
  • CH 5 quote: "The spatially isotropic and homogeneous cosmological models are described in §5.3, and their simplest anisotropic generalizations are discussed in § 5.4." CH 10 has "isotropic", but not "anisotropic". – Buzz Mar 17 '21 at 21:17