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I thought that it was generally agreed that the universe seems to be flat, according to the Planck data, but every once in a while I'll hear someone say that the universe is hyperbolic. I know this isn't a very formal question, but I'm wondering where this idea comes from? Is it just a common misinterpretation of something, or are they right?

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    $\begingroup$ I suspect the reference to a hyperbolic universe refers to the de Sitter geometry. If we are right about dark energy the universe is already approximately de Sitter and will become increasingly close to de Sitter as the expansion continues. Remember that while the universe is spatially flat in comoving coordinates it is certainly not flat in the sense of zero curvature. But unless you can cite specific examples of this terminology any answer is going to be a matter of opinion. $\endgroup$ Feb 28, 2021 at 19:04
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    $\begingroup$ Is it possible you mean that the metric in GR should be globally hyberbolic? If so, this is not a statement about spatial or space-time curvature, but rather a statement about causality; loosely speaking there should not be closed timelike curves. For example this is described on the wikipedia page: en.wikipedia.org/wiki/Globally_hyperbolic_manifold $\endgroup$
    – Andrew
    Feb 28, 2021 at 19:55
  • $\begingroup$ Thanks guys, John's answer sounds right to me. $\endgroup$ Feb 28, 2021 at 22:48

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Generally the characterization of the geometry of cosmological space (not space-time) is determined by the values of the four parameters in the Friedmann Equation presented in the section: https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation . These are the $\Omega$ density parameters which in a cosmological model have values for the density of radiation, matter, curvature, and dark energy (also known as the cosmological constant density). The corresponding subscripts are: R, M, k, and $\Lambda$. The sum of these four parameters equals 1. (I will use here just the subscripts to represent the $\Omega$ variables.) If R+M+$\Lambda$ > 0, then k < 0 corresponding to what is called a 3D hyper-spherical space. R+M+$\Lambda$ < 0, then k > 0 corresponding to what is called a 3D hyperbolic space. If k = 0 then the universe shape is called flat or Euclidean.

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