There is a beautiful argument, based on the spherical harmonics of the cosmic microwave background, which calculates the curvature of the universe.

Are there any other methods of computing the curvature of the universe?

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    $\begingroup$ The supernova observations (for which Adam Riess & Sol Perlmutter won a Nobel Prize) also predict nearly-zero curvature & are quite easy to find online. $\endgroup$ – D. Halsey Apr 2 at 14:41
  • $\begingroup$ Please note that all these methods assume the Friedman metric, which is non-physical for the flat universe due to the infinite mass-energy in any finite region. Thus these methods prove only that the Friedman solution is invalid, but not that the universe is actually flat. Secondly, note that the spatial curvature can be interpreted differently. For example, in the Milne model, the universe expands as an explosion in a flat Lorentzian spacetime (no gravity, no curved space). However, the same model expressed in the Friedman terms has a negative spatial curvature. $\endgroup$ – safesphere Apr 2 at 17:05
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    $\begingroup$ @safesphere: I had never heard that a flat Friedman universe was not valid (except in a previous comment of yours). Is this a personal theory, or can you cite other references? $\endgroup$ – D. Halsey Apr 2 at 23:17
  • $\begingroup$ It's not a theory, but a self evident and generally accepted fact that nothing infinite can be a physical observable (or exist in the physical reality), because (in simple terms) it cannot be measured. However, if you are just looking for a conventional answer within the Friedman framework, then you'd have to ignore the common sense like everyone else. $\endgroup$ – safesphere Apr 2 at 23:37
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    $\begingroup$ @ safesphere: I don't understand why the mass-energy in any finite region is infinite. $\endgroup$ – D. Halsey Apr 3 at 20:36

The figure below is redrawn by me from Perlmutter, 1998, arxiv.org/abs/astro-ph/ 9812133 and Kowalski, 2008, arxiv.org/abs/0804.4142.

plot of Omega_Lambda vs Omega_M

The error ellipse for the CMB observations happens to be oriented in such a way that it provides a very good constraint on spatial flatness. (BTW, "zero curvature of the universe" is kind of wrong -- it's only zero spatial curvature.) But the SN and BAO observations do also constrain the spatial curvature.

I have a longer and reasonably up to date discussion of this sort of thing at the end of section 8.2 in my general relativity book, which is free online.


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