We talked about the Forbes article in the previous thread. You were there too. We couldn't figure out where Siegel got those numbers from. It's hard to figure out how he got there, since he doesn't provide any hints as to his method.
As for the discrepancy in our numbers, it was explained in post #17. 170 Gly is erroneous. One gets it when adding instead of deducting the error bars on the curvature measurement.
What we're doing here is dividing the Hubble radius by the square root of the (absolute value of) curvature density parameter Omega k.
The values are taken from the Planck mission's data.
The Hubble radius is just the Hubble constant expressed differently, so that's what we need.
The result will be the radius of curvature Rk.
The closer Omega k is to zero, the larger the Rk. If it is zero, then Rk is undefined (can't divide by zero) and the universe is flat. If Omega k is negative, the curvature (k) is positive, and the universe curves onto itself - like a sphere. If Omega k is positive, the curvature is negative, and the universe curves the other way - like a saddle. In the latter case it is infinite despite having a radius of curvature. The smaller the Rk, the more curved the universe. In the case of positive curvature - also smaller in overall size.
So we want to look at the values of these two, (Omega k and the Hubble constant H0) and the uncertainty in their measurement, and sort of assume the actual value lies on the most extreme end of the uncertainty. This will give us the boundaries on how curved the universe could be.
For H0, as measured by Planck, the uncertainty is tiny. Something like one percent. So we can safely ignore it for our purposes.
For Omega k, we have 0.001 +/- 0.002. That's one sigma confidence. It's like saying 'we're reasonably sure the actual value lies somewhere between -0.001 and 0.003'. If we take three times the size of the error bars, +/- 0.006, we're saying 'it's almost certain the actual value lies between -0.005 and 0.007'. So that's what we do here.
If you then take the one extreme of the value: -0.005, you get a positively curved hypersphere of radius approx. 205 Gly.
If you take the other extreme value, you get the 170 Gly, but it's now a negatively curved space.
In other words, 170 Gly is the minimum radius of curvature of a universe that curves hyperbolically, into a saddle-like shape - the other way - instead of spherically.
I.e. it's the third non-exotic possibility: the universe, is either flat, a hypersphere of Rk > 205 Gly, or is hyperbolic with Rk > 170 Gly.
If we were to use one sigma error bars, we'd get different - larger - radii, but with lower confidence.
That's all that can be said with the data Planck provided. There are further caveats:
The stated numbers represent the current-best, but still rather coarse, model and a specific set of data (the CMB, ignoring e.g. the Hubble tension) and (within this framework) with high, but not perfect, confidence. Also, more exotic possibilities are discounted (cylindrical shape? toroidal? weirdo irregular something or other?).edit: corrected the wording to represent closed universe as k>0 when Ωk<0, while the open case as k<0 when Ωk>0, as per the posts below.
