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Based on the recent Planck Legacy 2018 release confirming the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB) power spectra compared to that predicted by the standard ΛCDM model, several authors (Di Valentino et al. (2022), Efstathiou & Gratton (2020), Handley (2019)) have proposed that the Universe is spatially closed, with a positive curvature 𝐾. A positive curvature of the Universe presents some tension with the standard model, which Di Valentino et al. (2022) couch in terms of a "possible crisis in cosmology".

The value of Ω𝐾 is of interest because it determines the form of the evolution of an expanding universe (Eingorn et al., 2019). Based on the ΛCDM model, Ω𝐾 will have a negligible effect on expansion for early times, but becomes predominate at successively later times. For 𝐾 < 0 expansion will continue forever. For 𝐾 = 0 expansion will continue forever, but the rate will approach zero asymptotically as 𝑡 goes to infinity. Indefinite expansion implies that the Universe must be unbounded and have an infinite volume, and thus posses no center. For 𝐾 > 0 expansion will halt at some time 𝑡 and reverse itself due to the inexorable tug of gravity. This implies that the Universe would have a finite volume. It is claimed that a bounded surface with a finite volume has a geometric center (centroid) which is the arithmetic mean of all points on the surface (Lovett, 2019).

Is this claim true?

References:

Di Valentino E., Melchiorri A., Silk J., 2022, Nature Astronomy, 4, 196

Efstathiou G., Gratton S., 2020, MNRAS, arXiv:2002.06892v1

Handley W., 2019, arXiv e-prints, arXiv:1908.09139

Eingorn M., Yukselci A. E., Zhouk A., 2019, Eur. Phys. J., 79

Lovett S. J., 2019, Differential Geometry of Manifolds. Barnes and Noble, New York, N. Y.

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    $\begingroup$ 2d analogy Is surface of sphere (imagine Earth). Is has finite area, positive curvature (which can be measured by Flatlanders without leaving the surface). But IT had no center located in this 2d world. $\endgroup$
    – Leos Ondra
    Feb 25, 2023 at 8:52

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It is quite possible for a manifold to be bounded and finite and not have a geometric centre (that is on the manifold)

To talk about an "arithmetic mean" you need a way to add up the points. That requires you to embed the manifold. Now, n-Manifolds may be embedded in ${\bf R}^{2n}$, and so the best you can say is that the "arithmetic mean" is a point in ${\bf R}^{2n}$. A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle.

But the embedding is a mathematical construction; it isn't "real". The manifold exists without any embedding. And so it is correct to say that a 1 dimensional circle doesn't have a centre.

And similarly, a curved, closed three dimension manifold doesn't have a centre (on the manifold).

What this means for the universe, suppose the universe does have positive curvature (That is not known) and suppose that the topology of space is a three-sphere, then it does not have a centre.

If we some how embed that three sphere in four dimensional euclidian space, then the universe would be the on the boundary of a four-dimensional disc, and that would have a centre. But this is a purely mathematical construction and isn't "real". There isn't a "real" four-dimensional euclidean space that the universe is really embedded in (or if there is, we have no access to it). The location of that "centre" is entirely dependent on your choice of coordinates. And you would end up with boring triviallities "The centre is at (0,0,0,0). Where is that? Nowhere." Because a "place" is understood to be a place in the universe, locations outside the universe aren't places.

As for the overall curvature of the universe. That is still not known to be positive or negative. Best estimates say it is zero, to within the margin of error, though the effect of dark energy seems to suggest unlimited expansion.

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    $\begingroup$ I imagine Flatlanders living on te surface of a basketball. They find that their world is curved, as they cannot construct parallel lines and their triangles have angles adding up to more than 180°. But they can't see a center of their world in three-dimensional space. $\endgroup$ Feb 25, 2023 at 10:57
  • $\begingroup$ " A simple example is a 1-manifold circle. It will embed in a plane and the centre of the circle is a point in the plane, and never a point on the circle." Sure, but we might still have a centroid on the 1-manifold itself. Eg, on a clock, if we have points of equal mass at 1, 2 & 3 o'clock, the centroid is 2 (but also its antipode at 10). OTOH, if we have n equally spaced points of equal mass, then there's no centroid (or every point is a centroid). $\endgroup$
    – PM 2Ring
    Feb 25, 2023 at 13:33
  • $\begingroup$ How do we know that the centroid in this example is at 2 o'clock? Not 8 o'clock? $\endgroup$ Feb 25, 2023 at 13:42
  • $\begingroup$ @Oscar Oops! In my example I should have said 8 o'clock rather than 10 o'clock. But we can (often) eliminate the antipode if we define the distance on the clock to be the shorter (absolute) distance. $\endgroup$
    – PM 2Ring
    Feb 25, 2023 at 15:33
  • $\begingroup$ Would it then be true to say that, given a surface with positive curvature, there will be a centroid, extrinsic to that surface? (i.e., that there should exist in 3-dimensional Euclidean space a point that represents the locus of the intersection of all lines perpendicular to the surface, and that this point represents the centroid of the volume enclosed by the surface. My goal is to find an expression for the center of the Universe, if it exists, given that recent findings claim that the Universe has a small but significant positive curvature.) $\endgroup$ Feb 26, 2023 at 0:36

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