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?
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.