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Cosmologists seem to not seriously consider the hypersphere as the best model for the universe even though they mention it as a candidate from time to time. If you look closely, it seems to be a very good fit.

The surface of a hypersphere is 3D and unbounded, just like our universe. And if one assumes the radius of the hypersphere is 13,820 Million Light-Years, and is increasing at the speed of light then one can calculate the expansion rate of the universe (which we all know as Hubble's constant). It would be the speed that the circumference of the hypersphere is increasing (2 pi c) divided by the length of the circumference of the hypersphere (2 pi R). So it would be (2 pi c)/(2 pi R) or (c/R).

Plugging in the numbers one gets 21,693 m/s/Mly, and after multiplying by 3.2615 to convert to mega-parsecs and dividing by 1,000 to convert to kilometers one gets 70.75 km/s/Mpc (which are the units for Hubble's constant astronomers like to use). The currently accepted value of Hubble's constant is around 71 km/s/Mpc.

So the hypersphere model fits the facts very well, but cosmologists do not embrace the model. What are the objections of the cosmologists to the hypersphere model of the universe? What facts or observations does the hypersphere model not fit?

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    $\begingroup$ A link-only answer: This was Einstein's opinion (though it would take me some time to dig up a citation), but it's not consistent with the observed flatness of the universe. $\endgroup$ – rob Mar 21 '18 at 16:57
  • $\begingroup$ The surface of the hypersphere is curving in the fourth dimension not in our three dimensions. Assuming light follows a curved path in 4D space along the surface of the hypersphere it would appear to be moving in a straight line to us, wouldn't it? I don't know how scientists have measured flatness, but it seems that if they use light, they might not be able to detect curvature in the fourth dimension. So until scientists show they can measure curvature in the fourth dimension, the flatness results cannot be used to refute the hypersphere model. $\endgroup$ – DG123 Mar 21 '18 at 17:45
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    $\begingroup$ @DG123: In your comment, you seem to be misunderstanding how general relativity works. It fundamentally uses spacetime curvature in 3+1 dimensions, it doesn't assume that anything is embedded in a higher-dimensional space, and the curvature it talks about is intrinsic, not extrinsic, so there is not part of it that's physically undetectable from within the spacetime. $\endgroup$ – Ben Crowell Mar 21 '18 at 19:30
  • $\begingroup$ Curvature can be measured for example by counting the number of galaxies within a certain distance of your position. In a flat universe it goes up as $r^3$. In s closed universe it rises more slowly with distance. In a universe with negative curvature it rises faster. (Not that galaxy counts are a great way of measuring curvature because of look back time: at very large distances you will see no galaxies because you are looking at a time before they formed, to give a crude idea). The point is there are lots of effects of curvature that can be measured in principle $\endgroup$ – PhillS Mar 23 '18 at 8:46
  • $\begingroup$ All three of the dimensions available to us on the surface of the hypersphere are parallel to the surface of the hypersphere. In order to measure the curvature of the surface of the hypersphere we would need to be able to measure distance in the fourth dimensional direction, but that direction is unavailable to us, so to believe one can measure curvature in the fourth dimension by making measurements parallel to its surface seems impossible to me. I would be interested in hearing (in detail) how you would accomplish it. $\endgroup$ – DG123 Mar 26 '18 at 6:30
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Why is the hypersphere not seriously considered by cosmologists as a model for the overall shape of the universe?

This is simply not true.

FLRW models are the most important models used by cosmologists, and they come in two flavors, open and closed (plus a borderline case, which is flat). The closed type has spatial cross-sections that are hyperspheres. If we fit parameters to a variety of cosmological observations, the universe is constrained to be very nearly spatially flat, but the error bars on the spatial curvature are big enough to allow both open and closed cosmologies. Therefore a hypersphere is currently a perfectly reasonable candidate for the spatial geometry of our universe, and cosmologists do take it seriously.

If this is the geometry, then models that fit the data constrain the radius of the hypersphere to be very large (IIRC orders of magnitude greater than the radius of the observable universe).

And if one assumes the radius of the hypersphere is 13,820 Million Light-Years, and is increasing at the speed of light then one can calculate the expansion rate of the universe (which we all know as Hubble's constant). It would be the speed that the circumference of the hypersphere is increasing (2 pi c) divided by the length of the circumference of the hypersphere (2 pi R).

Here you seem to be mixing up the observable universe with the entire universe. Even if we're talking about the observable universe, it doesn't expand at c.

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  • $\begingroup$ Actually, the radius of the observable universe is expanding at c. Multiply 21,693 m/s/Mly (Hubble's constant) by 13,820 Mly (radius of the observable universe) and see what you get. But, no, I'm not mixing up the radius of the observable universe with the radius of the hypersphere. The center of the hypersphere is in 4D space. The center of our observable universe and its entire volume is on the surface of the hypersphere. Every point in our world is a point on the surface of the hypersphere and is the same distance from the center of the hypersphere. Just saying, I think it needs more study. $\endgroup$ – DG123 Mar 21 '18 at 22:14
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    $\begingroup$ @DG123: Actually, the radius of the observable universe is expanding at c. You're mistaken. physics.stackexchange.com/questions/26549/… The center of the hypersphere is in 4D space. No, here you're showing further confusion about GR. You might want to post this as a separate question. $\endgroup$ – Ben Crowell Mar 22 '18 at 0:18
  • $\begingroup$ @DG123 IF the universe is closed then yes space is like a 3-sphere (a 4D hypersphere), but that doesn't necessarily mean that there's an extra spatial dimension: GR works fine with spacetime that's intrinsically curved, it doesn't need spacetime to be embedded in a higher dimensional manifold. $\endgroup$ – PM 2Ring Mar 22 '18 at 0:42
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    $\begingroup$ @DG123: You're falling prey to a couple of common misconceptions about cosmological expansion. See this answer physics.stackexchange.com/a/141219/4552 and/or the Scientific American article by Lineweaver that I linked to from there. $\endgroup$ – Ben Crowell Mar 22 '18 at 22:10
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    $\begingroup$ "If we fit parameters to a variety of cosmological observations, the universe is constrained to be very nearly spatially flat, but the error bars on the spatial curvature are big enough to allow both open and closed cosmologies." It should be noted that if the universe is flat, the error bars will always include the open and closed cosmologies. $\endgroup$ – PyRulez Mar 5 '19 at 11:41
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If you look closely, it seems to be a very good fit.

You didn't look closely, though – you only calculated one parameter from your model. If you calculate more, you'll find that the rest of them don't fit.

if one assumes the radius of the hypersphere is 13,820 Million Light-Years [...] one gets 70.75 km/s/Mpc

That value is just 1/(13820 million years) in km/s/Mpc.

It's true that the current value of the Hubble parameter is close to the reciprocal of the current age of the universe, but it's just an accident of the era we live in, like the Sun and the Moon having the same angular size. In this diagram, your model's prediction is the solid line labeled $\Omega_M = 0$, while the curve consistent with observations is the dashed one labeled $\Omega_M = 0.3, \Omega_\Lambda = 0.7$. They're normalized so they have the same zeroth and first derivatives at t=now, and they intersect the x axis at almost the same place – that's the coincidence. If you normalize them to match at any time significantly different from now, the x intercepts will be much farther apart.

Your model is also inconsistent with general relativity, although a similar model with a hyperbolic space instead of a sphere is consistent with GR (that's the $\Omega_M = 0$ model in that diagram).

There's nothing really wrong with your idea in principle, and it resembles cosmologies others have invented in the past, but it's very strongly excluded by current observations.

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