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The shape of the universe is the question of whether the universe is flat, has a positive curvature or a negative one. More recently astronomers have compared radiation coming from extremely distant points of the observable universe and have concluded that it is flat with a 0.4% margin of error, Based on articles I've read elsewhere this seems to have created the consensus that the universe is indeed flat. But isn't the universe also believed to be infinite in size? If it is really infinite then shouldn't such measurements be inconsequential as an infinite curved universe would still appear perfectly flat to a local observer?

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Yes, the universe is believed to be infinite in size. That's what you get if the curvature is zero or negative, assuming a simple topology. The curvature has to be positive for a finite universe, once again, assuming a simple topology, and no weird stuff like edges.

Now it's possible that the universe has a very small positive curvature, so that it's finite, but it's so large that it looks flat to us.

However, it's reasonable to assume that the observable universe is representative of the whole thing, and not just coincidentally a region of anomalous curvature. Of course, that's impossible to verify, but if the curvature were significantly greater outside our observable patch we'd expect the curvature to be a bit higher near the edges than it is in the middle of the patch, and we don't see that in the data.

But if the global curvature equals the maximum positive curvature consistent with the 0.4% margin of error of the WMAP, BOOMERanG, and Planck data, then the radius of curvature of the whole universe is (currently) around 150 times larger than the radius of the observable universe. And of course, in the future it will continue to expand.

That figure comes from How Big is the Entire Universe? by astrophysicist Ethan Siegel. That article has a great explanation of curvature, with lots of helpful diagrams.

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If it is really infinite then shouldn't such measurements be inconsequential as an infinite curved universe would still appear perfectly flat to a local observer?

No, it wouldn't necessarily. It could, but it's not mandatory.

Imagine an infinite line that just takes a sharp turn at some point. It would definitely be "curvy" at that point, even though it's infinite.

Take a parabola. It is infinite, but has a definite curvature in every point.

Same goes for our 3D universe.

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Why do scientists assume they can measure the shape of the universe if it is also widely believed to be infinite?

Because not everybody thinks it's infinite. Take a look at the timeline of the Big Bang written by Luke Mastin in 2009. He said “The linear dimensions of the early universe increases during this period of a tiny fraction of a second by a factor of at least 10$^{26}$ to around 10 centimetres (about the size of a grapefruit)”. You can find John Gribbin talking about a universe the size of a grapefruit in his 2008 book The Universe: A Biography. You can find Marcus Chown talking about a universe the size of a grapefruit in his review of Alan Guth’s 1997 book The Inflationary Universe. You can also find Jeremiah Ostriker and Paul Steinhardt talking about a universe the size of a grapefruit in The Quintessential Universe in SciAm in 2002. They weren't talking about the observable universe. They were talking about the whole universe. And if it was the size of a grapefruit 13.8 billion years ago, it can't be infinite now.

The shape of the universe is the question of whether the universe is flat, has a positive curvature or a negative one. More recently astronomers have compared radiation coming from extremely distant points of the observable universe and have concluded that it is flat with a 0.4% margin of error.

That's right. Two out of three options were always going to be wrong. IMHO if you've read the Einstein digital papers it's obvious that the universe was going to be flat. Yes, Einstein came up the idea of a closed curved universe, see his 2nd February postcard to Willem de Sitter. But by 1932 he and de Sitter had dropped the idea in favour of flat space in their Einstein-de Sitter universe. See John Baez and Emory Bunn’s preliminaries article dating from 2006: “Similarly, in general relativity gravity is not really a ‘force’, but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial”. Curved space is not the same thing as curved spacetime. Unfortunately some people confuse the two.

Based on articles I've read elsewhere this seems to have created the consensus that the universe is indeed flat.

I think that consensus is correct.

But isn't the universe also believed to be infinite in size?

A lot of people claim that a flat universe must be an infinite universe. But it's a non-sequitur. Going back to Einstein again, he thought of space as a thing. That’s why in his 1929 essay on the history of field theory, he described a field as a state of space. In his Nottingham lecture in 1930 he said space “remains the sole medium of reality”. So I think it's reasonable to envisage the universe as a sphere of space, with no space beyond it. Maybe light rays would undergo something akin to total internal reflection.

If it is really infinite then shouldn't such measurements be inconsequential as an infinite curved universe would still appear perfectly flat to a local observer?

That's what some people say. But they also say things like "the universe was infinite at the time of the Big Bang and it's even more infinite now". I'm not fond of that because it doesn't seem to fit in with Big Bang cosmology, and I just don't see how an infinite universe can expand.

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    $\begingroup$ This is pseudo-science. $\endgroup$ – Florin Andrei Jun 29 at 22:43
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    $\begingroup$ Steinhardt now argues that cosmic inflation creates more problems than it solves, but even assuming it's roughly the correct model and the pre-inflation Universe was the size of a grapefruit, can you point me to anything that confirms "They weren't talking about the observable universe"? My understanding is that it's precisely the observable Universe that was grapefruit-sized, and this is readily reverse-calculated using the current diameter, the 3 scale factors back to the end of inflation, and the $10^{26}$ expansion due to inflation. $\endgroup$ – Chappo Says Reinstate Monica Jul 1 at 8:21
  • $\begingroup$ @Chappo : follow my links and read what the authors actually said. Or google on universe size of a grapefruit. Big Bang cosmology always talked about the universe once being the size of a grapefruit. It's only relatively recently that this has been revised to the observable universe once being the size of a grapefruit. $\endgroup$ – John Duffield Jul 1 at 18:38
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    $\begingroup$ The text here is largely recycled from your website, meaning you're still looking for ways to use this site to get more eyes on your content. That is not a good faith use of this site. $\endgroup$ – called2voyage Jul 1 at 19:57
  • $\begingroup$ @JohnDuffield I followed your instructions, and found many references to the universe once being the size of a grapefruit. Winnowing out the unreferenced and uninformed rubbish, the remainder clearly identifies that the region being referred to is the observable universe. For example, I suggest you check out this 3-minute interview with Prof. Brian Cox, where he couldn't be more explicit on the matter. $\endgroup$ – Chappo Says Reinstate Monica Jul 2 at 0:44

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