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I've read various articles and books (like this one) stating that we are not certain about the geometry of the universe, but there were experiments on-going or planned that would help us find out.

Recently though, I've watched a lecture by cosmologist Lawrence Krauss where he seems to categorically assert that the universe has been proven to be flat by the BOOMERanG experiment. Here's the relevant portion of the talk.

I've looked around and there are still articles stating that we still don't know the answer to this question, like this one.

So, my question is two-fold:

  1. Am I mixing concepts and talking about different things?
  2. If not then is this evidence not widely accepted by some reason? What reason would that be?
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    $\begingroup$ The short answer is that the universe used to be within error bars of being flat, and it's still within error bars of being flat, but the error bars have gotten a lot smaller. When people say things like "is flat," "is proven to be flat," and so on, they're being sloppy with language by omitting the qualifier "to within error bars." $\endgroup$ – Ben Crowell Jan 4 '17 at 0:24
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    $\begingroup$ The trick with a flat universe is that we can never really definitely measure it to be flat. Think about it - if the universe was significantly spherical, we could be confident in it being spherical even with uncertain measurements (e.g. curvature of 1.5 ± 0.1 would still mean "yup, spherical"). But to make sure it's flat, you need to have infinitely precise measurements - any "error bars" turn a measurement of 1 into "maybe a tiny bit hyperbolic, maybe flat, maybe a tiny bit spherical". The best we can say is "it's at least this flat". $\endgroup$ – Luaan Jan 4 '17 at 13:27
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I think the reason you're suffering from conflicting sources is that you're mixing both new and old, out-of-date pieces of information. First off, the book you cited was published in 2001 - 15 years ago - and the other article you cite was published in 1999 - 17 years ago. There's been a lot of work done in the past 15 years, often under the term "precision cosmology", in an attempt to really nail down the precise content, shape, size, etc. of our Universe. By the early 2000's we pretty much knew the science behind everything (we knew about dark matter, dark energy, had well-developed theories on the Big Bang, etc.) but what we didn't have, were good, solid, believable numbers to put into these theories, explaining why the flatness of the universe was still contested in your sources.

I'll direct you to two incredibly important observatories which have been paramount in achieving our goal of having "good numbers". The first is the Wilkinson Microwave Anistropy Probe (WMAP), launched in 2001, and the second is the Planck satellite, launched in 2009. Both missions were designed to stare intently at the Cosmic Microwave Background (CMB) radiation and try to sort out the treasure trove of information which can be gleaned from it. In this vein, you might also come upon the Cosmic Background Explorer (COBE), launched in 1989. This satellite had a similar purpose as the other two, but was not nearly as precise as the later two missions as to provide us with good numbers and definitive statements by the early 2000's. For that reason I'll mostly focus on what WMAP and Planck have told us.

WMAP was a hugely successful mission which stared at the CMB for 9 years and created the most detailed and comprehensive map of its day. With 9 years of data, scientists were really able to reduce the observational errors on various cosmological quantities, including the flatness of the universe. You can see a table of their final cosmological parameters here. For the flatness, what you want to do is add up $\Omega_b$ (the baryonic matter density), $\Omega_d$ (the dark matter density), and $\Omega_\Lambda$ (the dark energy density). This will give you the overall density parameter, $\Omega_0$, which tells you the flatness of our universe. As I'm sure you know from your sources, if $\Omega_0 < 1$ we have a hyperbolic universe, if $\Omega_0 = 1$ our universe is flat, and $\Omega_0 > 1$ implies a spherical universe. From the results of WMAP, we have that $\Omega_0 = 1.000 \pm 0.049$ (someone can check my math) which is very close to one, indicating a flat universe. As far as I know, WMAP was the first instrument to give a truly precise measurement of $\Omega_0$, allowing us to say definitively that our universe appears flat. As you say, the BOOMERanG experiment also provided good evidence for this, but I don't think the results were nearly as powerful as WMAP's was.

The other important satellite here is Planck. Launched in 2009, this satellite has provided us with the best high precision measurements of the CMB to-date. I'll let you dig through their results in their paper, but the punchline is that they measure the flatness of our universe to be $\Omega_0 = 0.9986 \pm 0.0314$ (calculated from this result table), again extremely close to one.

In conclusion, recent results (within the past 15 years) allow us to definitively state that our Universe appears flat. I don't think, at this time, anyone contests that or believes it is still uncertain. As it usually goes with science, answering one question has only resulted in more questions. Now that we know $\Omega_0 \simeq 1$, we have to ask why is it one? Current theory suggests it shouldn't be - that it should be either enormously small or enormously large. This is known as the Flatness Problem. That in turn delves into the Anthropic Principle as an attempted answer, but then, I'm getting out of the scope of this question.

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    $\begingroup$ This answer contains a lot of good information, but a couple of things aren't quite right. In conclusion, recent results (within the past 15 years) allow us to definitively state that our Universe appears flat. Being within error bars of flatness doesn't mean that it is flat. This is known as the Flatness Problem. That in turn delves into the Anthropic Principle as an attempted answer,[...] The most popular/promising solution to the flatness problem isn't the anthropic principle, it's inflation. (And inflation is a testable scientific theory, whereas the anthropic principle isn't.) $\endgroup$ – Ben Crowell Jan 4 '17 at 0:21
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    $\begingroup$ Thank you for being careful with stating claims. The paradoxical phrasing of "... allow us to definitely state that our Universe appears flat" makes me smile =) $\endgroup$ – Cort Ammon Jan 4 '17 at 4:52
  • $\begingroup$ @BenCrowell I wasn't trying to state that the anthropic principle is the correct, or even the most viable answer, just simply pointing out an interesting response to the problem. (And really, the anthropic principle is applicable whether the answer is inflation or not - if the universe didn't turn out as it did, we wouldn't be here to observe it. Thankfully, inflation allowed for it to evolve as it did such that we'd be here to observe its present state). $\endgroup$ – zephyr Jan 4 '17 at 13:34
  • $\begingroup$ I would say that the experiments convincingly show that the universe can not be very far from flat. But they still leave it an open question whether it's exactly flat, and if not, which side it falls on. Much as before, only with a smaller window :) $\endgroup$ – hobbs Jan 5 '17 at 3:28
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The basic assumptions of the cosmological principle mean that space can only have constant scalar curvature. This can be positive, negative or zero and a flat Universe is one where the curvature is zero.

The curvature of space is something that can be measured and the current value is known to be close to zero, not just from BOOMERanG, but from subsequent observations. Vanilla FLRW cosmology has difficulty explaining this and it is known as the flatness problem. However the conventional view is that cosmic inflation does a very neat job in solving this problem.

However a truly flat Universe must have a spatial curvature of exactly zero on a large scale, so to truly determine if the Universe is flat, even using a number of reasonable assumptions requires an exact measurement, which is impossible. So observation can never rule out the possibility that the Universe might have a very small positive or negative curvature.

In addition, if you slightly relax the cosmological principle from its strictest interpretation, the scalar curvature does not completely determine the topology of the Universe, opening the door to so-called exotic topologies. For example a flat Universe could have a toroidal topology and be compact (of finite spatial volume).

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You ask "Am I mixing concepts and talking about different things?" I have no way of knowing whether you are or are not, but the title of your post and the first sentence are somewhat at odds. Your question "Is the Universe considered to be flat?" concerns curvature, which, by itself, does not completely determine geometry, while the statement "we are not certain about the geometry of the universe, but there were experiments on-going or planned that would help us find out" may be talking about something more general.

The first of your links is to Jeffrey Weeks' book The Shape of Space, which focuses a lot of attention on the topology of space. Table 19.1 on page 186 lists some possible topologies for the cases of positively curved, flat, and negatively curved space. The same page contains the surprising statement "When the first edition of this book appeared in 1985, many cosmologists were completely unaware of closed manifolds with flat or hyperbolic geometry." I am curious about whether that is a fair characterization.

On the preceding page of that book (page 185), the evidence, as of 2001, for a flat geometry is briefly outlined. In particular, there is the statement that "New data (coming from studies of distant supernovas and the cosmic microwave background radiation) make a strong case that the visible universe is not hyperbolic, but flat." The same page contains the question "Is the universe closed or open? In other words, is space finite or infinite?" and the answer "Put briefly, we don't know." The last two chapters of the book discuss "Cosmic crystallography" and "Circles in the sky", two proposed observational approaches to the topology of the universe.

Apparently work in the topology of the universe continues to be active. Scholarpedia contains a recent review.

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Yes, it's considered to be spatially flat on the largest scales we can observe, but we must remember that scientific measurements come with uncertainties, and our models can be replaced by better ones. At the present time, we have observations that say the universe is spatially flat to within a high degree of accuracy, but there is still some wiggle room there for it to be slightly curved that we cannot rule out. Also, we can only observe the part of the universe we can see, we cannot know that the rest of the universe has the same curvature as our portion does. We have a theoretical understanding that it would be very difficult for the universe to be close to flat, without being extremely close to flat, so we expect it is extremely close to flat. But theories can be replaced and usually are, and even if the theoretical expectation continues to pass tests way into the future, we still can never know the universe is exactly flat.

But the bottom line is, we have both very good observations, and a good theory (the theory of inflation, and the fact that flatness is unstable with age under general relativity), that agree the universe on the largest scales we can observe is very close to spatially flat. Hence, we can create a model in which is it is flat, and use that model successfully. That's all you ever get in science.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – called2voyage Jan 4 '17 at 1:54
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Just to add to @zephyr 's answer, LISA fired 3 lasers in space to form a triangle in order to measure the flatness of space: if the sum of the 3 angles exactly equals 180 degrees, then the space is flat; deviation from 180 degrees tells you how much the space is curved and the orientation of the curvature. But if the size of the space is too small, then the angles will sum to exactly 180 degrees; this is like looking at the surface of the Earth and thinking it looks flat when it is actually round. LISA measured exactly 180 degrees, so either space is actually flat or we can constrain the curvature of space on larger scales with error-bars.

EDIT: It was LISA, not WMAP, that did the laser experiment. Thanks to @zephyr for the correction.

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    $\begingroup$ Do you have citations for this? I've never heard of this experiment and am disinclined to believe it. For one thing, such an experiment would only measure a local curvature, not a universal curvature. For another, what was this laser triangle reflected off of such that WMAP could actually measure something? $\endgroup$ – zephyr Jan 4 '17 at 13:39
  • $\begingroup$ source 1 and source 2 $\endgroup$ – MPath Jan 4 '17 at 14:14
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    $\begingroup$ Neither of those sources supports your claim or mentions that WMAP "fired 3 lasers into space to form a triangle". I think you may be confused as to what the actual, physical measurements were and how information was derived from them. $\endgroup$ – zephyr Jan 4 '17 at 14:19
  • $\begingroup$ @zephyr That described the general concept of the experiment. Michio Kaku talked about it an interview and also wrote about it his book Physics of the Future. I googled to find an excerpt. Instead, I found that this link to Google Books worked for me; the page number is not shown in the book but the link takes you to the right page. $\endgroup$ – MPath Jan 4 '17 at 14:31
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    $\begingroup$ LISA is scheduled to launch in 2034. Its purpose will be to measure gravitational waves. $\endgroup$ – Will Orrick Jan 6 '17 at 8:33

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