### Is the universe considered to be flat?

• 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?

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."

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".

6 years ago

I think the reason you're suffering from conflicting sources is that you're mixing both new and old, out-of-date pieces of information.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.

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.)

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 =)

@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).

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 :)