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So I've read the Wikipedia articles on WMAP and CMB in an attempt to try to understand how scientists are able to deduce the curvature of the universe from the measurements of the CMB.

The Wiki article on CMB explains merely the following:

The peaks contain interesting physical signatures. The angular scale of the first peak [of the power spectrum of the cosmic microwave background radiation temperature anisotropy in terms of the angular scale] determines the curvature of the universe (but not the topology of the universe).

I don't really see, or simply do not understand, why.

I also searched Stackexchange for similar questions, like this one for example, but in that example, the marked answer simply says:

We can view the lumpiness of the Universe then by observing the cosmic microwave background radiation at high resolution.

...which doesn't really explain anything.

Can someone explain in non-mathematical or math-light terms why this data can tell us anything about the curvature of the universe?

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I posted an answer in non-mathematical terms(basically Wikipedia articles). I have some mathematical material which can be posted but that is of no relevance if you find my answer helpful enough. –  Cheeku Mar 7 '13 at 0:17

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OK, here's a brief outline that is very math light (and also somewhat oversimplified): The big idea is to compare the size of the lumpiness now to the size of the lumpiness back then when the CMB was generated, also called at the decoupling time. We measure the size of the lumpiness now by looking a galaxy superclusters and voids. We can then figure out the expansion rate of the universe, if we also measure the time and the distance back to the big bang (decoupling). From Einsteins GR we can convert the expansion rate, and also the sizes, distances and times into a curvature. (We may also assume and use some other things such as approximate uniformity, and spherical symmetry.) If you can truly explain all this, including the GR part, with a math-lite explanation, you are a much better man than I am, Charlie Brown.

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Thanks for this! The missing piece of the puzzle for me was that you need to compare the size of the clumps in the CMB with the size of "today's" galaxies and voids. Now I get it! :) –  user1459524 Mar 7 '13 at 12:04

The answer is in the same Wikipedia article but I feel the need to mention anisotropies:

Anisotropy /ˌænaɪˈsɒtrəpi/ is the property of being directionally dependent, as opposed to isotropy. An example of anisotropy is the light coming through a polarizer. An example of an anisotropic material is wood, which is easier to split along its grain than across its grain.

In general, anisotropies give us an idea about density fluctuations in the early universe which form out the basis(or seeds) of dense matter clusters(galaxies etc) in the universe. How big are these anisotropies(fluctuations)? That measurement is related to parameters of the universe. The answer is in the paragraph above the paragraph you mentioned. This :

The structure of the cosmic microwave background anisotropies is principally determined by two effects: acoustic oscillations and diffusion damping (also called collisionless damping or Silk damping). The acoustic oscillations arise because of a conflict in the photon–baryon plasma in the early universe. The pressure of the photons tends to erase anisotropies, whereas the gravitational attraction of the baryons—moving at speeds much slower than light—makes them tend to collapse to form dense haloes. These two effects compete to create acoustic oscillations which give the microwave background its characteristic peak structure. The peaks correspond, roughly, to resonances in which the photons decouple when a particular mode is at its peak amplitude.

Here are necessary details. Some information about the Primordial Density Fluctuations from Wikipedia:

The statistical properties of the primordial fluctuations can be inferred from observations of anisotropies in the cosmic microwave background and from measurements of the distribution of matter, e.g., galaxy redshift surveys. Since the fluctuations are believed to arise from inflation, such measurements can also set constraints on parameters within inflationary theory.

Note that it says, these measurements set constraints on the parameters, i.e. they must be less than or greater than some value. An exact opinion is and was given using different techniques and looking for the common areas in results.

If there's something that bothers you or I was unable to explain well, please mention. :)

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Figuring out how big the lumps are is the first part, then measuring how big the lumps look to us is the second part. If they look bigger or smaller than they should, you have curved space.

http://wmap.gsfc.nasa.gov/media/030639/index.html

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