I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says:(adsbygoogle = window.adsbygoogle || []).push({});

"A central feature of the microwave background fluctuations are randomly placed spots with an apparent size ~1 degree across. These are produced by sound waves that travel through the hot ionized gas in the universe at a known speed (the speed of light divided by the square root of 3) for a known length of time (375,000 years). By using the relation: distance = rate * time, we can infer the distance the sound travels, and thus the actual size of a typical hot (compressed) or cold (rarefacted) spot. By comparing the apparent size of the spots to their known actual size, we can measure a combination of the distance to the last scattering surface and the curvature of the light path between us and this surface, which depends on the geometry of the universe. Then If we independently know the Hubble constant, we can determine the distance to the last scattering surface and thus use the spot size to determine the geometry uniquely."

I was wondering how the "the actual size of a typical hot (compressed) or cold (rarefacted) spot" was calculated? How was the Hubble constant used? How did they ultimately show that the spots should be 1 degree across in a Euclidean universe? I understand that it should be basic Euclidean geometry, but I'm not quite understanding the problem. Please keep it simple for me to understand.

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# B How to do the calculations showing the Universe is flat?

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