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1.) The angular size of the temperature fluctuations that appear in the Cosmic Microwave Background. Our Universe was very uniform in the early stages of the hot Big Bang, but not perfectly uniform. There were tiny imperfections: regions that were slightly more or less dense than average. There’s a combination of effects that take place between gravity, which works to preferentially attract matter and energy to the denser regions, and radiation, which pushes back against the matter. As a result, we wind up with a set of patterns of temperature fluctuations that get imprinted into the radiation that’s observable, left over from the hot Big Bang: the cosmic microwave background. These fluctuations have a particular spectrum: hotter or colder by a certain amount on specific distance scales. In a flat Universe, those scales appear as they are, while in a curved Universe, those scales would appear larger (in a positively curved Universe) or smaller (in a negatively curved Universe). Based on the apparent sizes of the fluctuations we see, from the Planck satellite as well as other sources, we can determine that the Universe is not only flat, but it’s flat to at least a 99.6% precision. This tells us that if the Universe is curved, the scale on which its curved is at least ~250 times larger than the part of the Universe that’s observable to us, which is already ~92 billion light-years in diameter. We can look arbitrarily far back in the Universe if our telescopes allow, and the clustering of galaxies should reveal a specific distance scale – the acoustic scale – that should evolve with time in a particular fashion. If the Universe has positive, negative, or flat spatial curvature, this type of detailed analysis will reveal it. (Credit: E M Huff, the SDSS-III team and the South Pole Telescope team; graphic by Zosia Rostomian) 2.) The apparent angular separations between galaxies that cluster at different epochs throughout the Universe. Similarly, there’s a specific distance scale that galaxies are more likely to cluster along. If you put your finger down on any one galaxy in the Universe today, and moved a certain distance away, you can ask the question, “How likely am I to find another galaxy at this distance?” You’d find that you would be most likely to find one very nearby, and that distance would decrease in a particular way as you moved away, with one exceptional enhancement: you’d be slightly more likely to find a galaxy about 500 million light-years away than either 400 or 600 million light-years away. That distance scale has expanded as the Universe has expanded, so that “enhancement” distance is smaller in the early Universe. However, there would be an additional effect superimposed atop it if the Universe were positively or negatively curved, as that would affect the apparent angular scale of this clustering. The fact that we see a null result, particularly if we combine it with the cosmic microwave background results, gives us an even more stringent constraint: the Universe is flat to within ~99.75% precision. In other words, if the Universe isn’t curved — for example, if it’s really a hypersphere (the four-dimensional analogue of a three-dimensional sphere) — that hypersphere has a radius that’s at least ~400 times larger than our observable Universe. The quantum fluctuations that occur during inflation do indeed get stretched across the Universe, but they also cause fluctuations in the total energy density. These field fluctuations cause density imperfections in the early Universe, which then lead to the temperature fluctuations we experience in the cosmic microwave background. The fluctuations, according to inflation, must be adiabatic in nature. (Credit: E. Siegel/Beyond the Galaxy) All of that tells us how we know the Universe is flat. But to understand why it’s flat, we have to look to the theory of our cosmic origins that set up the Big Bang: cosmic inflation. Inflation took the Universe, however it may have been previously, and stretched it to enormous scales. By the time that inflation ended, it was much, much larger: so large that whatever part of it remains is indistinguishable from flat on the scales we can observe it. The only exception to the flatness is caused by the sum of all the quantum fluctuations that can get stretched across the cosmos during inflation itself. Based on our understanding of how these fluctuations work, it leads to a novel prediction that has yet to be tested to sufficient precision: our observable Universe should actually depart from perfect flatness at a level that’s between 1-part-in-10,000 and 1-part-in-1,000,000. The quantum fluctuations that occur during inflation get stretched across the Universe, and when inflation ends, they become density fluctuations. This leads, over time, to the large-scale structure in the Universe today, as well as the fluctuations in temperature observed in the CMB. New predictions like these are essential for demonstrating the validity of a proposed fine-tuning mechanism. (Credit: E. Siegel; ESA/Planck and the DOE/NASA/NSF Interagency Task Force on CMB research) Right now, we’ve only measured the curvature to a level of 1-part-in-400, and find that it’s indistinguishable from flat. But if we could get down to these ultra-sensitive precisions, we would have the opportunity to confirm or refute the predictions of leading theory of our cosmic origins as never before. We cannot know what it’s true shape is, but we can both measure and predict its curvature. This is one of the major goals of a series of upcoming missions and observational goals, with the new generation of Cosmic Microwave Background measurements poised to measure the spatial curvature down to 1-part-in-1000 or better, and with the Roman Telescope, the EUCLID mission, and Rubin Observatory all planned to come online and measure the baryon acoustic oscillation signature better and more precisely than ever before. Although the Universe appears indistinguishable from flat today, it may yet turn out to have a tiny but meaningful amount of non-zero curvature. A generation or two from now, depending on our scientific progress, we might finally know by exactly how much our Universe isn’t perfectly flat, after all, and that might tell us more about our cosmic origins, and what flavor of inflation actually occurred, than anything else ever has.