Our flat, fractal universe

In order to evaluate the curvature of a space, we drawn a triangle and measure its internal angles. If the value is approximately 180°, the space is flat; if it is greater than 180 degrees, the space is like a sphere; if less than 180°, the space is a kind of saddle. To evaluate the curvature of a space, however, we need to find sufficiently large triangles: if we try to draw a triangle on the ground, it will most likely be a flat triangle, but if we try to draw a triangle, from space, with the extremes of the Sicily, we will have a spherical triangle. Similarly, for the universe, we must determine a triangle as large as possible. At this point we could take three stars and draw a triangle: the only complication is finding three stars that are at the same time from the moment the cosmic expansion began, and this thing is not exactly easy to determine. This forces us to examine a widespread signal that we are certain is from the same period in the universe timeline: the cosmic microwave background.

Examining this radiation led us to the conclusion that the universe is flat⁽¹⁾, because taking a triangle on this map, it turns out to be a flat triangle. The cosmic background radiation, however, dates back to the so-called epoch of recombination, so it would be more correct to say that the universe was flat. Whether it has always been flat in the course of its evolution or whether it is now is a completely different story. Furthermore, it is not certain that a flat universe cannot have a curved shape: in the 1950s John Nash and Nicolaas Kuiper suggested the existence of a particular square flat torus, such as the Pac-Man plank, which was finally visualized as a three-dimensional surface in 2012⁽²⁾. So the universe could have a toroidal shape and be flat just like the figures of Nash and Kuiper, but it should also have a fractal structure, which at the moment has not been verified, but which would not be incompatible with cosmic inflation.
However, I wrote all this round of words to introduce an interesting article on arXiv, The equivalence principle and QFT: Can a particle detector tell if we live inside a hollow shell?:

We show that a particle detector can distinguish the interior of a hollow shell from flat space for switching times much shorter than the light-crossing time of the shell, even though the local metrics are indistinguishable. This shows that a particle detector can read out information about the non-local structure of spacetime even when switched on for scales much shorter than the characteristic scale of the non-locality.

But we must not forget that the portion of the universe that we are able to observe is much smaller than it could be, in particular if the model of cosmic inflation were correct. This would mean that the universe globally could easily not be flat, in spite of the flatness of the universe that we can observe and we should consider local, despite its dimension are a few tens of billions of light years.

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1. de Bernardis, P., Ade, P. A., Bock, J. J., Bond, J. R., Borrill, J., Boscaleri, A., ... & Ferreira, P. G. (2000). A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature, 404(6781), 955. doi:10.1038/35010035 (arXiv) ↩
2. Borrelli, V., Jabrane, S., Lazarus, F., & Thibert, B. (2012). Flat tori in three-dimensional space and convex integration Proceedings of the National Academy of Sciences doi:10.1073/pnas.1118478109 ↩