## Why is the Universe Spatially Flat? An argument from conformal symmetry.

##### Summary

The 't Hooft criterion states that smallness of a physical parameter can be considered \textit{technically natural} if setting it to zero would enhance the symmetry of the system. In this work we propose to extend this criterion to include space-time isometries and curvature. Within this framework, the observed smallness of the curvature of the universe can be understood as technically natural (and protected from quantum fluctuations) if setting curvature to zero increases the number of isometries of space-time.
In General Relativity, setting the curvature of an expanding space-time to zero does not lead to additional isometries. However, such a symmetry enhancement does happen if we consider Conformal Gravity, where we modify gravity to be conformally symmetric.
Studying the behaviour of conformal isometries leads us to formulate the Geometric Isolation conjecture. This asserts that a manifold that decomposes nontrivially into two or more isolated factors admits only a limited number of conformal symmetries. In such a setting, the conformal Killing equation can be greatly simplified and will yield fewer independent solutions than allowed by the dimension of the manifold.
We apply these principles to cosmological space-times that factor as $\RR \times \Sigma$, where $\RR$ indicates the time-like direction and $\Sigma$ an expanding, space-like 3-manifold. We will deviate from the usual FLRW space-times where $\Sigma$ is chosen to be one of $S^3$, $\RR^3$ or $\HH^3$ and instead allow $\Sigma$ to be one of the eight geometries from Thurston's Geometrization Conjecture. While the Conformal Gravity setting introduces extra isometries for any choice of $\Sigma$, we will show that only the flat geometry $\Sigma \simeq \RR^3$ restores the full conformal group in four dimensions.
As a result, flat space-time represents a point of exceedingly enhanced symmetry, thus satisfying the requirement for our extension of 't Hooft technical naturalness.