We discuss a possible scale of gravitational origin at around 10 MeV, or
cm, which arises in the MacDowell–Mansouri formalism of gravity due to the topological Gauss–Bonnet term in the action, as pointed out by Bjorken several years ago. A length scale of the same size emerges also in the Kodama solution in gravity, which is known to be closely related to the MacDowell–Mansouri formulation. We particularly draw attention to the intriguing incident that the existence of six compact extra dimensions originated from TeV-scale quantum gravity as well points to a length scale of
cm, as the compactification scale. The presence of six such extra dimensions is also in remarkable consistency with the MacDowell–Mansouri formalism; it provides a possible explanation for the factor of ∼
multiplying the Gauss–Bonnet term in the action. We also comment on the relevant implications of such a scale regarding the thermal history of the universe motivated by the fact that it is considerably close to 1–2 MeV below which the weak interactions freeze out, leading to Big Bang Nucleosynthesis.
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