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As I understand it the allowable spin states of elementary particles is tied to the number of space and time dimensions, so for any two universes with the same number of space and time dimensions the allowable spin states will be the same in both of them. In the Early Universe some of the fundamental forces that are now separate forces were the same force due to the energy density of the early universe, such as electromagnetism and the weak force being the same force. I read in an answer to another question that the number of fundamental forces is not related to the number of dimensions in spacetime.

My question is if two universes have the same energy density, in natural units, would this imply that they necessarily have the same set of fundamental forces, quantum fields and elementary particles associated with those fields, and quantum numbers associated with charge?

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I think this question is too broad to be answered in any specific way.

We suspect that there could be different interactions in addition to electroweak, strong, and gravitational, but they could act on an energy scales that are not currently possibly to achieve in the lab or on LHC. Even if a different universe has exactly the same set of interaction as ours (including the aforementioned unknowns) these interactions could act on energy scales different from our universe.

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