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Talking about asymptotics already assumes that the universe is infinite. Any smooth, finite spacetime will have finite energy. The interesting thing about asymptotically flat spacetimes is that they can be infinite in volume but have finite energy.

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But our universe is believed to be isotropic, and it has positive energy density everywhere, so it seems that should imply infinite total energy.

Why would this lead to infinite energy? Positive density summed over a finite volume does not equal infinity.

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