They realise that if their universe was a 4-torus, that would result in extra modes for fermionic vacuum energy that led to an overall negative energy density (fermionic vacuum energy is negative) which, in this kind of universe, would require space to be positively curved everywhere. But you can't have a space with the topology of a 4-torus that is positively curved everywhere. So what they conclude is that the universe being a 4-torus is self-contradictory.
Men ...
They have also known for a long time that the universe must be finite in all directions, to avoid exponentially growing solutions to the wave equation. So the simplest alternative topology to a 4-torus is a 4-sphere. In that case, the topology doesn't have the extra fermionic modes, and the overall vacuum energy is positive, which, in this kind of universe, requires space to be negatively curved.
Så ...
It's not that they conclude that the curvature actually is positive everywhere, and hence the universe can't be a 4-torus. It's that they see why a 4-torus both implies positive curvature and at the same time is ruled out by positive curvature, which eliminates the whole possibility.
Du frågar:
Also, why can't the curvature be negative everywhere, resulting in a hyperbolic universe?
Detta universum ...
can't be an infinite hyperbolic universe, which is what is usually meant by that phrase. However, you can have a finite universe with the topology of a 4-sphere but negative curvature. It just can't be uniform negative curvature, it has to vary in magnitude from place to place.
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