What caused the universe to start off expanding in the first place.
Was it negative pressure that drove the initial expansion or was it just something else?
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It's important to emphasize that nothing is needed to drive the expansion of the universe. For a given energy content, the Friedmann equations, which govern the evolution of a homogeneous and isotropic universe, have both an expanding and a contracting solution. We select the expanding solution simply because that's what we observe the universe to be doing.
If you are referring to the accelerated expansion associated with the epoch of primordial inflation, then as others have said, this is traditionally modeled as arising from the vacuum energy of some scalar field.
Mabe a theory of negative pressure is that before the big bang there was only dark energy/negative pressure, so if there was only negative pressure the pressure became so great that it could have reached infinite pressure but then there would wouldn't be any existence for infinity. The laws of reality then created the big bang, the opposite of dark energy which is gravity and if you think about it black holes are also more or less the same because it's caused by the laws of reality saying that only a certain amount of mass may be allowed before becoming a black hole. That brings me to my theory that eventually the universe wil be sucked into a black hole and when the black hole runs out of mass there wil only be dark energy left and the cycle starts again. Just a theory.