The young Universe was expanding because of the high of the inflaton field. It is natural to ask “What was it expanding into?” To answer this question we need to think about space and time.
Albert Einstein’s Special Theory of Relativity showed that space and time are intimately linked together. To us, there seem to be three dimensions of space, but Einstein showed that we live in a four-dimensional manifold called spacetime. Space and time are merely different aspects of the same thing.
Expansion of Spacetime
It is possible that the Universe is closed, that spacetime is curved around upon itself so that the Universe has no outside. So if you set out on a journey from the North Pole and traveled for long enough you would eventually come back to where you started but approaching the South Pole (ignoring any movement of the Earth in the meantime!).
It this is true then, when the Universe expands, it does not need anything to expand into. It simply creates more spacetime for itself.
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