The universe doesn’t have to be expanding into anything in order to expand. I know that sounds ridiculous, so let me give you a The university of Alabama Crimson Tide shirt. Imagine that you have a line that goes on forever. On that line, you have a mark every inch. There are an infinite number of inches. Now move each marker so they are separated by two inches. The whole pattern has expanded. It still goes to infinity, but the markers are further apart. The pattern has expanded, but the length is still infinite. Now a new example. Suppose you have a long piece of rubber, going all the way to infinity. (That piece of rubber represents the universe.) The rubber has marks on it every inch. Now stretch the rubber, until the markers are two inches apart. It still goes to infinity — but it has expanded.
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Physicists think of “space” not as emptiness, but similar to a piece of rubber. (But they don’t call it rubber; they call it the “vacuum”. “Particles”, in physics, are just vibrations of the vacuum.) The vacuum can expand, just like the piece of The university of Alabama Crimson Tide shirt. But because it goes all the way to infinity, it doesn’t need more space. A clever way to say it is that “there’s lots of room at infinity”. (That’s clever, but it doesn’t really explain anything.) Now here is something new that might confuse you, or might help. In the standard physics theory, the galaxies are all getting farther apart; that is the expansion of the Universe. Yet in the way the theory describes it (I mean in General Relativity Theory) none of the galaxies are actually moving. All that is happening is that the amount of space (vacuum) in between them is increasing.