If the universe is infinite, it always had to be spatially infinite since at the big bang. And I'm assuming that cosmologists don't think that a Big Bang singularity actually occur in the real universe therefore the universe has to always be infinite from the beginning.
In the big bang theory says that the universe evolved from a denser early stage at the big bang. But there was already infinite space even at the big bang. So how can the infinite space be dense?
According to current theories, space itself is expanding. The galaxies that were closer together are now traveling (moving away) faster than light away from each other because the space between them is more.
Imagine this: you traveled 1 km. But now you are 10 kms away from the point you started because the extra 9kms just came into existence. Now things are less denser than before.
But there is another hypothesis, that space was always infinite. But now even though it is still much bigger than before and it is still infinite.
