Having never heard of the universal balloon before this question I wanted to make sure I was going to answer the right question.I believe it's referring to the idea where you take a balloon, put marks on it, and then blow it up and observe the change in distance between the marks you've made.
This illustrates one way astronomers believe the universe is changing, which is expanding uniformly everywhere, radially. So if you start with a small ball and the ball grows and grows, while staying in the same shape, that's how the universe might be expanding.
The balloon helps you visualize how two spots that start close together, can end up very far apart.
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