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So, we're fairly sure that the universe is infinitely large. We're also fairly sure that it is expanding. My question is how do these two facts relate to the multiverse? My favorite interpretation was given by Michio Kaku, who stated that in the multiverse each individual universe is like a bubble, and all of these bubbles are floating around in some space (and sometimes interact?).

How can there be an infinitely large object in a space with other presumably infinitely large objects? Is this in like math where there are different "sizes" of infinities, where some are larger than others? What is the space that the universe is expanding into? Does that have a boundary, or is it also some infinity?

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One way for infinitely large objects to occupy the same space without intersecting/interacting is if the objects do not fill the space they are in.

Take for example a line on the XY plane. The XY plane is infinite in both directions, and in this infinite plane the line y=x is also infinite, you could keep walking along it and never reach the end.

But it is not the only line that could be described as such. The line y = $x^2+2$ also goes on for ever, but never intersects with the other line. In this case we had lines (infinite 1 dimensional objects) in a plane (infinite 2 dimensional objects). I know for euclidean geometry anytime your space has more dimensions than the objects you can fit an infinite number of them there. I am not as confident about the "always" part of that statement in non-euclidean spaces.

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These days the multiverse normally refers to the results of eternal inflation. If this is what you're asking about, the multiverse is really just a single universe with different bits of it behaving differently. By this I mean that you can in principle draw a continuous line in spacetime from one bubble universe to another. Note that nothing could ever follow this line because that would mean travelling (much!) faster than light, but the line exists.

A frequent model for an expanding (closed) universe is a rubber balloon being blown up. If the whole balloon is taken as the multiverse then the balloon is expanding exponentially i.e. doubling in size every 10$^{−43}$ seconds. A bubble universe forms when a patch on the balloon ceases exponential growth and starts the Hubble expansion we see today. So the bubble universe isn't actually infinite, though it can grow infinitely and is part of an infinite whole.

I've never understood the idea of collisions between bubbles, because the separation between any two bubble universes should be doubling every 10$^{−43}$ seconds along with the multiverse.

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