Does the expansion of space imply anything about the dimensionality of the Universe?

Question

I have read that we should visualize the expansion of the Universe as something like a balloon expanding with the Universe, which apparently is 3 dimensional, not being the balloon itself, but rather the surface of the balloon.

Does this imply that there is something else, the "balloon" in the analogy, which is at least potentially more than 3 dimensional? Something exotic is plainly happening if space itself can expand at faster than the speed of light.

So are astronomers at least entertaining the idea that, whether you call it "The Universe" or not, something that is not 3 dimensional into which the 3D Universe is expanding?

Or can the expansion of The Universe be modeled without extra dimensions?

Answer 1

No, not at all.

Balloons — good and bad

The balloon analogy is good for explaining how all distances expand at the same factor, and it is good for explaining how no observer is more central than any other, but it breaks down hard the moment you start thinking about where the balloon is.

A balloon has a 2D surface, but in reality it is a 3D object, embedded in your three well-known dimensions. In contrast to this, the 3D Universe is not, to our knowledge, embedded in a higher dimension, and its expansion does not require more dimensions to explain.

The scale factor

So yes, the expansion of the Universe is indeed modeled without extra dimensions. Mathematically, this is no problem at all, in that the metric — i.e. the function used to define distances — simply includes a factor that is multiplied on all coordinates. This factor is called the scale factor, written $a(t)$, where the $t$ is time. That is, it is time-dependent, and determining its value at different times throughout the history of the Universe is one of the prime goals of modern cosmology.

Its value is defined to be 1 today, and 0 at the Big Bang, but because the Universe hasn't expanded linearly, it wasn't simply 0.5 exactly halfway through the Universe's life (in fact it reached $a=0.5$ just under 6 billion years ago).

Intrinsic vs. extrinsic curvature

So we have no problem mathematically (or physically), only visually. A similar problem arises when we want to visualize the curvature of space due to massive objects. You typically see something like a rubber sheet with billiard balls depressing the sheet. But the depression bends the 2D surface of the rubber down in a third dimension. This dimension doesn't exist; there is no "down" or anything other direction that goes away from the sheet.

So, to conclude: expansion and curvature is intrinsic. It happens "in" the space. Balloons and rubber sheets show you extrinsic curvature.

Perhaps it would be better to visualize intrinsic curvature as a flat sheet with lines bending, sort of like this:

from John Rennie's answer on a similar question on physics.SE. The problem with this is that, in order to draw lines contracting in one place, you must draw them as expanding in another place. But for instance space can contract around a black hole without expanding elsewhere.

Perhaps it would be better to color-code the sheet according to a scale factor.

Perhaps it would be better not to visualize it at all.