Into what space is the universe expanding?
Over time, the universe is expanding in space. The words 'space' and 'universe', sometimes used interchangeably, have distinct meanings in this context. Here 'space' is a mathematical concept and 'universe' refers to all the matter and energy that exist. The expansion is in reference to internal dimensions only.
Finite space theory does not suppose space has an edge, but rather that space wraps around on itself. If it were possible to travel the entire length of space without going faster than light, one would simply end up back in the same place, not unlike going all the way around the surface of the balloon (or a planet like the Earth).
The notion of more space is local, not global; we do not know how much space there is in total. The embedding diagram
[here] has been arbitrarily cut off a few billion years past the Earth and the quasar, but it could be extended indefinitely, even infinitely, provided we imagine it as curling into a spiral of constant radius rather than a circle. Even if the overall spatial extent is infinite we still say that space is expanding because, locally, the characteristic distance between objects is increasing.
The expansion of space is often illustrated with models which show only the size of space at a particular time, leaving the dimension of time implicit.
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In the
"ant on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope which is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate—the ant's position over time will match the path of the red line on the embedding diagram above.
In the "rubber sheet model" one replaces the rope with a flat two-dimensional rubber sheet which expands uniformly in all directions. The addition of a second spatial dimension raises the possibility of showing local perturbations of the spatial geometry by local curvature in the sheet.
In the "balloon model" the flat sheet is replaced by a spherical balloon which is inflated from an initial size of zero (representing the big bang). A balloon has positive Gaussian curvature while observations suggest that the real universe is spatially flat, but this inconsistency can be eliminated by making the balloon very large so that it is locally flat to within the limits of observation. This analogy is potentially confusing since it wrongly suggests that the big bang took place at the center of the balloon. In fact points off the surface of the balloon have no meaning, even if they were occupied by the balloon at an earlier time.