Caption: An animation illustrating that the expansion of an IDEAL loaf of raisin bread when being baked is analogous to the expansion of the universe.
Yours truly is pretty sure real raisin bread does NOT just scale up when baked---but it can be eaten later.
Features:
So the structures defined by these cosmological proper distances just scale up with time.
By convention, a(t=t_0) = a_0 = 1, where t_0 is present cosmic time.
The theoretical time zero is the big bang singularity when the observable universe had zero size and infinite density.
Actually, no one believes that we can run our cosmological models back to the big bang singularity.
The observable universe is thought to track into one of those cosmological models a tiny fraction of second after the theoretical time zero (see Wikipedia: Graphical timeline of the Big Bang).
What came before that tiny fraction of second? That is a story for another figure caption.
If the spatial geometry of the whole universe is hyperspherical, then NOTHING. In this case, the universe has the geometry of a 3-dimensional surface of a 4-dimensional n-sphere (AKA hypersphere)---this object is called a 3-sphere (glome).
The 3-dimensional surface is a finite region with no boundary and the surface is growing with cosmic time.
The 2-dimensional analogue is the 2-dimensional surface of a growing ordinary sphere.
We do NOT notice the curved nature of space because we're microbes on a beach ball---it looks flat (i.e., Euclidean) to us.
What is off the 3-dimensional surface of the 3-sphere? General relativity does NOT say or imply in anyway that space is a 3-sphere in a 4-dimensional mathematical space. It says space has that geometry and nothing more. So There is NO reason to posit anything OFF the curved 3-dimensional surface that is like a 3-sphere. So there's nothing off the 3-dimensional surface and universe is NOT expanding into anything.
In this case, the universe is infinite and the expansion of the universe is a scaling up infinity. This makes mathematical sense---just accept it.
Once again universe NOT expanding into anything.
The most-considered version of this idea is that the whole universe is a multiverse, infinite and eternal, full of pocket universes that are embedded in a background universe. The most-considered version of the multiverse is called eternal inflation with a background universe that is a false-vacuum universe.
But we do NOT see any hint of the boundary of our (hypothetical) pocket universe. The observable universe seems to obey cosmological principle (i.e., it is homogeneous and isotropic). Thus, we do NOT know where we are in our pocket universe NOR where is its center of expansion if it has one in any sense.