I would consider this a philosophy of physics question since 'What is space? What is time? and What are space-time?' have undergone a conceptual evolution from the time of the Ancient Greeks through Einstein's special and general relativistic theories which is a primary topic of discussion in the philosophy of physics; aren't fully answered; and raise a host of questions like, if general relativity and quantum mechanics aren't unified, is our understanding of the universe incomplete? (Obviously yes.) The notion that physicists have a definition or explanation of the universe that has universal support is silly. One hundred years after the quantum revolution, not everyone even supports the Copenhagen Interpretation let alone is any work on unifying GR and QM decisive.
So what is the physical universe? Is it the same as the mathematical models we use to describe it? Is it possible to think of QM mechanics as describing the substance of the universe and the universe being a container for substance? Do we have a consensus on the fundamental nature of the universe? Is it part of a multiverse?
Lot's of question and we'll start with 'What does it mean for the physical universe to have a boundary and is it even meaningful?'
In traditional macroscopic experience, we can say an apple is a sack, and that there is space (a collection of points in the sack, outside the sack, and the sack) that is filled in one way or another, including empty space which apparently can be understood as being filled with quantum foam. But to say, what sort of space is all space contained in is a meaningless question, because it's a contradiction to have one space of which purports to be all space be a subspace of a yet larger space. There are mereological rules (SEP) at play here that might be violated. These "rules of containment" are dealt with by physicists by making two claims. That the universe is fundamentally understood with the mathematical abstraction of the curvature of space-time is done with sophisticated models like the tensor calculus and Minkowski space, and that such mathematical models do not require other models to contain them: that is, there is no embedding space which can be understood as that there is no sack or space that contains the sack. Such explanations often invoke the finite but unbounded surface of a balloon as an analogy. Two simple explanations about this are:
If you watch these arguments, the essence of them is that the universe by definition can't be contained by something else, so to understand the notion of expanding space time, you cannot use what your everyday sensory experience tells you about containment. To understand space expanding without filling something else means you have to your mathematical intuitions instead. Imagine the density property of the reals. Let's consider an analogy.
Between two points, one can find a midpoint, and then between the first and the midpoint, another, and another, and another, and if we were to construct a set, we can grow that set without having to move the endpoints. That is, our set expands by adding members (by summing and dividing by two using our metric) and yet we never move the endpoints. So, there are two ways we can expand a set of points. One is to expand the interval define on the reals (0,1), (0,10), (0,100), and the other is to expand the listed points in our extension of the interval {0,1}, {0, 0.5, 1}, {0, 0.25, 0.5, 0.75, 1}, ... Thus, what we have are differences in the definition of expansion.
Yes, but you might argue, the universe isn't the math we model it with. In which I would agree with you, but not all physicists would. There are some physicists who maintain that the universe is inherently information (the marginalian.org). John Archibald Wheeler is one such person:
Wheeler speculated that reality is created by observers in the universe. "How does something arise from nothing?", he asked about the existence of space and time.[82] He also coined the term "Participatory Anthropic Principle" (PAP), a version of a Strong Anthropic Principle.[83]
In 1990, Wheeler suggested that information is fundamental to the physics of the universe. According to this "it from bit" doctrine, all things physical are information-theoretic in origin:
So, if our infinite density property of reals is at the bottom of the what is fundamentally real about the universe, then it's the representations that determine the substance. These sorts of philosophical positions are considered alternate interpretations of quantum mechanics. Some of these are forms of philosophical idealism.
So, the conventional answer is that the mathematical physics we have are sufficient to suggest that, whatever the universe is, given a materialist interpretation of the universe, that it doesn't need to expand into anything because expansion is defined in terms of the increasing distance between galaxies, for instance, and one needn't have even a finite universe, let alone something outside the universe, to make that make sense mathematically.
