That's where the analogy breaks down. If you take Einstein relativity at face value, there's no priviledged spatial slicing (contrasting with, say, Lorentz ether theory), and the balloon is not really a thing.
The most obvious interpretation of general relativity is in terms of B-theory of time: Spacetime is some 'pre-existing', 'eternal' thing over mich matter is distributed. This also fixes the geometry (things like lengths and angles) via Einstein's equations, which more or less state that energy-momentum ∝ Ricci curvature.
In our universe, that distribution comes in layers, ie there's a spatial slicing where the matter distribution appears homogeneous. In that sense, there is a priviledged slicing, which has the unfortunate side effect of making people forget the lessons of special relativity.
Now, the average density of matter changes from layer to layer, and, if our universe were described by the 'closed' Friedmann model, so would the (finite!) volume of the slice. That change is not arbitrary: The layers can be labelled by cosmological time, and with its increase, the average proper distance between galaxies increases as well. That's called the metric expansion of space, because in our idealized model, the metric (a thing defining distances and angles) within a given slice is just a scaled version of the one in a different slice.
The most obvious interpretation of general relativity is in terms of B-theory of time: Spacetime is some 'pre-existing', 'eternal' thing over mich matter is distributed. This also fixes the geometry (things like lengths and angles) via Einstein's equations, which more or less state that energy-momentum ∝ Ricci curvature.
In our universe, that distribution comes in layers, ie there's a spatial slicing where the matter distribution appears homogeneous. In that sense, there is a priviledged slicing, which has the unfortunate side effect of making people forget the lessons of special relativity.
Now, the average density of matter changes from layer to layer, and, if our universe were described by the 'closed' Friedmann model, so would the (finite!) volume of the slice. That change is not arbitrary: The layers can be labelled by cosmological time, and with its increase, the average proper distance between galaxies increases as well. That's called the metric expansion of space, because in our idealized model, the metric (a thing defining distances and angles) within a given slice is just a scaled version of the one in a different slice.