Question

Given 2 objects moving at some velocity v relative to one another, is it possible to determine whether they are moving or whether the space between them is expanding?

Answer 1

Firstly, remember what it means for space to be expanding - it doesn't mean that space is some rubbery fabric that gets pulled, it means that the metric of space is expanding. That is, the scale factor (which essentially represents the relative distance of objects) is growing in time. So, if all galaxies were separated by some distance, this distance would then be larger at a later time.

So, how could we tell that the expansion is really metric expansion, and not relative velocity ? Well, the first reason is Hubble's Law. Since the universe is undergoing metric expansion, it appears that galaxies have an apparent velocity, given by V=H0D. So, we see that galaxies that are further away are moving away with a higher velocity. This makes sense - photons travelling from more distant galaxies must traverse more expanding space, and therefore have their wavelengths expanded by a larger amount, so that they are more redshifted. This wouldn't be the case if galaxies were just moving away from us, since we would see a wide variety of redshifts, not a pattern.

The second reason is general relativity. GR predicts that certain metrics that contain homogeneous distributions of matter (i.e. the galaxies that make up the universe) will cause metric expansion to occur. Since GR is supported by the evidence, so therefore is metric expansion.

Third is the cosmic microwave background. We observe a uniform microwave radiation that fills the universe, that has a temperature of 2.73 degrees Kelvin. Satellites that have studied it (COBE, WMAP) have determined that it has a blackbody curve, and that it has the redshift that dates it back to a time very soon after the big bang. We now know that this represents the first radiation ever emitted, 380,000 years after the big bang at the recombination, when hydrogen atoms formed.

Another reason is that when a quasar's spectrum shows absorption lines due to neutral hydrogen clouds, the redshifts of the hydrogen lines are always found to be less than the redshift of the quasar. Furthermore, examples have been found in which the absorption spectrum shows a feature called the Gunn-Peterson trough, which had been predicted earlier as a consequence of the reionization of hydrogen in the early universe.

However, your example considers only two galaxies. In that case, there is nothing that they can do to determine if they are moving apart just due to velocity, or expansion. Fortunately, we don't live in such a universe.