How can we be sure that gravity is not the reason for the perceived expansion of the universe? The question has been partially discussed here but I believe the following formulation is a bit different.
What experience could disprove the notion that the universe is not expanding despite the pull of gravity, but that every dense portion of the universe (like the one we live in) is in fact slowly increasing the curvature of space-time around it (therefore giving the impression of an expanding universe from our standpoint)?
 A: If an shining object was placed between two distant dense regions (so in a non-dense region) then the light coming from that object would be redshifted less than the light coming from galaxies at the same distance from us as the object. The problem is how to get the object there but in principle, you could bring it.
You could also bring a huge amount of shining objects to arbitrary distances, where space is flat. If the redshift of the light coming from them is different from that coming from the dense regions then your model could be right (while space could be expanding still globally). If not, then space is (only) expanding globally.
The big question would be though why the expansion of space would be different around dense regions. You could imagine the space around the object to be expanding too (in the same way as around the dense regions), in which case you couldn't tell. But then even objects around us would show a redshift.
Now it could be that space is expanding only near our dense region. This would give the impression that objects outside our region are receding from us. In that case, we couldn't tell from here. Only if we find ourselves in outer space, or by sending a rocket there with equipment to measure redshifts. It would read no redshifts (or redshifts different from ours if a global expansion is also present.
So it is possible to distinguish between all scenarios:
-Space expanding only globally
-Space expanding only around all dense regions (and maybe also globally)
-Space expanding only around our dense region (and maybe also globally)
It would be very strange though if dark energy (or whatever causes expansion of space) was be distributed in such a way to produce the expansion of space you have in mind. That is, the higher the mass, the stronger the expansion around it.
You could say that space expansion is caused by a mass (or energy) is constant over the whole of spacetime surrounding it. But why then should there be an accelerated expansion of the universe as we observe it today?
It is supposed that there was a hypothetical particle field once (the inflaton field) that produced the fast expansion of the early universe. This field had negative energy though, as opposed to mass. Mass cannot have both a negative value (to produce expansion) and a positive value (to produce attracting gravity). It either makes space contract or expand.
So the assumption that dense regions can make space expand, while at the same time the matter inside the region is held together by gravity cannot be made. Unless you state that the expansion is caused by the energy content of space itself (dark energy), rather than by the mass inside the region. In that case, why should there be only dark energy around these regions and not in the empty regions?
