What limits range of gravitational attraction concerning space expansion?

Question

Wikipedia says:

Metric expansion is a key feature of Big Bang cosmology and is modeled mathematically with the FLRW metric. This model is valid in the present era only on large scales (roughly the scale of galaxy clusters and above). At smaller scales matter has become bound together under the influence of gravitational attraction and such bound objects clumps do not expand at the metric expansion rate as the universe ages, though they continue to recede from one another.

...why?

All gravity equations I know are continuous and smooth - gravity never drops to zero. Reading the above though sounds as if there was a cut-off point somewhere; bodies far enough cease to be bound entirely. It's no longer several forces overlapping and struggling, like electromagnetic and strong, struggling between binding the atom nucleus and tearing it apart. The statement makes it sound as if there was a clear-cut border between where gravity works and where metric expansion works, clusters of galaxies with their gravitational field reaching a finite distance, like free-falling snowflakes able to clump together when they meet.

Is there a border limiting range of gravity? How is the separation between gravitationally bound and independent systems explained?

Answer 1

If you go back to the recombination era, when the cosmic microwave background was emitted, the universe was extremely smooth. Indeed the inhomogeneities in the CMB radiation are only one part in a hundred thousand. So if you pick out some test particle, the universe around that particle was very close to isotropic and hence the net gravitational force on that particle was very nearly zero. At this time the FLRW metric was a very very good description of the universe at even small scales.

But even though they were small, inhomogeneities did exist. That means in some parts of the universe the density was slightly higher than the average, and a particle near to one of these overdense areas would feel a net gravitational attraction towards the overdense area. This net gravitational field meant there was an escape velocity to get away from the overdense area, just as today there is an escape velocity to get away from the Earth.

If you start at the centre of the overdense area and work outwards, the velocities of the surrounding particles get larger as the distance from the centre of the overdense area gets bigger, as the FLRW describes. So at some distance the velocity due to the expansion of space is greater than the escape velocity, and particles beyond that point keep moving away. But nearer to the overdense area particles don't have enough velocity to escape the slightly higher gravitational field of the overdense area, and they start falling towards it.

But as more and more particles are attracted into the overdense area that area gets even denser, which attracts neighbouring particles more strongly, and you get a vicious cycle. Even though the density difference was initially extremely small it inevitably grows bigger and bigger. Eventually you end up with a gas cloud dense enough to form stars, and shortly thereafter the first galaxy.

So to get back to your question, the reason metric expansion doesn't affect objects at small scales (in this context small means the size of galaxy clusters!) is because those galaxy clusters grew from small overdense areas pulling in all the matter moving too slowly to escape. It's only when you get to scales beyond supercluster sizes that the inhomogeneities get averaged out and we get the distance:velocity relationship predicted by the FLRW metric.