I Transformation matrix for an expanding space

[PAllen]

If the individual dust grains all start out on comoving trajectories, i.e., expanding, and if the dust is far enough from all gravitating masses, and if the dust has negligible self gravity, yes. But in practice no real ball of dust will satisfy all these conditions.

Just to add a little more to this, if this idealized ball of dust starts with trajectories such that they are mutually at rest (i.e. some initial slice in fermi-normal coordinates based on one of them is 4-orthogonal to all of them), then:
1) if the second derivative of a(t) is positive (rate of expansion increasing), they ball will expand
2) if it is zero (the Milne cosmology), they will remain at mutual rest
3) if it is negative (decreasing rate of expansion, not decreasing scale factor), then this ball will contract.

The initial conditions of the ball matter a lot.

 

[PeterDonis]

you can consider a system of entangled atoms

What do entangled atoms have to do with anything? You have already got enough complications without introducing quantum mechanics.

since atoms are entangled they are not to be seemed as individuals but as parts of a whole

Again, this doesn't help your scenario at all, it just introduces further complications that are irrelevant to the question you are trying to answer.

 

[PeterDonis]

If you place the buoys so that they each see the CMB as isotropic (which is probably what you want - that's at rest with respect to matter near you, ideally anyway) then I think the diameter of the sphere at time $t$ is $a(t)/a(t_0)$ times the size it was at time $t_0$ (Peter will correct me if I'm wrong...)

No correction needed; this is right.

 

[PeterDonis]

Is that the scale factor?

You labeled this thread as "I" level. That means you should already know the answer to this question. But for the record, the answer is yes, $a$ is the scale factor.

the points on the surface won't have zero peculiar velocity

Yes, they will. At least, they will if you want every point on the sphere to be "expanding with the universe". The fact that you don't understand why this is true should be a huge red flag to you that you do not have the required background knowledge to investigate this topic yet.

You really, really, really, really, really need to learn some basics about the FRW models in cosmology. You are taking up several people's time with questions that an "I" level poster should already know the answers to.

 

[PeterDonis]

Since the OP does not have the required background knowledge for the topic, and plenty of information has been presented at this point to enable further research, this thread is now closed.