Mass in an expanding or static spherical distribution of matter

In a spherical distribution of matter - such as with clusters of galaxies - how to calculate how much mass there should be for it to not expand with the expanding universe - in other word, for it to be a bound, static system?

I think that for a "ball" not to expand, the escape velocity on its surface needs to be greater than the velocity of the expansion there as per Hubble law.

I don't think there's a general answer to this. For example, in a Big Rip scenario no systems are bound at all.

I think the McVittie metric is the thing you need to look at.

Various models are described in Jones, Bernard J. T.. Precision Cosmology: The First Half Million Years. It starts with this:

Hill said:

I think that for a "ball" not to expand, the escape velocity on its surface needs to be greater than the velocity of the expansion there as per Hubble law.

I think this is basically correct, at least as a heuristic. We had a previous thread on this some time ago that referenced a paper which did the calculation in more detail. I'll see if I can find it.

I claim this question is inherently complex, and that what approximations you make depends on what question you are really trying to answer. Cosmologically, the interesting question is what are the largest bound systems that might come to be in a universe broadly similar to ours, with various assumptions about initial inhomogeneities. But very different assumptions might be made as pure question of mathematical GR. To wit, I propose arguments that for one statement of the problem, the answer is completely determined by dark energy assumptions and can be answered without reference to an FLRW solution - using the same types of arguments used in the work referred to in post #4.

Consider an initially contracting ball of dust (pressureless perfect fluid) embedded in an empty, asymptotically flat spacetime. Basically, this is some initial state of an Oppenheimer-Snyder class of solution. Excise this just outside the ball, glue into an FLRW solution with a boundary shell where the FLRW perfect fluid density goes to zero (and there is no dark energy). This is needed for a smooth gluing. Now, by arguments based on Birkhoff, the evolution within the ball is unchanged, and it will contract to a BH no matter how large an instance of this you create.

Now consider dark energy. For simplicity, let's only discuss cosmological constant. Then the initial set up is an initially contracting dust ball in an otherwise empty universe with cosmological constant. I believe the result here is the for any choice of such constant and details of initial ball state, there is a minimum size such that the ball will eventually stop contracting and start expanding. Again, with the same gluing strategy as above, except that the at the inner glue shell boundary you have pure dark energy matching the ball solution (assumed to be the same as the universe at large), it is again true that the rest of the FLRW solution is irrelevant to the ball dynamics until well after reversal occurs (in the cases where it reverses). Thus, the question of whether the ball reverses and eventually joins the hubble flow is answerable with an isolated treatment of the ball.