Buzz Bloom said:
For a two body example, one can use the rule that if a body has a distance from and a velocity relative to another body, and this velocity is less than the escape velocity, then the two are bound.
Yes, this is the general rule.
Buzz Bloom said:
if one body has less mass than the other body the rule might say that the lighter body might be bound to the heavier body, but not necessarily vice versa. Or I might be mistaken about this.
Yes, you're mistaken. "Bound" is a state of the
system of bodies, not of one body or the other. The escape velocity criterion is the escape velocity from the
system, based on the total mass of the system. So, for example, the escape velocity from the Milky Way galaxy at the location of the solar system is due to the mass of the entire galaxy--or more precisely that portion of the galaxy that is closer to the center than the solar system is.
In other words, what "escape" means, and therefore what "bound" means, depends on what system you are looking at. The escape velocity from the solar system at the Earth's location is the velocity required to escape the solar system--but escaping the solar system is not the same as escaping the Milky Way galaxy, which in turn is not the same as escaping the Local Group, which in turn is not the same as escaping the entire galaxy cluster of which the Local Group is a part.
Buzz Bloom said:
Regarding (2), this might mean either
It means (a) as far as expansion itself is concerned. It means (b) as far as dark energy specifically is concerned. See further comments below.
Buzz Bloom said:
I gather the McVitte metric should be able to make a clear distinction between (2a) or (2b).
The McVittie metric itself is highly unrealistic for analyzing bound systems for the same reason FRW metrics in general are: it assumes that there is a uniform density of matter everywhere, except for (in the McVittie case) a single "mass" at the spatial origin. But that is not true of the actual universe. In the actual universe, any bound system, whether it's a planet, star, solar system, galaxy, or galaxy cluster, is surrounded by empty space. So any metric that assumes a uniform density of matter where there is actually empty space will lead to an incorrect model.
In an FRW-type spacetime (where "FRW-type" is meant to include the McVittie metric), the uniform density of matter everywhere, in the form of a perfect fluid, and the fact that the matter is everywhere expanding, does exert what amounts to a "pull" on any object that is embedded in the fluid. So any object embedded in the fluid, unless it is moving with exactly the "comoving" fluid velocity at its location, will feel a force that tends to make it move with the comoving fluid velocity.
However, this force is not exerted by "space"; it's exerted
by the perfect fluid matter that is of uniform density everywhere. The presence of that uniform density everywhere is what produces the force; so in the real universe, where there is not uniform perfect fluid matter everywhere but isolated bound systems surrounded by empty space, this "force" simply
does not exist. In other words, the FRW-type model that assumes a uniform density of perfect fluid everywhere is simply
an incorrect model for analyzing bound systems in the actual universe. And that is why the expansion of the universe, by itself, does not affect the motion of objects in bound systems
at all (your option 2 (a) above).
In a universe with dark energy, as we believe our universe to be, the dark energy is the sole exception to the above, because, as far as we can tell, dark energy
is present in a uniform density everywhere. So what we have been calling "empty space" up to now, within and surrounding all bound systems, is not actually completely empty; it has dark energy in it. And because it
is present in a uniform density everywhere, dark energy
does exert a force everywhere that tends to push things apart. So dark energy does have a (tiny) effect on, for example, the motion of objects in the solar system, or in the Milky Way galaxy, etc. (your option 2 (b) above). But it's far too tiny for us to be able to measure it, so for practical purposes we can ignore it.