Proportions and other Stuff about Expansion

[ComaBerenices]

I'm not too well read in cosmology, but I know the universe is presumed to be homogeneous in space but not in time, therefore the curvature of the universe can essentially be calculated under one mean.
I just have a few questions about how the universe expands and whether some proportions are legit.
Firstly, say you have two gravitationally attracted points in space that are expanding away from each other:
http://geometry.freehomeworkmathhelp.com/Geometry_Main_0/geometry_homework_help_line_segment.GIF
Because I have heard that DE does not affect the actual motion of galaxies in space, but simply expands the space between them, would these objects continually accelerate 'towards' each other until they reach the distance where a signal cannot be communicated between them due to the increasing expansion between them? If so, if the two objects were within the range where gravitation and expansion balance out, would the two objects soon come together due to acceleration of velocity because of gravity?

Secondly, in an expanding, homogeneous universe, as distances grow, so does gravitational potential energy, as the curvature of space becomes less steep, the gravitational potential energy grows more slowly. This has seemed to correlate with a decreasing Hubble rate of expansion over time, however it is currently decreasing at a decelerated rate. I find this proportion curious so I was wondering whether any cosmologist has looked into this.

Regards,

 

[Bandersnatch]

Because I have heard that DE does not affect the actual motion of galaxies in space, but simply expands the space between them, would these objects continually accelerate 'towards' each other until they reach the distance where a signal cannot be communicated between them due to the increasing expansion between them? If so, if the two objects were within the range where gravitation and expansion balance out, would the two objects soon come together due to acceleration of velocity because of gravity?

Yes, that's why overdense regions in the early universe collapsed to form gravitationally bound objects, like galaxies and groups of galaxies, and why the expansion is visible only on the large scales (e.g. the solar system doesn't expand).
You don't have to separate the two 'kinds' of velocities here, though - you might just as well treat the recession due to expansion as escape velocity w/r to the mass of the overdense region.

Also, this has got nothing to do with the ability to 'communicate signals' - the scale where groups of galaxies are too far away from each other to be gravitationally bound is much less than the distance at which recession exceeds the speed of light. In the same way, when you shoot a probe out of the Solar system with escape velocity, it is going to be continuously affected by the gravity of the Sun, even as you can be sure it'll never return.I'm not too comfortable with the second question, so I'll let somebody else take it (or get back to you later if nobody does).

 

[ComaBerenices]

Okay thanks, I would think there'd still be a gravitational effect creating acceleration, can gravity effect things that recede beyond the Hubble radius?
Apologies if the second bit sounds a bit weird, I read a bit of an arvix preprint that stated that expansion doesn't violate conservation of energy if you take into account gravity as a decreasing negative factor, so I was curious.

 

[Bandersnatch]

I'm saying Hubble radius has got nothing to do with massive systems being gravitationally bound or not. The recession velocities are high enough to exceed escape velocity of a typical overdense region (group of galaxies) on the scale as small as a few megaparsecs.

 

[ComaBerenices]

sorry, misinterpreted! C:

 

[PeterDonis]

in an expanding, homogeneous universe, as distances grow, so does gravitational potential energy

Not really. The concept of "gravitational potential energy" isn't well-defined in a spacetime that is not static. The expanding universe is not static.