Alright...back to the topic

Before I toss some numbers out, let me describe the problem qualitatively a little more clearly. And let me begin with some observations: We are gravitationally bound to the earth. The earth is gravitationally bound to the sun. The sun is bound to the galaxy. The galaxy is bound within the Local Group of galaxies. The Local Group is orbiting about some super-cluster, which itself is...wait. At some point, things go from bound, to unbound. We find, as we look around, that we are gravitationally bound within larger and larger groupings, but at some point, everything is "unbound," or falling away.

At what scale does this transition occur, and why?

I have already answered the scale-part: it occurs @ +/- 5 Mpc. The "why" part is subtle, however, and can be looked at from several angles.

Start by considering the solar-system. Imagine a bubble of space, centered on the sun, extending to the outer fringes. Clearly, as the sun burns its hydrogen fuel and shines, more energy leaves this bubble than enters it. But now, what if we expand the bubble to 10 parsecs? Well, in that case, the surface area of the bubble becomes pretty big, but now there are other stars radiating away energy, and we can still be pretty sure that more energy is being radiated away from the volume of space than is entering it. Now imagine a bubble 100 kpc in diameter, this time centered on the center of our galaxy. Such a bubble would now encompass a lot of empty space, but being centered on our galaxy (a relatively bright beacon in our neck of the universe), it would still radiate away more energy than enters it.

But when the size of the bubble gets to about 5 Mpc, a funny thing happens. The surface area of the sphere becomes so large, and the amount of radiative matter within the sphere becomes so small in proportion to it, that the amount of energy leaving the bubble and the amount of energy entering it become equal. Then what? What does that mean to say the amount of energy entering and leaving the bubble is the same?

What it means is that the energy balance has to be taken into consideration. When we considered just the solar-system, we could say, "The sun's energy is radiated into space," and be done with it. There is no accounting for it: energy goes "into space," and that is that. But when we get to 5 Mpc, because the surface area of the sphere is so large, as much energy is "coming in from space" as is being radiated "into space," so we arrive at a paradox. If as much energy is entering as is leaving, what is chainging?

What is changing is the distance between things. Within the 5 Mpc bubble, things tend to get closer together, i.e contract. Beyond it, things get further away, i.e. expand. Within the bubble, matter "loses energy" by radiating it "into space." With respect to matter beyond the bubble, however, matter "gains energy" on account of the expansion.

As for the rate at which matter radiates energy "into space," we can express it in joules/kilogram/year. Visible matter in our universe can be observed to be radiating, on-average, so-many j/kg/yr. Likewise, due to the expansion, matter can be said to be gaining so-many j/kg/yr. Determining the average rate that matter radiates energy is a matter of culling observational data. Calculating the energy gain due to expansion using GR...that gives me a headache just thinking about it!