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This paper dates to 1998:

Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," http://arxiv.org/abs/astro-ph/9803097v1

They show that systems such as the solar system, galaxies, and clusters of galaxies experience nonzero effects from cosmological expansion. They estimate these effects numerically, and show that they're undetectably small. They are, however, fun to think about.

At the time the paper was written, we didn't know about dark energy. It turns out to be a little tricky to adapt the results of the paper to modern cosmological models. Well, tricky for me, at least -- I messed it up repeatedly in this thread: https://www.physicsforums.com/showthread.php?p=4061068#post4061068

The effect of cosmological expansion is felt, in a circularly orbiting system, as an anomalous acceleration [itex](\ddot{a}/a)\mathbf{r}[/itex], where a is the cosmological scale factor. The direction of this effect depends on the sign of the cosmological acceleration [itex]\ddot{a}[/itex], not the sign of the cosmological expansion [itex]\dot{a}[/itex]. In an expanding matter-dominated universe, it's inward rather than outward!

However, the effect of this perturbation on an orbit is a whole different issue. If I'm understanding correctly how to retrofit the Cooperstock results to an a(t) that isn't what they anticipated, it looks like the sign of the secular trend [itex]\dot{r}[/itex] in the orbital radius r depends on the sign of [itex](d/dt)(\ddot{a}/a)[/itex]. This does not relate directly to the direction of the anomalous acceleration. In the matter-dominated cosmology that Cooperstock assumed, the inward anomalous acceleration produces, very counterintuitively, an expansion of the orbital radius. If you were free to pick any function a(t), you could devise examples in which the orbit would contract in response to cosmological expansion, which would be pretty crazy!

However, the FRW equations tell us this:

[tex]\frac{\ddot{a}}{a} = \frac{1}{3}\Lambda-\frac{4\pi}{3}(\rho+3P)[/tex]

That means that [itex]\dot{r}[/itex] is proportional to [itex]-(d/dt)(\rho+3P)[/itex], with a positive constant of proportionality. The cosmological constant term vanishes when you differentiate, so although a nonzero [itex]\Lambda[/itex] can affect the sign of the anomalous acceleration, it has no effect on the secular trend in r.

For simplicity, let's assume a universe in which dust and dark energy are dominant, so that we can take P=0. Then it looks to me like the secular trend in the radius of a circular orbit is proportional to [itex]-d\rho/dt[/itex].

So:

(1) In a matter-dominated closed universe, this tells us that r expands slightly while the universe expands, then contracts slightly while the universe contracts. Not too surprising. (And remember, the effect is much too small to measure, much tinier than the variation in a(t).)

(2) In a flat or negative-curvature matter-dominated universe, r expands.

(3) In a vacuum-dominated universe, the effect vanishes.

Does this all seem right? Am I still messing anything up?

It seems kind of funny that 1 and 2 are so much like what we would have naively expected from some kind of struggle between binding and cosmological expansion, even though the effect really comes from quantities like the third derivative of the scale factor, and not on the rate of expansion itself. Seems like some kind of voodoo that must have a deeper explanation. Meanwhile 3 suggests that all of this is just some silly coincidence that arises out of various grotty, complicated dynamical effects that almost vanish, and that only work in this particular way for circular orbits. (It really does all change if the system isn't a circularly orbiting one. E.g., if you scoop out a spherical vacuum from a cosmological spacetime, Birkhoff's theorem guarantees that local effects vanish exactly.)

Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," http://arxiv.org/abs/astro-ph/9803097v1

They show that systems such as the solar system, galaxies, and clusters of galaxies experience nonzero effects from cosmological expansion. They estimate these effects numerically, and show that they're undetectably small. They are, however, fun to think about.

At the time the paper was written, we didn't know about dark energy. It turns out to be a little tricky to adapt the results of the paper to modern cosmological models. Well, tricky for me, at least -- I messed it up repeatedly in this thread: https://www.physicsforums.com/showthread.php?p=4061068#post4061068

The effect of cosmological expansion is felt, in a circularly orbiting system, as an anomalous acceleration [itex](\ddot{a}/a)\mathbf{r}[/itex], where a is the cosmological scale factor. The direction of this effect depends on the sign of the cosmological acceleration [itex]\ddot{a}[/itex], not the sign of the cosmological expansion [itex]\dot{a}[/itex]. In an expanding matter-dominated universe, it's inward rather than outward!

However, the effect of this perturbation on an orbit is a whole different issue. If I'm understanding correctly how to retrofit the Cooperstock results to an a(t) that isn't what they anticipated, it looks like the sign of the secular trend [itex]\dot{r}[/itex] in the orbital radius r depends on the sign of [itex](d/dt)(\ddot{a}/a)[/itex]. This does not relate directly to the direction of the anomalous acceleration. In the matter-dominated cosmology that Cooperstock assumed, the inward anomalous acceleration produces, very counterintuitively, an expansion of the orbital radius. If you were free to pick any function a(t), you could devise examples in which the orbit would contract in response to cosmological expansion, which would be pretty crazy!

However, the FRW equations tell us this:

[tex]\frac{\ddot{a}}{a} = \frac{1}{3}\Lambda-\frac{4\pi}{3}(\rho+3P)[/tex]

That means that [itex]\dot{r}[/itex] is proportional to [itex]-(d/dt)(\rho+3P)[/itex], with a positive constant of proportionality. The cosmological constant term vanishes when you differentiate, so although a nonzero [itex]\Lambda[/itex] can affect the sign of the anomalous acceleration, it has no effect on the secular trend in r.

For simplicity, let's assume a universe in which dust and dark energy are dominant, so that we can take P=0. Then it looks to me like the secular trend in the radius of a circular orbit is proportional to [itex]-d\rho/dt[/itex].

So:

(1) In a matter-dominated closed universe, this tells us that r expands slightly while the universe expands, then contracts slightly while the universe contracts. Not too surprising. (And remember, the effect is much too small to measure, much tinier than the variation in a(t).)

(2) In a flat or negative-curvature matter-dominated universe, r expands.

(3) In a vacuum-dominated universe, the effect vanishes.

Does this all seem right? Am I still messing anything up?

It seems kind of funny that 1 and 2 are so much like what we would have naively expected from some kind of struggle between binding and cosmological expansion, even though the effect really comes from quantities like the third derivative of the scale factor, and not on the rate of expansion itself. Seems like some kind of voodoo that must have a deeper explanation. Meanwhile 3 suggests that all of this is just some silly coincidence that arises out of various grotty, complicated dynamical effects that almost vanish, and that only work in this particular way for circular orbits. (It really does all change if the system isn't a circularly orbiting one. E.g., if you scoop out a spherical vacuum from a cosmological spacetime, Birkhoff's theorem guarantees that local effects vanish exactly.)

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