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I was wondering if theoretically the universe expanding affects speed or acceleration due to gravity, compared to if space weren't accelerating.

The way I have it in my head, something is falling towards a planet with gravity G acting upon it. It's been falling for T amount of time and has gone X distance with Y left to go. Currently the universe is expanding, so the distance between its starting and ending point is a very tiny bit longer when it arrives than when it started, either affecting its distance traveled at time T, or its speed at time T. Compare this to a non-expanding universe where the distance between start and end remains the same, no matter what point in time you choose.

I know that local gravity forces and atomic forces greatly overpower expansion forces so galaxies, solar systems, planets, and watermelons don't just fly apart. I'm just wondering if the value of either gravitational acceleration, or an object's speed could theoretically be changed slightly due to cosmic expansion.

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    $\begingroup$ Good question, but local gravity forces not only overpower the expansion of the Universe on small scales, it actually prevents it. That is, inside galaxies the Universe doesn't expand at all. So the answer is no. $\endgroup$ – pela Jun 9 '16 at 14:42
  • $\begingroup$ @pela That's a bit misleading. They do counter it with gravity, but because the amount of dark energy in the Universe is increasing, they will eventually expand. Not to mention, as far as I know, space is still being created between stars in galaxies — but they are still held together by gravity. $\endgroup$ – Sir Cumference Jun 9 '16 at 16:54
  • $\begingroup$ @SirCumference: On second thought, I don't know to be honest, or at least I think it's impossible to differentiate observationally. Space expands because the initial kick (Big Bang) was larger than the mutual attraction of stuff (and lately because of dark energy). By Birkhoff's theorem, an overdensity (at least a spherical one) can be treated as a mini-universe, independent of the rest of the Universe. So in places with enough matter, space is "held together" by gravity, preventing expansion. That's the standard discription at least, but I'm not sure it's true… $\endgroup$ – pela Jun 10 '16 at 16:48
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    $\begingroup$ @pela If I recall, space itself is expanding; the objects aren't moving away from each other in the traditional sense, but are becoming more distant, since space is actually being created in between the matter. In fact, they can actually drift away from each other faster than light; Einstein said nothing can move though space FTL, but here, space is literally being created between them. Pretty much, gravity could hold the objects close together, but space between them will expand independently. Though, I'm not 100% sure if this info is correct, but this is from what I've heard. $\endgroup$ – Sir Cumference Jun 10 '16 at 17:07
  • $\begingroup$ @SirCumference Completely true; space expands, and galaxies follows along, and for sufficiently large distance, galaxies recede faster than light. But until space was small enough that dark energy played no significant role, the expansion was decelerated because of the mass in space. So matter prevents expansion. Whether or not it prevents it completely inside sufficiently dense regions is debatable, or not known. This is at least the conclusion I and a few colleagues reached today (but none of us are general relativists, only galaxy/dark matter astrophysicists, so I better stop talking :) ) $\endgroup$ – pela Jun 10 '16 at 19:28
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Right now the expansion of space is only observed on scales like that of galaxies and the entire universe. We say that with the fancy term globally. Expansion of space does not take place within galaxies at least for now.

As pela has rightfully noted in the comments, for now the gravitation of galaxies, stars etc. is strong enough to overcome the expansion of the universe. But as General Relativity tells us gravity is local, that is, it only affects objects that are considerably close. So, on large scales expansion continues. (I am no expert, but I think that the reason for expansion not happening within galaxies out smaller systems is that the matter and energy is more dense, resolution by in stronger gravitational fields).

It is hypothesized that the universe may end with a big rip: this means that the expansion eventually would start to affect galaxies. Galaxies, star systems, planets and eventually atoms will be ripped apart due to the expansion of space. When and if that happens, not only gravity but not even electromagnetism or the strong force would work to overcome the mighty expansion.

But this situation is highly hypothetical and maybe it won't even happen. So for now, the answer is no: the expansion of the universe could not affect gravitational acceleration.

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