Does the expansion of the Universe affect orbiting bodies?

[Cato]

TL;DR Summary

Does the expansion of the universe affect orbits? Would the orbits of the Magellanic Clouds, for example, be different if the universe were not expanding?

Does the expansion of the universe affect orbits? Would the orbits of the Magellanic Clouds, for example, be different if the universe were not expanding? If orbits are affected, at what scale do we first detect the effects?

 

[lomidrevo]

No, the bound systems like galaxies or local groups are not affected

 

[Ibix]

The "expansion of the universe" (more precisely, the increasing scale factor) is just Newton's First Law, complicated by curved spacetime. Things that are moving apart continue to move apart. Things that are bound together, however, won't start moving apart just because other things are moving apart.

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[davenn]

...….Does the expansion of the universe affect orbits? Would the orbits of the Magellanic Clouds, for example, be different if the universe were not expanding?

No, the bound systems like galaxies or local groups are not affected

lomidrevo should have added a word to qualify his answer so thatr you had a better understanding

No, gravitationally bound systems like galaxies or local groups are not affected.

The Magellanic Clouds are small galaxies that orbit the Milky Way Galaxy and are grav. bound to itDave

 

[Ibix]

No, gravitationally bound systems like galaxies or local groups are not affected.

I'd modify this a bit. Yes, local groups are bound together by their mutual gravity and you're right that it's probably worth pointing that out. However, any bound system will not expand. An isolated asteroid, for example, may be bound by electromagnetic interaction between its atoms not by its gravity. It will not expand any more than the gravitationally bound cluster will.

 

[Cato]

Thanks, all, for your responses. I guess I do not understand the concept of “being bound” in a gravitational interaction. I do not see how orbiting bodies are in any way gravitationally bound – they are simply falling toward each other. In a hypothetical nonexpanding universe, even far distant bodies will fall toward each other and eventually meet. In an expanding universe, those far distant bodies, though gravitationally attracted to each other, will nevertheless recede from each other because the effect of the expansion of space is greater than the effect of gravitational attraction.

Is this discussion of the Big Rip in Wikipedia just wrong: “In physical cosmology, the Big Rip is a hypothetical cosmological model concerning the ultimate fate of the universe, in which the matter of the universe, from stars and galaxies to atoms and subatomic particles, and even spacetime itself, is progressively torn apart by the expansion of the universe at a certain time in the future.”

 

[PeterDonis]

I guess I do not understand the concept of “being bound” in a gravitational interaction.

"Bound" simply means "cannot escape without energy being added to it". If you are in orbit about the Earth, you can't escape unless energy is added to you--for example by firing a rocket. That means you are gravitationally bound to the Earth.

This sense of "bound" is the relevant one for this discussion.

 

[PeterDonis]

those far distant bodies, though gravitationally attracted to each other

"Gravitationally attracted to each other" is not the same as "gravitationally bound". The latter is a much stronger condition.

 

[PeterDonis]

Is this discussion of the Big Rip in Wikipedia just wrong

No. The Big Rip is not the same as "expansion of the universe", which is what you asked about in the OP.

 

[Cato]

OK, thanks for that.

 

[Ibix]

I guess I do not understand the concept of “being bound” in a gravitational interaction.

As Peter says, two things are bound if they can't escape each other without some injection of energy. You are correct that we are (in principle anyway) gravitationally attracted to stars billions of light years away, but there are stars in all directions a long way away and it averages out. Locally it doesn't, because there are over-dense and under-dense regions and things near an over-dense region will tend to be pulled towards it.

Is this discussion of the Big Rip in Wikipedia just wrong

The "expansion of space" is a phenomenon in a lot of (all?) cosmological models. But looking in detail at the data we actually see, general relativity cannot model the expansion of our universe if it consists solely of matter, dark matter, and radiation. You need to add a little something extra. Depending on how you specify the dynamics of that extra bit, it's either called dark energy, a cosmological constant, or quintessence. Only the last of these would lead to a Big Rip, which would indeed rip apart bound systems.

I would say that it isn't the expansion of space that leads to the Big Rip in that model, though. It's the presence of quintessence, since that's what leads to runaway growth of the scale factor.

 

[Cato]

Thank you, Ibix. Just so that I am clear about what you and the others are in uniform agreement on -- Imagine a purely hypothetical nonexpanding universe empty of matter except for two earth-sized bodies orbiting each other at a distance of, say,10 billion light years. Every answer I have gotten in this discussion seems to say that even these two orbiting bodes are gravitationally bound and that, if that universe were changed instead to one that was expanding at the rate ours is, those two bodies would still be orbiting each other exactly as before because 1) they are gravitationally bound and 2) no energy has been added to them in order to cause them to move apart. Can that really be correct? If it is, I'll try to understand it. Maybe the orbits are quantized...

 

[PeterDonis]

Every answer I have gotten in this discussion seems to say that even these two orbiting bodes are gravitationally bound

In this particular case, yes, they would be. But the energy required for one of them to escape would be extremely small.

if that universe were changed instead to one that was expanding at the rate ours is, those two bodies would still be orbiting each other exactly as before because 1) they are gravitationally bound and 2) no energy has been added to them in order to cause them to move apart. Can that really be correct?

Yes. "Expansion" in itself just means objects are moving apart due to inertia. It doesn't mean there is a force pushing them apart.

It happens to be the case that in our actual universe, according to our best current model, there is a force pushing objects apart; that's what dark energy does. That force increases linearly with distance, so while it is much too small to matter for planets and stars and solar systems and galaxies and even galaxy clusters, it would in fact cause two earth-sized bodies 10 billion light-years apart to not be gravitationally bound, since the dark energy force would provide enough energy to them to allow them to escape.

But dark energy does not equate to "expansion" by itself; it only equates to "accelerated expansion". It is not causing the universe to expand; that is due to inertia from the Big Bang. Dark energy is only causing the expansion to accelerate.

 

[PeterDonis]

Maybe the orbits are quantized...

QM is not involved in anything I have said.

 

[Ibix]

Every answer I have gotten in this discussion seems to say that even these two orbiting bodes are gravitationally bound

Probably. We can't actually test that GR is accurate for such a weak interaction, but that's certainly what we would predict for such a scenario, given your stipulation that they are orbiting each other.

if that universe were changed instead to one that was expanding at the rate ours is,

There are some subtleties hiding here. You can actually do this just by a change of coordinates on the (near enough) flat spacetime these two planets inhabit from Einstein to Milne coordinates. And changing coordinates doesn't change anything except your description of the situation. But Milne's model, while it could match the current expansion rate of our universe, can't match the expansion history we see in our universe. For that, you actually need a universe more or less uniformly full of matter, and whether two objects at some distance apart can be gravitationally bound in such a universe depends on the scale of density fluctuations (edit: and the presence or absence of dark energy type phenomena, as @PeterDonis notes). I think ten billion light years is too long a scale for our universe; I don't know if two planets in an otherwise uniform universe on that scale is plausible or not. It's a completely different model, though.

So, in summary, I think if you are happy with the coordinate change then the answer is trivially "yes they are still bound". But if you want a universe more like ours then it depends on how you make your new universe look.

 

[Cato]

Yep. I'm happy. What a great site Physics Forums is. Thanks, all.

 

[alantheastronomer]

Summary:: Does the expansion of the universe affect orbits? Would the orbits of the Magellanic Clouds, for example, be different if the universe were not expanding?

Does the expansion of the universe affect orbits? Would the orbits of the Magellanic Clouds, for example, be different if the universe were not expanding? If orbits are affected, at what scale do we first detect the effects?

I have to emphatically disagree with every other response posted on this thread! Being gravitationally bound does not make a system immune to the universe's expansion, so the answer to your question would be "yes"...as long as the distances involved are a significant fraction of a megaparsec - which the Andromeda Galaxy and the Magellanic Clouds are.

To see just how significant the effect is, the distance to Andromeda is two and a half million light years. The current value of the expansion rate is 70km/sec per megaparsec, so the expansion contributes 55km/sec to Andromeda's recession from the Milky Way. But these two galaxies are supposed to collide in five billion years, approaching each other with a velocity of 110km/sec. What this means is that, were the universe not expanding, the two galaxies would be approaching each other at a speed of 165km/sec and collide in three billion years instead. I have no idea whether or not this expansion has been taken into account in the calculation of the collision's timescale.

But what does this mean for the Magellanic Clouds? They're at distances of 46 and 63 kiloparsecs, respectively, which means they're receding from the Milky Way at velocities of 3.2 and 4.4 km/sec. This translates to them having receded 30 and 45 parsecs over ten billion years, to put that into perspective...

 

[PeterDonis]

I have to emphatically disagree with every other response posted on this thread!

You have given no proper support for your claims. In fact you are simply assuming your conclusion: assuming that you can calculate a "recession velocity" due to the universe's expansion for any distance and then simply apply it to nearby galaxies. You are certainly not using the actual math that physicists use when they say that gravitationally bound systems are not affected by the universe's expansion.

The expansion of the universe is not a force and does not change the relative velocities within gravitationally bound systems.

What this means is that, were the universe not expanding, the two galaxies would be approaching each other at a speed of 165km/sec

It means no such thing. The relative velocity of the Andromeda galaxy and the Milky Way is unaffected by the universe's expansion.

I have no idea whether or not this expansion has been taken into account in the calculation of the collision's timescale.

It isn't, because it shouldn't be. See above.

They're at distances of 46 and 63 kiloparsecs, respectively, which means they're receding from the Milky Way at velocities of 3.2 and 4.4 km/sec.

No, they're not. You do know that we have used telescopic observations to measure the actual speeds of the Magellanic Clouds, right? Perhaps you should look up the results of those observations before making obviously erroneous claims.

 

[alantheastronomer]

... you are simply assuming your conclusion: assuming that you can calculate a "recession velocity" due to the universe's expansion for any distance and then simply apply it to nearby galaxies.

So at what length scale does universal expansion cease to apply?

You are certainly not using the actual math that physicists use when they say that gravitationally bound systems are not affected by the universe's expansion.

Which is?

The relative velocity of the Andromeda galaxy and the Milky Way is unaffected by the universe's expansion.

Now you're the one making an unsubstantiated claim...

No, they're not. You do know that we have used telescopic observations to measure the actual speeds of the Magellanic Clouds, right? Perhaps you should look up the results of those observations before making obviously erroneous claims.

You're right, I should have said "the contribution to their motion due to universal expansion is..." though I know you disagree with that as well!

 

[PeterDonis]

at what length scale does universal expansion cease to apply?

It's not a matter of "what length scale". It's a matter of whether a particular object is part of a gravitationally bound system or not.

Which is?

In any textbook on cosmology. This is the sort of background knowledge you should already have if you are going to post in an "I" level thread on cosmology.

Now you're the one making an unsubstantiated claim...

No, I'm the one making a claim that, unlike yours, actually takes into account our best current knowledge of this subject area, which includes theoretical models that have been well confirmed by experiment. It's impossible for us to get a second copy of the universe which is not expanding and compare relative velocities in that universe to this one, so any claim about whether or not that expansion affects the relative motion of objects cannot be based solely on observation. It has to take into account theory. Mine does. Yours does not; it's just an erroneous use of a simple calculation in a case where that simple calculation does not apply.

You are making the common mistake of viewing "the expansion of the universe" as a force, a thing that can cause other things to move. It's not. It's just an effect of the prior history of the universe: objects which are not gravitationally bound to each other are moving apart due to inertia--they were moving apart in the past. There is no force involved that is pushing objects that are gravitationally bound to each other apart. Nor is there some magical "expansion velocity" that contributes to the relative motions of gravitationally bound objects. That is what our best current understanding says, based on our best current theoretical model, which, as above, has been well confirmed by experiment.

Your "unsubstantiated claim" argument basically amounts to: we can't actually directly measure that there is zero contribution of "expansion velocity" to the relative motions of the Andromeda Galaxy and the Milky Way, or the Magellanic Clouds and the Milky Way (since, as above, we can't get a second copy of the universe that is not expanding to compare to, and the timescale for it to affect the distance to those objects in a measurable way is much too long), so we have no way of refuting the claim that there is such a contribution. In other words, you are saying "since we can't actually directly refute my claim, I'm going to stick to it, even though it contradicts our best current model of the universe, which is well confirmed by experiment". That's not a viable position to hold.

 

[stefan r]

It's not a matter of "what length scale". It's a matter of whether a particular object is part of a gravitationally bound system or not...

What happens at the boundary between the Milky Way and Maffei I? Suppose, for example, one particle is on the Milky Way side and is falling towards the Milky Way and another particle is one parsec further and is falling towards Maffei 1. Is the space between them growing at 70 mm/s? Is there a gap of space between the Milky Way and Maffei 1 which is not bound to either galaxy? Where is the boundary at which expansion starts to apply?
I had the impression that a spaceship leaving the local group at escape velocity plus Maffei 1's recessional velocity would be bound to Maffei 1. Once the Milky Way and local group pass further away Maffei 1's gravity would start to accelerate the ship. That means the ship is gravitationaly bound to Maffei group even though inter-galactic gas particles around the ship are still bound to the Milky Way. Is this wrong?

 

[PeterDonis]

What happens at the boundary between the Milky Way and Maffei I?

There is no "boundary". Galaxies aren't objects with well-defined boundaries. Nor is there a sharp cutoff in the gravitational influence of galaxies. You can't go to a certain distance outside the Milky Way, or any galaxy, and say that now you are "outside" it and its gravitational influence no longer counts.

Suppose, for example, one particle is on the Milky Way side and is falling towards the Milky Way and another particle is one parsec further and is falling towards Maffei 1.

The dynamics involved is nowhere near as simple as this. A galaxy is not a single object with a single center of attraction. Also you are ignoring dark matter halos, which complicate things even further.

Even more important, the Milky Way, its satellite galaxies, and the Andromeda galaxy are all part of a larger gravitationally bound system, the Local Group, which in turn is part of a galaxy cluster, which in turn is part of a supercluster. So the idea that you have somehow escaped from being in a gravitationally bound system just by being at some particular point "outside" the Milky Way, even if it were valid with respect to the Milky Way (which it isn't, see above), would not be valid with respect to the larger gravitationally bound systems that include it.

Is the space between them growing at 70 mm/s?

No. "Space growing" is not a good description of what the expansion of the universe does, although it unfortunately is a commonly used term in pop science discussions. The fact that it has misled you here into an invalid inference is a good example of why it's not a good description.

 

[alantheastronomer]

You are making the common mistake of viewing "the expansion of the universe" as a force...

I never intended to give that impression; I was simply illustrating what effect universal expansion would have on their relative motion, interpreted as a velocity of recession.

Is the space between them growing at 70 km/s?

No. "Space growing" is not a good description of what the expansion of the universe does..."

then let's just say "increasing"...
Suppose for a moment that the relative motion of Andromeda and the Milky Way were due to a chance encounter rather than to their being in a bound system. In that case would they be subject to universal expansion or no?

This is the sort of background knowledge you should have...

Are you saying "if you don't know then I'm not going to tell you"? It would be nice to see the math that shows that gravitationally bound systems remain unaffected by the universal expansion, not only for myself but for others who view this thread...every child knows that being told "because I said so!" is a very unsatisfying answer, and so far no one has laid out the reasoning behind it...

You are simply assuming your conclusion. You have given no proper support for your claims.

Yes it's true. I'm beginning with making the assertion that universal expansion applies throughout the universe, at all length scales, and under all circumstances.

The expansion of the universe is not a force.

Exactly! In fact it's an unprecedented physical phenomena that can't be explained...it can be described mathematically, it can be modeled by computer, but there's no underlying physical principle to show why it should be so.
The scale factor, a(t), is used to describe the expansion of the universe as a function of time. Solely as a function of time, not of time and mass, or time and density. So simply by extrapolation it seems to me it should apply to all length scales regardless of what's occurring with the mass or energy density within it. Which is not to say that the motions of galaxies cannot be modified by their gravitational interactions, but that would be on top of the underlying expansion.

The standard cosmological model invokes Birkhoff's Theorem - which states that any gravitationally bound system is described by the Schwarzschild metric and is static, therefore cannot be affected by the general expansion. But that too is a bit of circular logic; it's saying because there is no time dependence of the metric therefore the system must remain static and uninfluenced by any expansion.

The reason Birkhoff's Theorem was introduced was to resolve the question of how it was possible for galaxies and large scale structure to form to begin with, in a uniform, expanding universe. Since bound systems are governed by the Schwarzschild metric and not the FLRW metric, they are not affected by the expansion.

But it's not necessary to invoke Birkhoff's Theorem in order explain large scale structure formation. One simply has to look as far as the critical density, 3H^2/8piG. Usually used to describe the amount of matter needed to halt the universe's expansion and close the universe, it can also be described as the critical density needed for an element's self-gravity to overcome the expansion and undergo gravitational collapse.

This is not to say that Birkhoff's Theorem doesn't apply, just that it doesn't supersede FLRW. They are both operative in gravitationally bound systems.

 

[PeterDonis]

I was simply illustrating what effect universal expansion would have on their relative motion, interpreted as a velocity of recession.

Universal expansion has no effect on relative motion unless you assume the objects are comoving with the universal expansion. But that is precisely what you cannot assume for objects that are gravitationally bound to each other.

Suppose for a moment that the relative motion of Andromeda and the Milky Way were due to a chance encounter rather than to their being in a bound system.

There's no need to "suppose". You can calculate whether the two are gravitationally bound based on their relative motion, their distance apart, and their masses. Such a calculation shows that they are indeed gravitationally bound.

Are you saying "if you don't know then I'm not going to tell you"?

No, I'm saying that I'm not going to give you a course in cosmology. That's what cosmology textbooks are for.

I'm beginning with making the assertion that universal expansion applies throughout the universe, at all length scales, and under all circumstances.

And this is, once again, precisely what you cannot assume, because this assumption is equivalent to assuming that no objects in the universe are gravitationally bound to each other, anywhere, at any length scale, and that the universe has uniform density everywhere, on all length scales. Which is obviously false. See further comments below.

In fact it's an unprecedented physical phenomena that can't be explained...

This is nonsense. The expansion of the universe is perfectly well explained by inertia.

You are getting very close to a warning for personal speculation.

The scale factor, a(t), is used to describe the expansion of the universe as a function of time.

As a function of time in a particular set of coordinates, yes.

by extrapolation it seems to me it should apply to all length scales regardless of what's occurring with the mass or energy density within it

Not at all. The reason why the model admits such a nice set of coordinates, in which the scale factor is only a function of coordinate time, is that the stress-energy present is entirely in the form of a perfect fluid, with uniform density everywhere at a given instant of coordinate time. This is a reasonable model for the average content of the universe on large scales, such as hundreds of millions of light-years and up. But it is obviously not a reasonable model for the content of the universe on smaller scales, like the scales of galaxy clusters, galaxies, solar systems, stars, and planets. And no cosmologist claims that it is. Cosmologists are well aware that the universe is lumpy on small distance scales, and that this means the simple model of an "expanding universe" simply cannot be applied on such scales.

This is the sort of thing that you would already know if you took the time to look even briefly at a cosmology textbook. Or even a decent brief online treatment of cosmology, such as Chapter 8 of Carroll's online lecture notes on GR:

https://arxiv.org/abs/gr-qc/9712019

The standard cosmological model invokes Birkhoff's Theorem

It does no such thing. Cosmology models the universe using FRW spacetime, not Schwarzschild spacetime.

which states that any gravitationally bound system is described by the Schwarzschild metric and is static

It states no such thing. Birkhoff's Theorem states that any spherically symmetric vacuum spacetime must be described by the Schwarzschild metric. Cosmologists do not model the universe as a vacuum spacetime; they model it as containing a perfect fluid.

The reason Birkhoff's Theorem was introduced was to resolve the question of how it was possible for galaxies and large scale structure to form to begin with, in a uniform, expanding universe. Since bound systems are governed by the Schwarzschild metric and not the FLRW metric, they are not affected by the expansion.

This is all nonsense. See above. Cosmologists' models of structure formation in the universe have nothing whatever to do with Birkhoff's Theorem or the Schwarzschild metric. I don't know where you are getting all this from.

it can also be described as the critical density needed for an element's self-gravity to overcome the expansion and undergo gravitational collapse

Please give a reference for this claim.

This is not to say that Birkhoff's Theorem doesn't apply, just that it doesn't supersede FLRW. They are both operative in gravitationally bound systems.

Nonsense. See above.

 

[alantheastronomer]

Please give a reference for this claim.

If you set the escape velocity from the surface of a spherical fluid of radius R equal to the expansion velocity at that distance, you get the expression for the critical density.

I don't know where you are getting all this from.

Harwit, Astrophysical Concepts, chapter 12.

 

[PeroK]

Harwit, Astrophysical Concepts, chapter 12.

The overall expansion of the universe currently is dominated by the vacuum energy density. This is about 0.7 compared to the mass density of 0.3. If you put those into the appropriate equation(s) you get the current expansion rate.

A system such as the local group does not have those densities. The vacuum energy density is almost negligible. My question is: what equations would you use to show that the behaviour of the local group is not dominated simply by its mass density? Isn't it the case that only the vacuum energy of the local group (the part of the universe under consideration) is relevant?

How would the average vacuum energy density throughout the entire universe affect a system that doesn't share that average density?

 

[Buzz Bloom]

No, gravitationally bound systems like galaxies or local groups are not affected.

No, the bound systems like galaxies or local groups are not affected

Things that are bound together, however, won't start moving apart just because other things are moving apart.

Universal expansion has no effect on relative motion unless you assume the objects are comoving with the universal expansion. But that is precisely what you cannot assume for objects that are gravitationally bound to each other.

Hi dave, lomidrevo, Ibix, and Peter:

I confess I have been trying (unsuccessfully) to get my head around this concept (expansion has no effect on components of a bound system) for quite a while now. My best guess now about what the root of my problem is relates to the lack of clarity of the definitions involved. There are two terms that seem to me to be ambiguous.

1. bound - this seems to imply that there is a conceptually measurable rule which can be applied to the a collection of mass objects which distinguishes unambiguously whether or not they are gravitationally bound together.

2. not affected - this seems to mean that there would be no difference at all in the characteristics of an orbit with respect to two universe configurations: (i) the universe is expanding, and (ii) the universe is not expanding.

In the discussions I have been reading, regarding (1) there has not been any completely clear and unambiguous definition of what it means for a collection of bodies to be bound to each other. 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. Note: 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.

Regarding (2), this might mean either:
(a) if a lighter body has a velocity relative to a heavier body and this velocity is less than the escape velocity for the distance between the bodies, then orbits would be identical whether or not the universe is expanding,
or
(b) the orbit might be different between the two situations, but the difference would be so small as to be negligible for any practical consideration.

Some of the posts seem to be saying (a), but some are unclear whether (a) or (b) is intended.

I gather the McVitte metric should be able to make a clear distinction between (2a) or (2b). If someone could show by a thought experiment's example what McVitte's metric makes clear regarding this ambiguity, that would be very helpful. I am not confident that my math skills are up to this challenge, but I am working on it.

Regards,
Buzz

 

[PeterDonis]

Harwit, Astrophysical Concepts, chapter 12.

That book makes the same claims you are making about Birkhoff's Theorem?

 

[Ibix]

I confess I have been trying (unsuccessfully) to get my head around this concept (expansion has no effect on components of a bound system) for quite a while now. My best guess now about what the root of my problem is relates to the lack of clarity of the definitions involved. There are two terms that seem to me to be ambiguous.

If the universe were everywhere filled with a perfect fluid (this is made up of genuinely continuous matter, not of atoms or anything - don't ask too many questions) that was everywhere of uniform density and pressure, then that universe would be accurately described by an FLRW spacetime. There is nowhere special in this universe. Every point is exactly like every other, always was, and always will be. Every point in that fluid is moving away from every other point in the exact same way.

The real universe is rather like this at very large scales. In the same sense that I can treat a glass of water as a continuous fluid, not worrying about atomic-scale effects, I can talk about the universe being isotropic and homogeneous, not letting the fact that this isn't true at scales below a hundred million light years bother me - the observable universe is around 450 times larger than that. However, at small scales, it's nothing like that. There are galaxies and stars and planets and stuff, which are much denser than the vacuum around them. These are "over-dense" and "under-dense" regions, which occur on many scales. Looked at in detail, matter in these regions is moving more or less randomly - but if you average the motions over a hundred million light year volume and compare it to the next hundred million light year volume, both of those are moving apart. Below that scale, though, things are not uniform and never were. Things are not flying apart from their neighbours and never were. And that's why we are saying that the expansion of the universe has no effect on small systems that are bound together by their gravity - they are not flying apart and never were.

There is a caveat. At least one variant on dark energy, called "quintessence", does eventually tear apart even atoms in the so-called "Big Rip". However, I'd say that in this case it isn't the expansion of the universe that tears the atoms apart so much as the presence of quintessence, which also dictates the particular form of expansion of the universe.

I think your attempt to understand "bound systems" in an expanding universe is probably the wrong way to look at this. It's a bit like trying to understand a fluid by focussing on Brownian motion. The fundamental fact is that, at small scales, the universe is not FLRW and never was, so at small scales the matter in the universe is not flying apart and never was. At large scales, on average, it's flying apart because of its own inertia and always has been. The presence of dark energy-type phenomena modifies that slightly because that does exert an additional outward pressure on everything. But we don't know enough about it to know if it's affecting small scale motion - we certainly have never detected it in the motion of the planets.

Now to see how many mistakes @PeterDonis points out in the above...

 

[PeterDonis]

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.

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.

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.

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.

 

[PeterDonis]

The fundamental fact is that, at small scales, the universe is not FLRW and never was

The "never was" part isn't necessarily true. If you go back early enough, the universe was filled with matter (and radiation) everywhere, with a density that was close to uniform everywhere. (But not exactly uniform--the small inhomogeneities that existed then ended up leading to the much larger inhomogeneities we have now.) So an FRW model actually was a reasonable model even on small scales in the early universe. It got less and less reasonable on small scales (and what scale counted as "small" got larger and larger) as the universe developed more and more structure.

 

[Ibix]

So an FRW model actually was a reasonable model even on small scales in the early universe. It got less and less reasonable on small scales (and what scale counted as "small" got larger and larger) as the universe developed more and more structure.

So both the degree and scale of inhomogeneity grow with time. Is it correct to say something like "there is always a scale (which shrinks as you go back in time) below which the universe is not FLRW to some precision (which gets tighter as you go back in time"?

 

[PeroK]

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.

How would you describe the solution when it comes to the local group of galaxies and, in particular, the collision of the Milky Way and Andromeda? The distances here are significant if we plug in the current universe expansion rate. But, is this valid? That expansion rate is based on the overall universal balance of mass/vacuum. How would the universe expansion rate get into the equations for the dynamics of the local group?

Instead, should we model the local group using the specific mass/vacuum density we find in this region? This would then include a small element due to the dark energy, but would still be largely negligible.

Can these two models be tested against the observed separation velocity?

 

[PeterDonis]

Is it correct to say something like "there is always a scale (which shrinks as you go back in time) below which the universe is not FLRW to some precision (which gets tighter as you go back in time"?

Probably, at least back to the end of inflation, since our best current model says that it was tiny inhomogeneities at the end of inflation (due to quantum fluctuations in the inflaton field) that eventually grew into the structures we see today.