Does expansion exist on the micro level as well as the galactic level?

[Axuality]

Does expansion exist on the micro level as well as the galactic level?

I've never read about it, but it occurs to me that there can be no uniqueness about where the expansion of the universe exists.

I have to envision an atom as a 'kind' of solar system, and a molecule therefore as a 'kind' of galaxy, and an ore sample of this molecule as a galactic cluster. With this simplistic view in mind, it seems to me that the space between ALL particles, not just large objects, must be increasing- and frankly, that the particles themselves (since they are not "solid" objects, but rather consist of "smaller" entities (smaller expanding "universes")) must be expanding also.

Why have I read nothing of this? I'm not well-read admittedly.

By the way, if this were the case, wouldn't we feel the expansion of the physical object beneath our feet as a force pushing up against the bottom of our feet? Come to think of it, our feet would also be expanding down toward the ground, thereby creating some sort of feeling.

 

[Janus]

Expansion does not extend down to the microscopic level. In fact, it doesn't affect things smaller than clusters of galaxies. The reason for this is that the forces holding these objects together (gravity, electromagnetic and nuclear forces) are stronger than universal expansion.

 

[Axuality]

Expansion does not extend down to the microscopic level. In fact, it doesn't affect things smaller than clusters of galaxies. The reason for this is that the forces holding these objects together (gravity, electromagnetic and nuclear forces) are stronger than universal expansion.

Are we accepting that clusters of galaxies are expanding along with the space between them or not?

You implicitly say that stronger forces stop the expansion on levels smaller than clusters. You mean that these forces stop THINGS from moving.

When I stand on the surface of the earth, is it not said that I am accelerating at 32 ft/sec/sec, although I'm not MOVING toward or away from the center of the earth? Can there not be accleration or even expansion without movement?

It does not make intuitive sense to me that at some arbitrary level, expansion would no longer exist.

We may have to discover that at some relevant frame of reference, the forces of which you speak are not stronger than one another.

 

[Chronos]

In the present epoch, as Janus noted, it safe to say expansion does no operate at levels below intergalactic space. The future is less certain. Google on 'the big rip' for discussions.

 

[bwana]

...
It does not make intuitive sense to me that at some arbitrary level, expansion would no longer exist.

I agree and have puzzled about this contradiction as well. I explain it thusly:

Hubble parameter measures expansion of space and it is greater at increasing distances. Since these distances are HUGE and measured in light years, the information provided by these observations are VERY OLD. I conclude that the universe is no longer expanding since we do not see a Hubble shift in our neighborhood, although we could detect it.

here is what i wrote:
https://www.physicsforums.com/showthread.php?t=335946
http://forums.anandtech.com/messageview.aspx?catid=50&threadid=2320348&enterthread=y

 

[Janus]

Are we accepting that clusters of galaxies are expanding along with the space between them or not?

You implicitly say that stronger forces stop the expansion on levels smaller than clusters. You mean that these forces stop THINGS from moving.

When I stand on the surface of the earth, is it not said that I am accelerating at 32 ft/sec/sec, although I'm not MOVING toward or away from the center of the earth? Can there not be accleration or even expansion without movement?

It does not make intuitive sense to me that at some arbitrary level, expansion would no longer exist.

We may have to discover that at some relevant frame of reference, the forces of which you speak are not stronger than one another.

The forces of which I speak all get stronger as the distance between the objects get closer together. The expansion of the universe does not. So yes, there is a boundary between expansion and nonexpansion. it is that point where galaxies are close enough for their mutual gravitational attraction is greater than their tendency to move apart. Galaxy clusters do not expand because the individual galaxies are close enough to each other for gravity to hold them together. Galaxy clusters move apart because they are not close enough to each other.

An analogy: You are standing on a tile floor in your stocking feet. The floor has 1ft square tiles. The floor starts to expand (the 1 ft tiles eventually become 2 ft tiles, etc.)

You will see a person standing a few tiles away recede from you. (as the tiles expand, the distance between the center of the tiles increases. )

Now imagine a third person standing next to you, and you are holding hands. The person few tiles away will still recede from you, but the person standing next to won't. This is because your grip is stronger than the friction between both of your stockings and the floor. You are like two galaxies in a local cluster gravitationally bound to each other. The tiles still expand under your feet, you just don't move apart with them.

 

[bwana]

---

You are like two galaxies in a local cluster gravitationally bound to each other. The tiles still expand under your feet, you just don't move apart with them.

If gravity has this effect, then measurements of the Hubble parameter in the vicinity of intense gravity wells (black hole, quasar,etc) should be 0. So, 2 galaxies that are very far away might have a certain redshift but a supermassive black hole next to a different galaxy in another part of the universe (but just as far away) should have much less redshift. Has this been seen?

 

[herbert]

Expansion does not extend down to the microscopic level. In fact, it doesn't affect things smaller than clusters of galaxies. The reason for this is that the forces holding these objects together (gravity, electromagnetic and nuclear forces) are stronger than universal expansion.

Whilst i appreciate that i am not at the high level of a PF mentor,

Surely the above cannot be correct.
If so how did inflation come about?
Space must expand at every level.
However, we have gravity too. this causes things to collapse.
There is a battle between expansion and gravity/electrical forces
On the microscopic, gravity and electromagnetic forces win and so whilst space itself expands the particles, planets etc don't.
One has to get far away and into what is called the 'Hubble flow' before expansion effects are greater than 'local effects'.
For instance the andromeda galaxy is close by and will eventually collide with our own. Gravity wins because the expansion effects are smaller that gravity effects.
BUT... all space exapnds no matter on what scale.

 

[herbert]

If gravity has this effect, then measurements of the Hubble parameter in the vicinity of intense gravity wells (black hole, quasar,etc) should be 0. So, 2 galaxies that are very far away might have a certain redshift but a supermassive black hole next to a different galaxy in another part of the universe (but just as far away) should have much less redshift. Has this been seen?

Intense gravity wells etc have gravitational redshift. this will be added on to cosmological redshift.
IE a photon of light escaping a black hole, gains lots of gravitational potential energy. Since E = hf, its energy reduces, frequency reduces wavelength increases.
Gravitationally redshifted.
Not zero.

 

[Chronos]

Gravitational redshift is a non contributor to H0. The vast majority of photons generated by black holes, etc., originate too far from the gravity well to be noticeably redshifted.

 

[PRDan4th]

If Newton were alive today and knew what we know, I think he would have added another term to his gravity equation. This would be a negative term (repulsive) and have a very very small constant times distance squared and perhaps independent on their masses. This term applies all the way from short distances to extremely long distances but is so small that it is not detectable at distances shorter than inter galactic distances. Perhaps something like:

F=G*(m1*m2)/r^2-R*r^2 and may have a m1*m2 relationship as well.

R would be the expansion or repulsive constant.

If this were the case at some distance the R term would balance the G term and gravity force F would be zero, beyond that distance expansion would overwhelm attraction.

Does this make any sense?

PRD

 

[herbert]

Gravitational redshift is a non contributor to H0. The vast majority of photons generated by black holes, etc., originate too far from the gravity well to be noticeably redshifted.

But surely one needs a 'standard candle' to determine Ho?
Cepheid variables work as do supernovae 1a.
Is there a standard candle black hole? ie a black hole that is equally bright (or dark?) no matter where it is in the universe?
Otherwise, surely it is not a matter of the photons 'not being noticeably redshifted' but that they just cannot be used?

 

[sylas]

Does expansion exist on the micro level as well as the galactic level?

No.

To a first approximation (and I must thank https://www.physicsforums.com/member.php?u=51971" as a major help for my understanding this) expansion simply means that things are moving apart from each other. It isn't a force pushing things apart. The universe expands because things are moving apart from each other ... not the other way around.

That's the first approximation. The second approximation is that there are actually forces that act to slow down or to increase the rate at which things are moving.

Gravity works to pull things together, and hence to slow down expansion.

Dark energy works to push things apart, and hence to speed up expansion. As I understand it, "dark energy" is just a name we give for the apparent existence of something we don't really understand but which is the cause of expansion to be speeding up.

But on sufficiently small scales (a cluster of galaxies or less) you can just consider the nice simple model you are all used to, of a bunch of stuff moving around in space, and subject to gravity.

A galaxy isn't expanding, because its constituent parts are not moving apart from each other. No matter what motions you see elsewhere, the galaxy is all hanging together very nicely, and there is no particular "force" to counteract that, or to pull it apart. (Dark energy might in principle; but in the present epoch dark energy is far too weak to pull apart a galaxy or a galactic cluster.)

That's the bottom line. The only thing that might push things apart is a "dark energy", and what we infer about dark energy now is that it is way WAY too weak to matter at small scales. Apart from that, there's nothing at all about the expansion of space that would lead one to expect anything to be pulled apart, on any scale, unless it is already moving apart right now.

Cheers -- sylas

 

[herbert]

No.

To a first approximation (and I must thank https://www.physicsforums.com/member.php?u=51971" as a major help for my understanding this) expansion simply means that things are moving apart from each other. It isn't a force pushing things apart. The universe expands because things are moving apart from each other ... not the other way around.

Don't think so, Space expands carrying all with it. The expansion we see now is just the dying embers of inflation and its scalar fields.
On top of that we have the peculiar motion of the heavenly bodies within it.
On the small scale, gravitational effects are far bigger than expansion of space effects so they dominate.
But on the large scale expansion effects dominate and things move away from each other.
Its a good thing that objects themselves don't get any bigger as otherwise we would not see the expansion.
If distances got bigger and our 'ruler' got bigger at the same rate then we would measure the distances as constant. Its like a thermometer. The only reason the mercury rises is that mercury expands but the glass doesn't (hardly). If everything expanded then the mercury level would stay the same.
Congratulations on the edit Sylas

 

[Axuality]

Don't think so, Space expands carrying all with it. The expansion we see now is just the dying embers of inflation and its scalar fields.
On top of that we have the peculiar motion of the heavenly bodies within it.
On the small scale, gravitational effects are far bigger than expansion of space effects so they dominate.
But on the large scale expansion effects dominate and things move away from each other.
Its a good thing that objects themselves don't get any bigger as otherwise we would not see the expansion.
If distances got bigger and our 'ruler' got bigger at the same rate then we would measure the distances as constant. Its like a thermometer. The only reason the mercury rises is that mercury expands but the glass doesn't (hardly). If everything expanded then the mercury level would stay the same.
Congratulations on the edit Sylas

I was under the impression that the expansion of galaxies is NOT measured by rulers, but by the red shift. In which cases it would be allowable for the rulers to be getting bigger at the same rate as distances.

 

[Axuality]

Expansion does not extend down to the microscopic level. In fact, it doesn't affect things smaller than clusters of galaxies. The reason for this is that the forces holding these objects together (gravity, electromagnetic and nuclear forces) are stronger than universal expansion.

I should ask, how do we know that the universe is "expanding"? Is it not the red shift? And of course we make the assumption that red shift means that things are getting more meters apart. But the red shift would still be observable if all things, including space, were expanding, would it not?

And are we certain that what you call the strength of "universal expansion" is not greater in small distances (close quarters), than it is at the galactic cluster level?

 

[Chronos]

Redshift tells us the universe is expanding. Relativity tells us it is not a classical doppler effect. Spectral line shifts do not lie. We do not see discordant redshift in spectral lines. That tells us it is a global [spacetime] effect.

 

[v2kkim]

Another way to look at the expansion effect in small scale is to start with Earth and sun orbit. Th expansion and the gravitation will happen at the same time, though gravitation is much larger. Here, discrete modeling may help, that is say the sun-earth distance increases a little instantly due to expansion, then the Earth orbit will change slightly and as a result the sun-earth distance increase will be diminished considerably after a while (actually the Earth orbit will be slightly more elliptical.). and this diminishing effect will be greater in a mutibody case like in a galaxy.
I think mathematically we can make equation and even can find a solution for a simple case, and the result will be in line with above rough thinking.
In much smaller scale like our body cells, the dominant force is electric or inter-atomic force which will adjust each atomic motion continuously, diminishing expansion effect considerably.
This is a more detail thinking of the expansion effect to small scale.

 

[Axuality]

Redshift tells us the universe is expanding. Relativity tells us it is not a classical doppler effect. Spectral line shifts do not lie. We do not see discordant redshift in spectral lines. That tells us it is a global [spacetime] effect.

d

I'm not disagreeing with you and let me ask you:

Does what you're saying agree or disagree with the proposition that while according to the red shift the universe is expanding, it is also possible that at the same time all objects and the space in between them are expanding also?

 

[v2kkim]

d

I'm not disagreeing with you and let me ask you:

Does what you're saying agree or disagree with the proposition that while according to the red shift the universe is expanding, it is also possible that at the same time all objects and the space in between them are expanding also?

The explanation by advisor and mentor helped me a lot to understand. On this Axuality's question, I 'd say the answer is yes in infinitely short moment.. i. e. the expansion applies even in subatomic scale, but this expansion does not accumulate as time goes on in short distance.
In long distance beyond galaxy cluster, the expansion accumulates as time goes on. Thanks.

 

[Flatland]

If expansion does indeed occur on microscopic scale, then shouldn't we detect a small repulsion force between bonding atoms and molecules?

 

[v2kkim]

If expansion does indeed occur on microscopic scale, then shouldn't we detect a small repulsion force between bonding atoms and molecules?

In principle you may be right , something reflecting space expansion effect in atomic scale,
on which I can just speculate based on my limited knowledge of quantum physics. The basic equation of quantum mechanics is Schrodinger wave equation, and well maybe you can try yourself, because: First I do not know that much though I used that equation a lot, secondly the expansion magnitude is really really small in atomic scale. Roughly the expansion magnitude is (expansion rate* distance), and do some math on the distance ratio of Galaxy cluster to galaxy cluster vs 1E-10 meter, so practically impossible to detect something at atomic level, and scientist are busy to deal with many other important subject.
But I respect your thinking, and you may try something theoretically at least. Let me know if you can develop some new idea of new equations.

 

[sylas]

If expansion does indeed occur on microscopic scale, then shouldn't we detect a small repulsion force between bonding atoms and molecules?

Expansion isn't a force. It's just things moving away from each other. What forces do is change the rate at which things move away from each other. On large cosmological scales, forces can work to accelerate or decelerate expansion. But expansion itself doesn't need a force, in the same way that ordinary motion of individual objects is maintained without any force.

Small things are not expanding. They've already been pulled together and are no longer moving apart. Small, in this case, is much bigger than microscopic. For example, galaxies are not expanding. All the stuff in a galaxy has been pulled together and is no longer moving apart.

Cheers -- sylas

 

[Flatland]

I never meant that expansion was a force, just that matter on the microscope scale needs to resist the expansion of space, otherwise atoms and molecules would be pulled apart.

 

[sylas]

I never meant that expansion was a force, just that matter on the microscope scale needs to resist the expansion of space, otherwise atoms and molecules would be pulled apart.

Um... if it is not a force, then what is "pulling"? Expansion doesn't "push", or "pull" or "force" things. It is a description of motions, not a tendency towards certain motions.

As with motion in local space, expansion of things proceeds by inertia. Once you pull things together, you've already altered the expansion, and (in the case of our galaxy, let alone atoms and molecules) they simply aren't expanding any more. "Expansion" isn't a force, or a "pull" or anything else like that which needs to be resisted.

Cheers -- sylas

 

[bapowell]

Perhaps it's easiest to think in terms of energy/matter distributions here. Einstein's Eqs give us an expanding universe, the Friedmann Robertson Walker Universe, when the energy density (the source in Einstein's Eq) is uniform and isotropic. This is approximately true of the universe only on the largest scales (ie the scale of galaxy clusters). On smaller scales, the universe is far from homogeneous. The spacetime metric in such regions is not of the FRW form, and is not expanding. This is essentially equivalent to many of the posts above that say that strongly bound systems (solar systems, galaxies, atoms) don't expand.

As far as inflation goes, there's nothing wrong with space expanding on all scales. The energy density during inflation was necessarily homogeneous on all scales!

 

[Ich]

The spacetime metric in such regions is not of the FRW form, and is not expanding. This is essentially equivalent to many of the posts above that say that strongly bound systems (solar systems, galaxies, atoms) don't expand.

It's equivalent to these posts, but wrong - or, at least, not the whole story.
Even if the universe were perfectly homogeneous, and could be decribed by the FRW metric down to the smallest size, two test particles at rest wrt each other won't follow the expansion.
Their bahaviour is completely independent of H, and is dictated only by the energy and pressure density between them.
In a "perfect" LCDM universe, right now, they would actually slowly start accelerating outwards. But a few billion years ago, they would fall inwards, despite H being much bigger then.

 

[George Jones]

It's equivalent to these posts, but wrong - or, at least, not the whole story.
Even if the universe were perfectly homogeneous, and could be decribed by the FRW metric down to the smallest size, two test particles at rest wrt each other won't follow the expansion.
Their bahaviour is completely independent of H, and is dictated only by the energy and pressure density between them.
In a "perfect" LCDM universe, right now, they would actually slowly start accelerating outwards. But a few billion years ago, they would fall inwards, despite H being much bigger then.

For such observers, the universe does not obey the cosmological principle, i.e., from the point of view of at least one of these observers, the universe is not isotropic.

Are you saying that this effect is more relevant for "holding galaxies together" than the stuff that bapowell wrote? If so, why?

The stars in a galaxy cannot be approximated as test particles as they clearly make a non-negligible contribution to the stress-energy tensor for the spacetime region of the galaxy. Consequently, a region of spacetime large enough to contains a few galaxies but not much larger is not homogeneous, so this region won't be decribed by a FRW metric, and there is no reason to expect that a galaxy will expand.

 

[Ich]

For such observers, the universe does not obey the cosmological principle, i.e., from the point of view of at least one of these observers, the universe is not isotropic.

Yes. In FRW coordinates, at least one of them has nonzero peculiar velocity. That's an initial condition.

Are you saying that this effect is more relevant for "holding galaxies together" than the stuff that bapowell wrote? If so, why?

There's no question that galaxies are tightly bound and can withstand any "drag" from the expansion with a safety margin of many OOM.
The questioni is: is there such a "drag effect" from expansion in the first place?

What sylas is saying, and what I'm supporting, is: expansion is an initial condition rather than an ongoing effect. If two objects have relative velocity, they will move apart. If not, they won't. You don't need the gravitation of a galaxy to justify that, that's simply inertia.

Of course, if there is matter or pressure between the objects, its gravitation will change their velocity.

Consequently, a region of spacetime large enough to contains a few galaxies but not much larger is not homogeneous, so this region won't be decribed by a FRW metric, and there is no reason to expect that a galaxy will expand.

The point is: even if the region were perfectly homogeneous, and described by the FRW metric, is there a reason to expect that the things (say, test particles) in that region have a tendency to "expand" (i.e. change distance) when they're not starting with relative velocities?
This question can be decided by looking at the local background density at any time, without referencing to H.
At any given time, it is irrelevant whether the rest of the universe (or even the local background) is expanding or contracting, as long as the background's kinetic energy may be neglected.

 

[JDStupi]

I must say that I know nothing, expecially compared to some of the well-informed responses posted as, I would say, successfull refutations of the matter expanding idea, but here is my two cents: to OP, let's assume that the deeper knowledge of astrophysics/galactric dynamics that these posters display is uknown (it is to you, and it is to me), but we do still know some properties of matter! True, matter is not definatley "point-particle like" as is commonly conceptualized, but what is it? It is the interaction of the electromagnetic (and other) forces on the small scale, and they occupy volume because of the properties of electrons (which is generalized to a class of particles) on the small scale. Using that information, we may be able to infer that matter would not "get bigger" because if the space inside it was expanding (assuming it was strong enough to break the incredibly strong electromagnetic forces) the "electron cloud" spread out through space's radius would get bigger, thus decreasing the electromagnetic forces between the atoms/molecules by Coulomb's law, and so rather than get "bigger" if anything matter would simply break apart starting with the largest molecules with the weakest bonds, something maybe like some type of weak molecular solid. This is unobserved, and seems unlikely, expecially when considering that space or, in a loose way, distance from an atom's nucleus,c an define the quanitzed energy level of a particle. What would happen there? There would be some funky things going on with energy. Adn we can also consider the yet unkown intimate relationship between space/matter.

 

[sylas]

What sylas is saying, and what I'm supporting, is: expansion is an initial condition rather than an ongoing effect. If two objects have relative velocity, they will move apart. If not, they won't. You don't need the gravitation of a galaxy to justify that, that's simply inertia.

Exactly. The issue is also considered in this paper:

• Davis, T.M. et. a. (2003) Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects, in Am. J. Phys, Vol 71, p 358, arXiv.org:astro-ph/0104349

Cheers -- sylas

 

[cepheid]

Now imagine a third person standing next to you, and you are holding hands. The person few tiles away will still recede from you, but the person standing next to won't. This is because your grip is stronger than the friction between both of your stockings and the floor. You are like two galaxies in a local cluster gravitationally bound to each other. The tiles still expand under your feet, you just don't move apart with them.

I didn't read much past this post by Janus, so I'm not commenting on the rest of the thread. I just wanted to say I thought this was a really great analogy. This is the crux of the answer to the OP.

 

[sylas]

I didn't read much past this post by Janus, so I'm not commenting on the rest of the thread. I just wanted to say I thought this was a really great analogy. This is the crux of the answer to the OP.

One of the problems with simple answers is that always go wrong at some point. Janus' analogy is good for sorting out some issues, but it still might give the misleading impression that there's actually a force involved, and that if you let go hands, you'll start moving with the floor again.

To get a better idea, you'd have to imagine everyone initially moving with the floor... and then the floor becomes completely frictionless. Everyone then just keeps drifting, and because there's no acceleration involved in moving with the floor, everyone stays above the same spot on the floor as all the tiles continue to expand.

NOW reach out and grab hold of someone else. You lock positions with them... and then you can release hands... and you will STAY locked in position with them, although the both of you are still expanding away from everyone else.

Cheers -- sylas

 

[twofish-quant]

One other way of thinking about it is to point out that the basic calculation that gives an expanding universe assumes that everything in the universe is this nice homogenous gas and there are no lumps in the universe. If you start with this assumption, then you get this nice uniform expansion. Now at large scales, the universe is close enough to this approximation, but as small scales, these assumptions no longer work.

One other analogy is the equations that people use to model the universe basically assume that the universe consists of a "gas" of galaxies. If you look at the air around you, it seems to be a continuous fluid, and for a lot of calculations you can assume that it is a continuous fluid, but if you look at very small scales (i.e. the individual atoms) the assumptions you are making break down.

The other thing is that I think the answer changes if there is a cosmological constant.

 

[Flatland]

Um... if it is not a force, then what is "pulling"? Expansion doesn't "push", or "pull" or "force" things. It is a description of motions, not a tendency towards certain motions.

You seem to be arguing semantics over here. And how can it be a description of motion? Nothing actually moves during the expansion of space.

As with motion in local space, expansion of things proceeds by inertia. Once you pull things together, you've already altered the expansion, and (in the case of our galaxy, let alone atoms and molecules) they simply aren't expanding any more. "Expansion" isn't a force, or a "pull" or anything else like that which needs to be resisted.

Cheers -- sylas

how do you alter expansion by pulling things together?

 

[sylas]

You seem to be arguing semantics over here. And how can it be a description of motion? Nothing actually moves during the expansion of space.

I'm explaining something about expansion of the universe. You proposed earlier:

I never meant that expansion was a force, just that matter on the microscope scale needs to resist the expansion of space, otherwise atoms and molecules would be pulled apart.

That's actually incorrect. I'm trying to explain why. Expansion doesn't pull on things.

As for "nothing moves during the expansion of space"; that's really misleading. You can make it true by choosing to define positions using the co-moving distance co-ordinate; but that's effectively a convention; a way of defining "moves" to make the statement true.

But in fact, expansion of space simply reduces in a Newtonian limit to a cloud of dispersing dust. Whether you define it so that space between non-moving particles is increasing, or particles are all moving apart from one another, is a matter of convention.

how do you alter expansion by pulling things together?

Almost by definition. Expansion means things are moving apart. A force that tends to pull things together, like gravity, slows expansion. In the FRW equations, you will find that the more matter you have, and hence the more gravitational attraction pulling things together, the more quickly expansion slows down. Give it enough matter and you can even reverse the expansion altogether and end up with a Big Crunch.

Defining a co-ordinate system in which all the particles remain at the same "position" while the space between them increases or reduces doesn't change any of this. It's just a different set of conventions for describing the same thing. Either way... gravitational attractions slow down expansion.

Cheers -- sylas

 

[Buckethead]

I'm surprised that no one has brought up that an expanding universe allows for objects very far off to be traveling faster than light with respect to us (as gets mentioned in the Cosmology sub forum quite often). This alone is all that needs to be considered in realizing that expanding space is not just objects moving away from each other. It implies that the space in-between space is ripping apart and filling up with new space (or some other similarly cryptic analogy that practically makes no sense). Ordinary matter cannot move apart from each other faster than light, so something much different and undigestible is going on. Am I wrong?

 

[sylas]

I'm surprised that no one has brought up that an expanding universe allows for objects very far off to be traveling faster than light with respect to us (as gets mentioned in the Cosmology sub forum quite often). This alone is all that needs to be considered in realizing that expanding space is not just objects moving away from each other. It implies that the space in-between space is ripping apart and filling up with new space (or some other similarly cryptic analogy that practically makes no sense). Ordinary matter cannot move apart from each other faster than light, so something much different and undigestible is going on. Am I wrong?

This get tricky. It all depends on what you mean by "distance"... which also indicates something much different is going on from what you may be used to.

The difficulty is that to describe the universe on large scales, you can't really use Newtonian physics, in which "distance" is not so problematic. You have to use General Relativity.

This also suggests that you can't understand these things without understanding General Relativity -- but that goes a bit too far, in my view. Better say that pretty much anyone can improve their understanding, and this process can go on for years. At some point along that track you'll need general relativity. That's about where I am, at present. But you don't have to drop everything and learn general relativity before starting down the road of understanding the universe.

The experts often argue about the best way to explain these concepts, or how they should be taught. Even if they have no disagreements on the actual technical details!

There's one very pithy little remark by John Wheeler, one of the great cosmologists and teachers. He once said:

Matter tells space how to curve.
Space tells matter how to move.
​

This is a very brief non-technical summary of general relativity. (Some pedants replace "space" with "spacetime", to be more technically correct but possibly less accessible.)

So, I would say, the expanding universe does involve objects moving apart from each other. On sufficiently small scales, this expansion is no different from a cloud of dispersing dust with Newtonian physics.

The expansion of "space" comes into it on larger scales because you can't actually separate expanding stuff from expanding space. And the notion of "distance" becomes problematic because there are a number of different definitions you could use, all of which give different numbers. They are all perfectly precise, and they all work out to be the same thing when you look on the smaller scales we are used to.

In all likelihood someone else will pop in shortly, who describes it all slightly differently, all in an attempt to help.

But here's the thing. The expansion we are talking about doesn't try to "pull" things apart on smaller scales. In that sense at least, expansion is like movement of things apart from each other. In so far as there is any "pull", or "push", it turns out to be gravity (which pulls) or dark energy (or pressure, or push) and on large scales that is slowing or speeding up the expansion respectively. It is not the expansion itself that is pushing or pulling things, and our galaxy is not expanding. It HAS been pulled together, and so also have other nearby galaxies in our local group, which are not expanding away from us either.

Our galaxy and those to which we are gravitationally bound will continue to sail on through the universe without ever expanding apart from each other, until long after all the stars have faded; while the rest of the universe continues to recede indefinitely. That is how the current best fit models of the universe work.

Cheers -- sylas

 

[Flatland]

I'm explaining something about expansion of the universe. You proposed earlier:That's actually incorrect. I'm trying to explain why. Expansion doesn't pull on things.

As for "nothing moves during the expansion of space"; that's really misleading. You can make it true by choosing to define positions using the co-moving distance co-ordinate; but that's effectively a convention; a way of defining "moves" to make the statement true.

But in fact, expansion of space simply reduces in a Newtonian limit to a cloud of dispersing dust. Whether you define it so that space between non-moving particles is increasing, or particles are all moving apart from one another, is a matter of convention.
Almost by definition. Expansion means things are moving apart. A force that tends to pull things together, like gravity, slows expansion. In the FRW equations, you will find that the more matter you have, and hence the more gravitational attraction pulling things together, the more quickly expansion slows down. Give it enough matter and you can even reverse the expansion altogether and end up with a Big Crunch.

Defining a co-ordinate system in which all the particles remain at the same "position" while the space between them increases or reduces doesn't change any of this. It's just a different set of conventions for describing the same thing. Either way... gravitational attractions slow down expansion.

Cheers -- sylas

Okay, consider this thought experiment. Let's say we have an extremely long string that's billions of light years across. I attach one end of the string to Earth and the other end to another planet in a distant galaxy that's moving away from us. Ignoring the motions of planets and stars (and their gravity), would this string eventually break due to expansion? Now let's suppose this string is made of some exotic material that's unbreakable. Would the two planets eventually be pulled out of their respective galaxies?

 

[sylas]

Okay, consider this thought experiment. Let's say we have an extremely long string that's billions of light years across. I attach one end of the string to Earth and the other end to another planet in a distant galaxy that's moving away from us. Ignoring the motions of planets and stars (and their gravity), would this string eventually break due to expansion? Now let's suppose this string is made of some exotic material that's unbreakable. Would the two planets eventually be pulled out of their respective galaxies?

Great question! It turns out that the problem will be to set up the string in the first place.

Thinking about it sounds fine. But let's say you try to put it into practice. Is the string already there? Or are we going to construct it? I appreciate this is a thought experiment, so we can imagine all kinds of unrealistic ways to do this.

If we construct it, what will we make it out of? If we have a whole bunch of engineers all spread through space to link it together... where will we find engineers who are at rest with respect to each other? And what does that mean anyway? We have the same problem there as with defining distance.

Method one. Moving the pieces of the string into position.

Here's one method. We want to construct this enormous string, and so we imagine a whole bunch of engineers who are going to do the job. They are going to have to start moving relative to their local circumstances in order to get to a position where they aren't expanding away from each other any more... and in doing so, they'll be moving closer together. So the string won't be as long as originally thought. The further apart all the engineers are to start with, the longer it will take for them to accelerate out of the initial Hubble flow and into a position where they aren't expanding away from each other anymore. There are all kinds of problems with defining the terms here, being at rest, measuring distances, measuring velocities. It's a mess. But you won't be able to make the string of the length you require.

Method 2. Extending the string as we build it.

How about this. We build the string in stages. Each stage adds another meter to the string, using a light clock to keep a constant meter distance, and all the light clocks are required to keep end-to-end with each other, like a piece of string. We add new stages to the string at the speed of light. That is, about 300,000,000 new meters are added on every second; or one new meter added every 1/300000000 of a second. Of course, this time interval is measured by the immediately preceding clock locally fixed onto the string.

We do this with fantastic precision. The Hubble expansion rate of 71 km/s/Mparsec implies that with each additional meter, the Hubble flow means there's about 2.3*10^-18 m/s of expansion velocity. We're going to have to ensure each additional extension to the string is moving relative to the local Hubble flow so as to remain at rest with respect to the adjacent ruler.

We construct this string for a billion years... but according to which clocks? The ones on the string, of course; but by the time you get out any distance, those clocks are now moving with respect to any local observers, moving with the Hubble flow. And hence the clocks are Lorentz contracted from the perspective of those local observers. And also the clocks on the string are running slow with respect to the local observers.

End result: it simply isn't possible, even speculatively, to set up the string so that it extends out of the Hubble volume and up to those galaxies which we can see that are receding faster than light.

"Tethered" galaxies

There is a way to think about what you are proposing in more precise terms. Not everyone loves this account, but you can make it precise with a bit of care. The account is

• Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects, by Davis, Lineweaver and Webb, in Am.J.Phys. 71 (2003) 358-364, also arXiv:astro-ph/0104349v3.

This paper is proposing something like what you suggest. It imagines two galaxies that are initially "tethered" with a long unbreakable string (the tether) and then released. Where do they go after that? Now there are all kinds of issues sorting out that informal description, and the paper does give a precise account of what this means. I'll skip over that, and just give the motivation, from the paper:

The general expansion of the universe is known as the Hubble flow. A persistent confusion is that galaxies set up at rest with respect to us and then released will start to recede as they pick up the Hubble flow. This confusion mirrors the assumption that, without a force to hold them together, galaxies (and our bodies) would be stretched as the universe expands. The aim of this paper is to clarify the nature of the expansion of the universe, including recession velocities and cosmological redshifts, by looking at the effect of the expansion on objects that are not receding with the Hubble flow. This paper is an extension of previous discussions on the expansion of space.[/color]​

The upshot is as follows.

• In the "empty universe" model, in which expansion neither accelerates nor decelerates, the "tethered" galaxy remains at the same "proper" distance. There's nothing about expansion of space to make it start to move closer, or further away.
• In a universe with accelerating expansion, the "tethered" galaxy, once "released" will start to move gradually further away. Effectively, it is being "pushed" by the same "forces" that are accelerating the expansion.
• In a universe with decelerating expansion (which is what we all thought up until the 1990s) the effects of gravity and matter are slowing expansion down. In this case, when you "release" a "tethered" galaxy, it does something unexpected. Rather than starting to move further away, as you would expect if expansion was to "pull" on things, it actually starts to get closer and closer, gradually moving more and more rapidly. Ignoring collisions, it eventually comes right up and passes through to the other side of the sky, where it moves off and (in the limit) rejoins the "Hubble flow" in the opposite direction.

Clear as mud, I presume? It's a fun paper. It also shows, pretty definitely, that the expansion of space is NOT something that tries to pull things apart. Once two objects are at rest with respect to each other, there's nothing about expansion to change that. The only thing that can start you moving apart again, or closer together, are forces of some kind. The expansion itself is not something that can pull you apart and start you moving again.

Cheers -- sylas

 

[Ich]

End result: it simply isn't possible, even speculatively, to set up the string so that it extends out of the Hubble volume and up to those galaxies which we can see that are receding faster than light.

Geez, now Davis and Lineweaver got you.
It is impossible to stay at a constant cosmological proper distance outside the Hubble radius. But that has nothing to do with tethered galaxies.
It is perfectly possible to tether any two objects as long as you don't cross an event horizon. The Hubble radius is not a horizon, it's an artifact of the definition of cpd.
Cosmological coordinates are explicitly time dependent (except in de Sitter space). Thus, stationary objects will always be at a changing cpd, and objects at a fixed cpd are necessarily in relative motion. Davis and Lineweaver completely fail to recognize that difference in their paper.

As an extreme case, objects "stationary" at cpd Hubble distance are in fact approaching us at the speed of light. While "really" stationary objects there have significant "recession velocity" in cpd.
Here, my definition of an ideal tether or "stationary" is: a chain of observers, each with zero two-way redshift wrt its neighbour. You could use such a chain to construct coordinates that are as close to our SR intuition as possible, at least in almost static (underdense or DE dominated) spacetimes.

That said, if you somehow bring such a tether in place (be it longer than the Hubble distance or not), and catch the planet, there will be tension in the tether which accelerates the planet away from the "Hubble flow". Once the system has settled down (which would take necessarily several billion years), the tension in the tether will depend on the matter and pressure density in the way sylas describes.


Flatlander, this is what sylas wrote:

Once you pull things together, you've already altered the expansion, and (in the case of our galaxy, let alone atoms and molecules) they simply aren't expanding any more. "Expansion" isn't a force, or a "pull" or anything else like that which needs to be resisted.

Your proposed thought experiment will behave exactly as predicted: you need force to "brake" the planet. Once this is done, you need a much smaller force - if any - to keep it in place.

 

[sylas]

Geez, now Davis and Lineweaver got you.
It is impossible to stay at a constant cosmological proper distance outside the Hubble radius. But that has nothing to do with tethered galaxies.

Shrug. Depends what you mean by tethered. I've tried to give the proper cautions in my previous post about definitions.

If you use the definitions in the paper, the one follows from the other. One can, of course, dispute that this is a sensible way speak of being "tethered" -- and this seems to be your position. But for whatever it is worth, the definition applied in D&L is that the rate of change of proper distance with cosmic time is zero.

It is perfectly possible to tether any two objects as long as you don't cross an event horizon. The Hubble radius is not a horizon, it's an artifact of the definition of cpd.

It depends what you mean by "tethered". It isn't a technical term; but you can make it precise if you give a formal definition.

By the definition in D&L, you CAN'T "tether" objects beyond the Hubble radius. That's not saying it is a horizon! But it is, of course, an artifact of the particular definition of "tethered" used, which follows also from the cpd definition.

Now of course there are different notions of distance one can apply, and (IMO) it is a mistake to think anyone of them is the "real" notion of distance. But the various definitions have sensible motivations, cdp not the least.

Cosmological coordinates are explicitly time dependent (except in de Sitter space). Thus, stationary objects will always be at a changing cpd, and objects at a fixed cpd are necessarily in relative motion. Davis and Lineweaver completely fail to recognize that difference in their paper.

You have helped me considerably in the past explaining an error in another paper of theirs; with respect to incorrect definitions of the SR model. (See [post=2144968]msg #71[/post] of thread 'Why "expanding space"?', where I acknowledge.)

In this case, I don't understand what you are saying. How do you define "relative motion"? My understanding is that there ISN'T a single definition, just as there isn't a single definition of distance in cosmology. So I can't see that your criticism makes sense, as yet. Your definitions seem fine; but that's not the same thing, I think.

I'm open to correction, as I have been corrected before.

As an extreme case, objects "stationary" at cpd Hubble distance are in fact approaching us at the speed of light. While "really" stationary objects there have significant "recession velocity" in cpd.
Here, my definition of an ideal tether or "stationary" is: a chain of observers, each with zero two-way redshift wrt its neighbour. You could use such a chain to construct coordinates that are as close to our SR intuition as possible, at least in almost static (underdense or DE dominated) spacetimes.

Well, sure, you can use different definitions to those used in the Davis and Lineweaver paper. IMO this is not a good basis for criticism of the paper unless you have a stronger reason for saying there's something actually wrong with the consequences they give from those definitions.

You DID show that for me last time, wrt to the "SR" model of the other paper. But so far, I don't see it here.

Your proposed definition, if I understand it correctly, is that a distant galaxy is "tethered" if it has a peculiar velocity such that the redshift is 0. Is that right? I'll think about that, and maybe post again on the matter.

That said, if you somehow bring such a tether in place (be it longer than the Hubble distance or not), and catch the planet, there will be tension in the tether which accelerates the planet away from the "Hubble flow". Once the system has settled down (which would take necessarily several billion years), the tension in the tether will depend on the matter and pressure density in the way sylas describes.

Yes, this is one point I glossed over. Flatlander's thought experiment involved yanking something out of the Hubble flow with a tether.

Cheers -- sylas

 

[Buckethead]

Reading this thread is really helping. One thing that helps is understanding the difference between comoving distance and proper distance and that when seen from the viewpoint of comoving distance, space is not expanding at all as the measure of distance expands with the comoving coordinates. It still defies understanding, but what happens in this scenario:

Let's say during the early years some stars began to form a long string due to gravitational attraction that eventually ended up being billions of light years long. There is no redshift between any particular star and it's neighbor up or down the string. At the furthest reaches of the string, the end star is moving through it's comoving coordinate at nearly the speed of light (since it's been "tugged" out of it's natural stationary (in comoving terms) position) but is stationary in proper distance terms with respect to us.

Now what happens if this chain goes on beyond the Hubble radius? This would mean the last star would be moving faster than light with respect to stars in its neighborhood that are stationary with respect to comoving coordinates.

 

[sylas]

Let's say during the early years some stars began to form a long string due to gravitational attraction that eventually ended up being billions of light years long. There is no redshift between any particular star and it's neighbor up or down the string. At the furthest reaches of the string, the end star is moving through it's comoving coordinate at nearly the speed of light (since it's been "tugged" out of it's natural stationary (in comoving terms) position) but is stationary in proper distance terms with respect to us.

Now what happens if this chain goes on beyond the Hubble radius? This would mean the last star would be moving faster than light with respect to stars in its neighborhood that are stationary with respect to comoving coordinates.

I believe the solution is as Ich suggested. This is not actually the definition of "proper distance". This is the key to why Ich doesn't like the word "tethered" being used in the way Davis and Lineweaver use it. I think. I've been trying to sort through Ich's contribution and this is how far I have come so far.

I don't know if there is a term for a distance defined along the lines you are proposing, but I suspect it might be a "Milne distance". But that is just a term that popped into my head and which others can use to see if I am on the right track or not! Defined in the way you suggest, the "string" is able to stretch out indefinitely within the observable universe, I believe; and any galaxy we can actually see could conceptually be part of the string... except, of course, that the string at that time and place would be pulling past the observed galaxy at a sufficient velocity to have no redshift observed.

If that remote galaxy had a jet of matter being ejected at relativisitic velocities in our direction (there are such jets observed!) than the material in that jet would, I think, be material "at rest" with respect to the hypothetical string! if the redshift of the jet of matter is zero. On the other hand, the "proper distance" co-ordinate of the string at that point would be increasing.

Ich can tell me if I am understanding this right; I'm jumping in now because I think I probably have got it right, but can use a bit of supervision from a more experienced "science advisor"! And sticking my neck out is a good way to get that.

Cheers -- sylas

 

[Ich]

EDIT: sylas, I didn't see your post when I wrote this.

It depends what you mean by "tethered".

Well, yes, if you mean by "tethered" something with feathers and a beak, it might well start to quack or to even meet Davis' and Lineweaver's definition.

It isn't a technical term; but you can make it precise if you give a formal definition.

You can't give it just any formal definition that you like. Instead of defining a GR-compatible tether (a formidable task, imho), I'll show you why I insist that D&L's definition is not an option.
Whenever things get complicated, first try a simple toy model. My preferred one is the empty universe: It is a valid FRW solution, so FRW concepts do apply. And it's good old Minkowski spacetime, so we understand the physics.

There is definitely no problem with the definition of a tether, if you allow some handwaving:

A long (ideally one-dimensional) object, all parts of which are at rest wrt each other (there's even a global definition of "at rest"), which has a constant proper length. An approximate physical realization is a rope with a little constant tension.
That's a tether.

or

A family of observers in a line, with different relative velocities, each originatin from the same event, where their clocks have been synchronized. The distanc to the next neighbour is small at any time. One of the observers is called the starting point. Now, at a specific proper time, each observer measures the distance to her neighbour. They add the distances, beginning with the starting point, and mark (in retrospective, as the adding takes some time) the event where the sum reached a definite value, the "cosmological proper length" of the construct. This is the end point. An approximate physical realization, good planning provided, coul de a rocket that tries to stay on a worldline through all these events. Which could prove difficult, as their separation is not guaranteed to be timelike.
That's not a tether.

If we agree, I've made my point. If not, I'd like to hear a definition and realization of D&L's "tether" that has at least a vague resemblance to

something (as a rope or chain) by which an animal is fastened so that it can range only within a set radius

Another viewpoint:
We know what "at rest wrt each other" means in a flat spacetime. There is no disambiguity, and no amount of coordinate definitions will change it.
A stationary source will have no redshift, as the only source could be doppler shift. That coincides with constant proper Minkowski distance, and with the end of our tether. It doesn't coincide with the end of D&L's tether.

We can extend a similar global principle to every static spacetime, like the empty expanding FRW model, exponentially expanding de Sitter space, or any steady state cosmology.
There, stationary objects are defined by the lack of two-way redshift (a reflected signal has the same frequency). De Sitter space is indeed the only case where the two above definitions agree. The difference is in the explicit time dependence of cosmological coordinates, which vanishes if H=const.

For an arbitrary FRW spacetime, I do not claim to have a "corrrect" definition of a tether. However, here's an unambiguous definition of special "static" coordinates:
Starting with a comoving worldline, we choose all observers to be "stationary" if they are at rest with their immediate neighbour (which is well-defined), as long as said comoving worldline is also part of the family.
I think that's quite close to a "tether" in a dynamic spacetime, but it's not really satisfactory.

Anyway, we agree that D&L are definitely not considering tethers, but events of constant cosmological proper distance?

 

[Buckethead]

If that remote galaxy had a jet of matter being ejected at relativisitic velocities in our direction (there are such jets observed!) than the material in that jet would, I think, be material "at rest" with respect to the hypothetical string! if the redshift of the jet of matter is zero. On the other hand, the "proper distance" co-ordinate of the string at that point would be increasing.
Cheers -- sylas

Increasing from that point, but from our vantage point, would not be changing. right?

It would be good to clear up the definition of proper distance since I thought it meant as you did, simply the coordinate system that shows no redshift.

 

[Buckethead]

Could someone clear up whether understanding the actual nature of expansion is related more to SR, in that distances and time change with the relativistic speeds of the moving distant galaxies, or does it have to do more with GR in that the spacetime over large distances is stretching, or both, or neither?

 

[Chronos]

The Guth model of expansion is generally considered the best fit to modern cosomological models. It solves more problems than it raises. It is, however, an effective theory - one that works, but, lacks a definitive mathematical explanation.

 

[sylas]

EDIT: sylas, I didn't see your post when I wrote this.

I had guessed that already... No problem.

[...]

For an arbitrary FRW spacetime, I do not claim to have a "corrrect" definition of a tether. However, here's an unambiguous definition of special "static" coordinates:
Starting with a comoving worldline, we choose all observers to be "stationary" if they are at rest with their immediate neighbour (which is well-defined), as long as said comoving worldline is also part of the family.
I think that's quite close to a "tether" in a dynamic spacetime, but it's not really satisfactory.

Anyway, we agree that D&L are definitely not considering tethers, but events of constant cosmological proper distance?

Yes; that is, they use the word "tethered" to mean "at a constant proper distance". The term "tether" is not a precise one... as you note! What Davis and Lineweaver are talking about is established by their mathematical definitions. I can see that the word "tether" might be a poor choice of word.

I attempted to warn readers about these kinds of interesting discussions even before mentioning this paper:

The experts often argue about the best way to explain these concepts, or how they should be taught. Even if they have no disagreements on the actual technical details!​

I don't think there's an error in their analysis (this time). But a poor choice of terms for their definition -- quite likely.

What they are really motivated to explain, I think, is precisely the question of this thread. The extract I gave from their paper, in [post=2617081]msg #40[/post], was intended to show this. Here's the passage again:

The general expansion of the universe is known as the Hubble flow. A persistent confusion is that galaxies set up at rest with respect to us and then released will start to recede as they pick up the Hubble flow. This confusion mirrors the assumption that, without a force to hold them together, galaxies (and our bodies) would be stretched as the universe expands. The aim of this paper is to clarify the nature of the expansion of the universe, including recession velocities and cosmological redshifts, by looking at the effect of the expansion on objects that are not receding with the Hubble flow.[/color]​

Essentially, they are using a certain kind of test particle to show that expansion is not what actually pulls or pushes things. Their analysis applies for all the basic FRW models in the ΛCDM models, determined by (Ω_m, Ω_Λ) at epoch. They show that the rate of change of proper distance remains at zero in the empty universe, that it increases if the expansion is accelerating, and it decreases if expansion is decelerating.

The basic method of their analysis depends on this interesting axiom. The momentum of any freely moving test particle in the frame of any local observers that are moving with the Hubble flow will be inversely proportional to scale factor at that time. In other words, the momentum of the particle is eroding, in the same way as a photon is redshifted -- although of course it can't be called a change in the particle in some identified frame. It is rather the smaller and smaller momentum in the different frames of all the co-moving observers it passes along the way; just as a photon is measured to be more and more redshifted in all the different frames of the co-moving observers it passes along the way.

They don't actually derive this axiom however. I have been referred in the past to MTW for the derivations, but at the time I didn't follow it. I'll have to try again some time.

I'd also like to repeat their analysis for myself, in the same class of ΛCDM models, but using your proposed definition. However, I have another big project underway at present, so I'll have to put it off for a bit.

[strike]: :[/strike]​

Chronos mentions Guth. The Guth model is a model for inflation, in the VERY early universe. It's useful for explaining flatness and the horizon problem and so on, but not really relevant for the question of this thread. During the inflationary epoch there WAS expansion at the microlevel, and the expansion was basically de Sitter expansion. But now, the inflation proposed by Guth is long finished, leaving its parting signature in the flatness of the universe and ripples of the background radiation.

In the present epoch there is a tiny dark energy factor, which does act to accelerate expansion and hence to push things apart. But it is far far too weak to detect on small scales. Furthermore -- and this is one of the major points in Davis and Lineweaver, all concerns of terminology aside -- it is not "expansion" which drags things along for the ride, so much as dark energy which pushes things apart and hence accelerates the rate of expansion; along with mutual gravitational attraction which draws things together and hence decelerates the rate of expansion.

Cheers -- sylas

 

[Ich]

I can see that the word "tether" might be a poor choice of word.

Davis and Lineweaver:

...by "Duck" we mean here an ostrich, struthio camelus. ... thus we have seen that ducks can't fly ... To our knowledge this is the first explicit derivation of this counter-intuitive behavior.

I mean, I never said their analysis is wrong (I also didn't say that when we discussed the paper for the first time).

But we had here someone asking: "can ducks fly?" And you answered, with reference to D&L, that they can't. Now this is wrong, but who's to blame, you or D&L?
In my opinion, it's definitely D&L who are to blame, not you. I claim that they did not fully understand what they were doing there, otherwise they would have mentioned that, in real life, ostriches are not ducks.

(If it isn't clear what I'm talking about: from D&L's analysis it follows that you can't tether a galaxy behind the Hubble distance. This is "true" because they stated explicitly that what they mean by "tether" is something completely different, as I showed in my last post.)

Yes; that is, they use the word "tethered" to mean "at a constant proper distance". The term "tether" is not a precise one... as you note!

That's not my point. "Tether" may not be precise, but it definitely can't be stretched to include "at constant cpd"! They call it "tether", but no amount of exegesis will make it one.

They don't actually derive this axiom however. I have been referred in the past to MTW for the derivations, but at the time I didn't follow it. I'll have to try again some time.

Did I already tell you that you can "derive" it from the https://www.physicsforums.com/showthread.php?t=294690"? (I guess I did, as I think it's a fun exercise.)

I'd also like to repeat their analysis for myself, in the same class of ΛCDM models, but using your proposed definition.

If you're talking about their derivation that ä/a ist the determining factor for the behaviour of initially stationary test paticles: there's no difference between cooordinates at the relevant order. What's interesting there is that it's not ä/a, but the local density that determines the behaviour.