Is the Big Bang Expanding into a Preexisting Void?

[TrickyDicky]

No. You've asserted that the Milne cosmology only works if the universe is completely empty. And that was based on claiming that Milne's metric was not the Minkowski metric, which I already told you was not true.

You seem to be muddled on this.

Let's see Minkowskian and Milne spacetimes(they are the same spacetime with a change in coordinates that shouldn't affect the physics) are both empty, meaning there is no gravity sources therefore no gravitational field. So the Milne cosmology is defined that way, as empty, and that has nothing to do with anyone's claims.

You are yourself admitting that Minkowski and Milne metrics are equivalent so you are implicitly admitting that the Milne universe is empty, so I don't know exactly where you disagree.

 

[JDoolin]

You seem to be muddled on this.

Let's see Minkowskian and Milne spacetimes(they are the same spacetime with a change in coordinates that shouldn't affect the physics) are both empty, meaning there is no gravity sources therefore no gravitational field. So the Milne cosmology is defined that way, as empty, and that has nothing to do with anyone's claims.

You are yourself admitting that Minkowski and Milne metrics are equivalent so you are implicitly admitting that the Milne universe is empty, so I don't know exactly where you disagree.

While Milne was attempting to show how ridiculous Eddington's ideas were, he gave an equation which would map comoving world-lines to world-lines that were moving away from a single event at a constant velocity. The equation was nonsense, and Milne's point was that it was nonsense.

However, because his point was also that Eddington's ideas were ridiculous, the Eddington followers latched onto the very equation that Milne was describing as nonsense, and began calling it The Milne Model.

I admit that the Minkowski metric and the real Milne metric are equivalent.

ds^2=dt^2-dx^2-dy^2-dz^2​

However when you map in the nonsense equation, and use the metric given on Wikipedia for the Milne Model:

ds^2 = dt^2-t^2(dr^2+\sinh^2{r} d\Omega^2)​

where

d\Omega^2 = d\theta^2+\sin^2\theta d\phi^2​

... this metric is no longer equivalent to the Minkowski Metric.

 

[Chalnoth]

And we're back to my previous question: why are you so absurdly opposed to a simple change of coordinates?

 

[TrickyDicky]

However when you map in the nonsense equation, and use the metric given on Wikipedia for the Milne Model:

ds^2 = dt^2-t^2(dr^2+\sinh^2{r} d\Omega^2)​

where

d\Omega^2 = d\theta^2+\sin^2\theta d\phi^2​

... this metric is no longer equivalent to the Minkowski Metric.

Well they are not exactly the same if that is what you mean, if you have an aesthetic repulsion towards FRW metrics applied to Milne's model (why? maybe because you see the introduction of a scale factor as artificial or arbitrary in a spacetime that is essentially static? I could understand that, but Milne actually also introduced artificially an explosion in his special relativistic universe that gave particle tests their speeds up to c) that's OK, but you must realize that physically from the point of view of these particles(from their proper time and length)the metric with the scale factor and the Minkowski metric are indeed equivalent, the Minkowski metric is called the "private" frame in Milne's universe and the FRW metric would be the "public" view as seen from an outside point of view.

 

[TrickyDicky]

Quite the contrary. That is no coincidence at all. Every particle in the system is, in its original trajectory, in the center of the explosion.

What is your velocity? In your own reference frame, your velocity is zero. The particles around you have velocities anywhere from zero to the speed of light. Therefore no matter which particle you pick, it is going to be approximately in the center.

This is basic relativity. Think about it.

This is indeed an interesting property of Milne's model shared by standard cosmology, and shows that isotropy does not necesarily always imply homogeneity, so the cosmological principle is indeed as has been said here before, a philosophical preference that ultimately will have to be confronted empirically since isotropy without homogeneity is also a possibility.

I think the key here is that our cosmology based in GR is basically telling us that there is no center( in this forum"where is the center of the universe?" is a frequent question), so no observer can be in the center, so isotropy without homogeneity doesn't imply any privileged point of view and therefore perhaps the cosmological principle is not philosophically valid in a universe ruled by the theory of relativity.

 

[Chalnoth]

This is indeed an interesting property of Milne's model shared by standard cosmology, and shows that isotropy does not necesarily always imply homogeneity, so the cosmological principle is indeed as has been said here before, a philosophical preference that ultimately will have to be confronted empirically since isotropy without homogeneity is also a possibility.

At the very least, void models to explain the accelerated expansion without any dark energy have been ruled out already:
http://arxiv.org/abs/1007.3725

I think the key here is that our cosmology based in GR is basically telling us that there is no center( in this forum"where is the center of the universe?" is a frequent question), so no observer can be in the center, so isotropy without homogeneity doesn't imply any privileged point of view and therefore perhaps the cosmological principle is not philosophically valid in a universe ruled by the theory of relativity.

Well, pretty sure that isotropy without homogeneity does imply a privileged point of view. It's just that so far there's no reason to believe our universe isn't homogeneous, as the homogeneous models work, but the inhomogeneous ones so far do not.

 

[TrickyDicky]

Well, pretty sure that isotropy without homogeneity does imply a privileged point of view. It's just that so far there's no reason to believe our universe isn't homogeneous, as the homogeneous models work, but the inhomogeneous ones so far do not.

The context of the quoted paragraph seems to indicate you meant to say does not imply.
In that case, I agree, perhaps the problem lies not in the cosmological principle in itself, which is quite reasonable and seems to agree with observation so far, but in the interpretation some cosmology books make of the principle.

 

[Chalnoth]

The context of the quoted paragraph seems to indicate you meant to say does not imply.
In that case, I agree, perhaps the problem lies not in the cosmological principle in itself, which is quite reasonable and seems to agree with observation so far, but in the interpretation some cosmology books make of the principle.

Hmm, perhaps there was some miscommunication here, as I am saying that isotropy without homogeneity does imply a privileged location.

Now, bear in mind that the statement of homogeneity is not an absolute statement. Rather it's just a statement that there is a potential choice of coordinates for which the universe appears homogeneous. If it isn't possible to select such a coordinate system, but the universe still looks isotropic to us, then that says we live in a special location.

One way to look at this is that if you can find some small number observers for whom the universe is isotropic, then the universe is also necessarily homogeneous for some choices of observers (IIRC the minimum is three non-colinear observers).

 

[TrickyDicky]

One way to look at this is that if you can find some small number observers for whom the universe is isotropic, then the universe is also necessarily homogeneous for some choices of observers (IIRC the minimum is three non-colinear observers).

How far apart would they have to be?

 

[Chalnoth]

How far apart would they have to be?

In principle any distance would do, if we're talking about a hypothetical situation where we have perfect isotropy. Clearly this isn't the case, so you'd want them to be about as far apart as is required to smooth out the small-scale fluctuations, so I'd place them at around 80Mpc or so in our universe, at a minimum.

Obviously we can't do this explicitly, but this isn't the point I'm trying to make. The point I'm trying to make is that isotropy plus no homogeneity equals a special location. The reason being that if you have isotropy at many points, you also necessarily have homogeneity. So the only way you can have isotropy and no homogeneity is if there are only a tiny fraction of the available points that have isotropy, which means that the isotropic location is a special location.

 

[TrickyDicky]

Obviously we can't do this explicitly, but this isn't the point I'm trying to make. The point I'm trying to make is that isotropy plus no homogeneity equals a special location. The reason being that if you have isotropy at many points, you also necessarily have homogeneity. So the only way you can have isotropy and no homogeneity is if there are only a tiny fraction of the available points that have isotropy, which means that the isotropic location is a special location.

Ok, unless (this is of course a thought experiment,not meant to describe our actual universe) the whole universe was bigger than the observable universe, and the 3 observers fields of view don't overlap, in that case each of them could comply with isotropy, not be in any special location wrt the total universe and this universe could be inhomogenous (but the observers would never know).

The observable part of the universe of each of these observers may or may not be itself homogenous, in case it was confirmed not to be homogenous, they could always hope that a sufficiently bigger sample of the total universe confirmed homogeneity in case it could be observed but they wouldn't be able to prove it ever since it would be outside their limit of observability.

 

[Chalnoth]

Ok, unless (this is of course a thought experiment,not meant to describe our actual universe) the whole universe was bigger than the observable universe, and the 3 observers fields of view don't overlap, in that case each of them could comply with isotropy, not be in any special location wrt the total universe and this universe could be inhomogenous (but the observers would never know).

Correct, but the point remains that it's a special position within the observable universe. That is enough, I think.

The observable part of the universe of each of these observers may or may not be itself homogenous, in case it was confirmed not to be homogenous, they could always hope that a sufficiently bigger sample of the total universe confirmed homogeneity in case it could be observed but they wouldn't be able to prove it ever since it would be outside their limit of observability.

Well, I don't think most cosmologists think that homogeneity is likely to be more correct on extremely large scales. That is, in order for a region to become nearly homogeneous, it really needs to have some time to reach some sort of thermal equilibrium. But once your distances get large enough, there won't have been any chance for such widely-separated regions to reach thermal equilibrium, and so you expect wildly different sorts of behavior.

Of course, given current observations, we expect this distance to be much larger than the size of our observable universe, but I think we expect things to get less homogeneous eventually as you go beyond our observable region.

 

[TrickyDicky]

Correct, but the point remains that it's a special position within the observable universe. That is enough, I think.

Right, it is a special location inside the observable universe since in a truly inhomogenous universe, observers on the edge of the observable universe of the original observer would probably loose the isotropy(unless isotropy lies in the eye of the observer, like beauty :) and is intrinsic to relativistic observers no matter what).
I am not sure if that is really enoug, though, as the cosmological principle is applied to the whole universe, not only to the observable part.

Well, I don't think most cosmologists think that homogeneity is likely to be more correct on extremely large scales. That is, in order for a region to become nearly homogeneous, it really needs to have some time to reach some sort of thermal equilibrium. But once your distances get large enough, there won't have been any chance for such widely-separated regions to reach thermal equilibrium, and so you expect wildly different sorts of behavior.

Of course, given current observations, we expect this distance to be much larger than the size of our observable universe, but I think we expect things to get less homogeneous eventually as you go beyond our observable region.

I tend to agree with you, but if I were to speculate about what cosmologists opine on this subject I'd say they pretty much don't think about it and when they do they favor homogeneity from a certain scale all the way to the extreme.
But the more I think of it the more convinced I am that homogeneity cannot be empirically confirmed, only suspected.

 

[Chalnoth]

But the more I think of it the more convinced I am that homogeneity cannot be empirically confirmed, only suspected.

Well, first of all, I tend to think that homogeneity should be the default assumption, because it is the simplest one in accordance with observation, and that unless pursuing an inhomogeneous universe can explain some observations, it shouldn't be considered reasonable.

One approach that has appeared recently is the attempt to explain dark energy as the result of us living in a large void. But as I linked a few posts back, this has been shown not to work when you look carefully at the details. So it seems we're back to the assumption that fits the data: homogeneity.

 

[JDoolin]

Chalnoth,

I will try to give you a demonstration of the Minkowski-Milne Model under Lorentz Transformation sometime soon.

I'm not real good with the definitions; but I think what you mean by isotropy is that it looks the same in all the directions (change theta or phi, and it looks about the same), but what you mean by homogeneity is that it looks the same at all distances, (change r, and it looks the same)

The thing is, if you ignore the relativity of simultaneity, then Chalnoth is right. Isotropy without homogeneity would imply a privileged point of view. However, if everything is flying apart from the same event, then you have to do the full analysis with the lines (or planes) of simultaneity. You'll find that every plane of simultaneity for every particle intersects the worldlines of the other particles in such a way that you DO have isotropy from the Point of View of every particle.

But you don't have homogeneity in any particle's point-of-view, because each observer sees the density tend towards infinity at the outer edge of the sphere.

Jonathan

 

[Chalnoth]

The thing is, if you ignore the relativity of simultaneity, then Chalnoth is right. Isotropy without homogeneity would imply a privileged point of view. However, if everything is flying apart from the same event, then you have to do the full analysis with the lines (or planes) of simultaneity. You'll find that every plane of simultaneity for every particle intersects the worldlines of the other particles in such a way that you DO have isotropy from the Point of View of every particle.

But you don't have homogeneity in any particle's point-of-view, because each observer sees the density tend towards infinity at the outer edge of the sphere.

This isn't what homogeneity means. Homogeneity means that if I move to a different location, I see the same thing as if I stay put.

 

[JDoolin]

This isn't what homogeneity means. Homogeneity means that if I move to a different location, I see the same thing as if I stay put.

There is some ambiguity in the statement "I move to a different location," but if we define our term "different location" to mean "landed on another inertial particle" then by your definition the Minkowski-Milne model does turn out to be both homogeneous and isotropic.

A physical change in r would represent an instantaneous change in position without changing time, or velocity. What I meant by a change in r was simply to ask what the density of particles was at a distance of r.

But you are saying you want to actually move the observer to a the new position, r. If you mean to do this literally, then you will have to increase your velocity toward the "position" where you want to go, then wait until you arrive at the "position" and then change your velocity again to stay at that "position."

This process is fairly straightforward if you have a set of comoving particles. You can take away the finger-quotes around the word "position." Since the worldlines are all parallel, the "position" as defined in the frame of the first particle, and the "position" as defined in the frame of the second particle are the same.

You will, of course, invoke the "Twin Paradox" so the traveler finds on both journeys that the particles have aged more.

However, in the Minkowski-Milne* model, an ambiguity arises; one which can be quickly cleared up by considering the intersection of world-lines, and you will need to use one of the following two definitions of position:
(1) The world-line associated with r="particle distance" which is parallel to your own, before you change velocity.
(2) The world-line of the actual particle.

And the final velocity that you wish to achieve once you get to that position could be either of the following.
(1) return to your own original velocity.
(2) match velocities with the particle and land on it.

If you use idea #1 for both, then you would not see the same thing as if you stayed put. The distribution of matter would still be a sphere, but you would no longer be in the center.

If you use idea #2, you would see essentially the same thing as if you had stayed put. You would be at the center of the sphere after you matched speed with the other particle.

Once again, accelerating and decelerating invokes the twin-paradox, but in the Minkowski-Milne model, the twin-paradox also manifests itself as "inflation" in the experience of the accelerating twin.

Jonathan

 

[Chalnoth]

There is some ambiguity in the statement "I move to a different location," but if we define our term "different location" to mean "landed on another inertial particle" then by your definition the Minkowski-Milne model does turn out to be both homogeneous and isotropic.

Yes, I would agree. The only way in which homogeneity is sensible is as statement that it is possible to choose a set of spatially-separated observers which all see the same properties of the universe. Not every potential cosmology has this property. But yes, I would agree that the Milne model does.

A physical change in r would represent an instantaneous change in position without changing time, or velocity. What I meant by a change in r was simply to ask what the density of particles was at a distance of r.

Well, there is no non-arbitrary way to connect velocities at one point with velocities at another point. So you are free to choose a different "rest" at every point in space-time, if you wish.

One way to think about it is that in General Relativity, one can move a vector at one point to another point through a method called "parallel transport". This basically consists of moving the vector along a line so that it is continuously parallel with itself. The problem is that if the space-time has any curvature, then the specific path you use to get from point A to point B changes the answer you get.

 

[TrickyDicky]

Well, first of all, I tend to think that homogeneity should be the default assumption, because it is the simplest one in accordance with observation, and that unless pursuing an inhomogeneous universe can explain some observations, it shouldn't be considered reasonable.

I agree that in practical terms the assumption of homogeneity as default makes things simpler, (the math treatment for instance) but as long as we don't have direct observations that clearly point to either homogenous or inhomogenous distribution of matter at large scales so far we just find the homogenous option more likely for philosophical, historical, model-dependent and practical reasons, not direct observational reasons, that still permit both assumptions.
When I say direct observation I mean that up to the largest range our telescopes allow currently, we haven't yet found strict homogeneity, and instead some disquieting large voids and unexpected distributions of clusters that can still be explained by statistical reasons so they don't point to an inhomogenous universe either. So it is still an open subject from the purely direct observational perspective.

Certainly, though, according to the standard model of cosmology the homogeneity assumption is mandatory and that is why we consider it as the only reasonable assumption allowed by the whole collection of observations about the universe.

For instance in an inhomogenous universe since there is no constant matter density, there is no such thing as a critical density that is ncesary to our model calculations of fundamental parameters. There wouldn't even be a mean density for the universe since it would be a function of location.

 

[Chalnoth]

I agree that in practical terms the assumption of homogeneity as default makes things simpler, (the math treatment for instance) but as long as we don't have direct observations that clearly point to either homogenous or inhomogenous distribution of matter at large scales so far we just find the homogenous option more likely for philosophical, historical, model-dependent and practical reasons, not direct observational reasons, that still permit both assumptions.

Well, I'd disagree on that. We do definitely have clear observations of isotropy. Given isotropy, we would have to live in a very special location for homogeneity to not also be true, therefore even without additional knowledge, homogeneity is very likely given isotropy.

The fact that we've been able to rule out some specific inhomogeneous models is just icing on the cake, really.

When I say direct observation I mean that up to the largest range our telescopes allow currently, we haven't yet found strict homogeneity, and instead some disquieting large voids and unexpected distributions of clusters that can still be explained by statistical reasons so they don't point to an inhomogenous universe either. So it is still an open subject from the purely direct observational perspective.

Well, obviously when we talk about homogeneity and isotropy, we're talking about statistical homogeneity and isotropy. The exact deviations from this are interesting, but don't undermine the statement that our universe is, on average, highly homogeneous and isotropic.

For instance in an inhomogenous universe since there is no constant matter density, there is no such thing as a critical density that is ncesary to our model calculations of fundamental parameters. There wouldn't even be a mean density for the universe since it would be a function of location.

Well, it's not quite that bad, because you can still talk about a mean density of the universe. This is how we deal with inhomogeneities that exist: consider the universe to be made of some mean distribution plus deviations from the mean. This separation would allow you to model any universe, in principle. The main difficulty is that the Friedmann equations start to give you the wrong answer if your universe gets too inhomogeneous.

 

[TrickyDicky]

Well, I'd disagree on that. We do definitely have clear observations of isotropy. Given isotropy, we would have to live in a very special location for homogeneity to not also be true, therefore even without additional knowledge, homogeneity is very likely given isotropy.

Yes, I was restricting my analysis to purely direct empirical confirmation. If we add a philosophical assumption (the special location issue) we obviously ge homogeneity.

Well, it's not quite that bad, because you can still talk about a mean density of the universe. This is how we deal with inhomogeneities that exist: consider the universe to be made of some mean distribution plus deviations from the mean. This separation would allow you to model any universe, in principle. The main difficulty is that the Friedmann equations start to give you the wrong answer if your universe gets too inhomogeneous.

I think this comment is purely argumentative . Now you accept inhomogeneity as long as it's not too much? How much inhomogenous can a universe be for you to be acceptable?
In my opinion the universe as a whole is either homogenous or inhomogenous and our preferred model tells us it is the former. There is no in between.

 

[Chalnoth]

I think this comment is purely argumentative . Now you accept inhomogeneity as long as it's not too much? How much inhomogenous can a universe be for you to be acceptable?
In my opinion the universe as a whole is either homogenous or inhomogenous and our preferred model tells us it is the former. There is no in between.

There most definitely is in between, though, because our universe is absolutely not completely homogeneous (planet Earth is a huge departure from homogeneity, for instance). It is only approximately homogeneous, as near as we can tell.

So it becomes a huge grey area as to whether or not a certain amount of inhomogeneity is "enough" to call our universe inhomogeneous.

Personally, I would approach it from this point of view: the CMB itself offers a natural scale for the inhomogeneities, namely that at the time the CMB was emitted, the universe at that distance from us was homogeneous to within one part in one hundred thousand. If this is an accurate statistical representation of the overall level of inhomogeneity throughout the visible universe at that time, then we can call our universe homogeneous.

An alternative measure might be from the dynamical point of view, where we can say that our universe is homogeneous if the Friedmann equations are accurate within our observable unierse.

 

[TrickyDicky]

There most definitely is in between, though, because our universe is absolutely not completely homogeneous (planet Earth is a huge departure from homogeneity, for instance). It is only approximately homogeneous, as near as we can tell.

You know I'm referring to large scale.

So it becomes a huge grey area as to whether or not a certain amount of inhomogeneity is "enough" to call our universe inhomogeneous.
Personally, I would approach it from this point of view: the CMB itself offers a natural scale for the inhomogeneities, namely that at the time the CMB was emitted, the universe at that distance from us was homogeneous to within one part in one hundred thousand. If this is an accurate statistical representation of the overall level of inhomogeneity throughout the visible universe at that time, then we can call our universe homogeneous.

This is all quite arbitrary, makes almost pointless to talk about homogeneity of the whole universe because it almost leaves the concept empty of meaning.
In an infinite universe those departures from homogeneity would become infinite, making to call such universe homogenous meaningless.

 

[Chalnoth]

You know I'm referring to large scale.

It doesn't actually matter. There are inhomogeneities on all scales. At some point you have to make a more-or-less arbitrary cutoff for how big the inhomogeneities can be before you call the observable universe inhomogeneous.

This is all quite arbitrary, makes almost pointless to talk about homogeneity of the whole universe because it almost leaves the concept empty of meaning.
In an infinite universe those departures from homogeneity would become infinite, making to call such universe homogenous meaningless.

Arbitrary doesn't mean meaningless, though. Such arbitrary distinctions are found all over science, and are actually quite useful.

 

[JDoolin]

Well, there is no non-arbitrary way to connect velocities at one point with velocities at another point. So you are free to choose a different "rest" at every point in space-time, if you wish.

One way to think about it is that in General Relativity, one can move a vector at one point to another point through a method called "parallel transport". This basically consists of moving the vector along a line so that it is continuously parallel with itself. The problem is that if the space-time has any curvature, then the specific path you use to get from point A to point B changes the answer you get.

This parallel transport is only necessary if you reject the use of Minkowski Space.

In Minkowski space, just because there is more than one way does not mean there is no non-arbitrary way. If you are talking about the non-modified Minkowsiki-Milne model, where all the objects move at constant velocity, in fact, all ways of determining the relative velocity in Minkowski space will be the same.

But if the particles are accelerating, still, in Minkowski space, an object has a clear velocity at any given event, determined as d\vec r /dt; This is the slope of its worldline.

The only ambiguity when you ask, "what is the velocity of a distant particle, now?" is to determine what you mean by now. Should you use a line of simultaneity, and try to match what velocity the particle is going now? Or should you use an inverted light-cone so you can try to match the velocity the particle was going when the image you are now seeing was produced.

When I say direct observation I mean that up to the largest range our telescopes allow currently, we haven't yet found strict homogeneity, and instead some disquieting large voids and unexpected distributions of clusters that can still be explained by statistical reasons so they don't point to an inhomogenous universe either. So it is still an open subject from the purely direct observational perspective.

It appears to me, though that this data has not yet been tabulated in any consistent manner, because all of the studies are done using different metrics. For instance, Hubble's Constant is treated as a universal constant. When one set of data disagrees with another, the astronomers are compelled to find some way to fudge the numbers, or take some kind of average. Wouldn't it be better to assume that the different Hubble-Constants are due to expulsion from different events?

Well, it's not quite that bad, because you can still talk about a mean density of the universe.

Put together with your earlier statement, "there is no non-arbitrary way to connect velocities at one point with velocities at another point" why would there be a non-arbitrary way to connect the density at one point with the density at another point? Wouldn't you need to do the same thing with parallel transport of the meter-stick?

In the Milne-Minkowski model, talking about the mean density of the universe only makes sense if you are talking about the mean density near the center of the sphere at a particular proper time.

(from http://en.wikipedia.org/wiki/Talk:Milne_model)

n dx dy dz = \frac{B t dx dy dz}{c^3<br /> \left(t^2-\frac{x^2+y^2+z^2}{c^2}\right)^2}​

This is derived as equation (9), in section 91 of Relativity, Gravitation, and World Structure, and repeated in a summary in section 112 as equation (36). In section 94, Milne proves that this distribution is Lorentz Invariant.

Though the Milne model is homogeneous and isotropic, it's density is not constant in either time or space.

 

[TrickyDicky]

Arbitrary doesn't mean meaningless, though. Such arbitrary distinctions are found all over science, and are actually quite useful.

Sure, I'm not arguing they are not useful, they help us construct models, my point is that some concepts lose their meaning when this distinction is too vague.

 

[Chalnoth]

Sure, I'm not arguing they are not useful, they help us construct models, my point is that some concepts lose their meaning when this distinction is too vague.

Well, I don't think comparing the inhomogeneities to the observed anisotropies in the CMB is too vague, though. Basically this just comes down to the assumption that the universe is statistically isotropic, and isotropic in the same way no matter where you are within the visible universe. That's a pretty specific statement about homogeneity.

 

[Chalnoth]

In Minkowski space, just because there is more than one way does not mean there is no non-arbitrary way. If you are talking about the non-modified Minkowsiki-Milne model, where all the objects move at constant velocity, in fact, all ways of determining the relative velocity in Minkowski space will be the same.

No, all ways of determining relative velocity in Minkowski space-time will not be the same. However, because the space-time curvature is identically zero, parallel transport gives the same answers no matter which path you take, which in turn means that you can use parallel transport to give a unique answer to the velocity at any other point in the space-time.

This is all academic, though, because Minkowski space-time doesn't describe our universe.

It appears to me, though that this data has not yet been tabulated in any consistent manner, because all of the studies are done using different metrics.

This is irrelevant. The coordinates we apply to reality don't change the behavior of reality. This means that the particular choice of coordinates is irrelevant, and since the Milne metric is actually a special case of the FRW metric, we actually test the Milne cosmology every time we perform an observation using the FRW metric, and we find that the Milne cosmology just doesn't fit observation.

Wouldn't it be better to assume that the different Hubble-Constants are due to expulsion from different events?

If it worked, perhaps. But it doesn't work.

Put together with your earlier statement, "there is no non-arbitrary way to connect velocities at one point with velocities at another point" why would there be a non-arbitrary way to connect the density at one point with the density at another point? Wouldn't you need to do the same thing with parallel transport of the meter-stick?

Yes, this is true. The way it's done in FRW coordinates is you define a set of observers that are stationary with respect to the CMB and all see the same CMB temperature as having the same value of the time coordinate. This is clearly an arbitrary choice, but it is a convenient one given the symmetries of our universe. Those symmetries allow us to express coordinate-dependent quantities such as the matter density in a much simpler fashion.

Though the Milne model is homogeneous and isotropic, it's density is not constant in either time or space.

It all comes down to the coordinates you use. If you use the "right" coordinates, the density is constant in space, but not in time.