# Expansion Limit of Universe

I have been sucessful in calculating the limit of expansion of universe.

Conclusion - Universe has expanded to its maximum

Here is my research

Expansion Limit

According to Hubble's law
$$v=H_0r$$

Where r is distance of Galaxies with respect to Center of Galaxy

As $$v$$ approaches speed of light $$c$$

i.e $$v\to c$$

$$r=c/{H_0}$$

$$r \approx10^26 m$$

Where $$d = 2 * 10^{26} m$$ is Layman Universe Expansion Limit

Observed Time to Reach Expansion limit

$$T = r/{H_0.r} \approx$$ 14 billion years

Actual time of Existence of Universe

$$T_0 = \frac 1 {H_0. \sqrt{1 - v^2/c^2}} = \infty$$

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marcus
Gold Member
Dearly Missed
I have been sucessful in calculating the limit of expansion of universe.
you have succeeded in calculating an important distance in cosmology---which astronomers call the "Hubble radius" and which is not the limit of expansion.
sometimes people mistakenly say it is the present size of the observable universe but it is not that either.

Conclusion - Universe has expanded to its maximum
you have drawn a false conclusion. The observed universe extends beyond the Hubble radius. However you have done a good calculation and found that the Hubble radius is approx 14 billion LY (a useful distance to know even if not the limit of expansion)

According to Hubble's law
$$v=H_0r$$

Where r is distance of Galaxies with respect to Center of Galaxy

As $$v$$ approaches speed of light $$c$$

i.e $$v\to c$$

$$r=c/{H_0}$$

What you have found here is the usual formula for the Hubble radius. Congratulations for finding it.
However typical spatial expansion speeds far exceed c---they are not governed by the speed limit of 1905 special relativity.

$$T = r/{H_0.r} \approx$$ 14 billion years

You have discovered the usual formula for the so-called "Hubble time" and you have correctly calculated it to be approx 14 billion years.

There was a good article in the Scientific American about spatial expansion and the confusions people have about it. It was by Charles Lineweaver and Tamara Davis. Free online. Very clear illustrations. You might like it. I will see if I can get a link.

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Maxwells Demon
the 14 billion years you've found is just the age of the universe right? I've just done this in astronomy class and found it to be 13,8 billion years

Maxwells Demon
I calculated H0 and then used t = 1/H0 for the age..

marcus
Gold Member
Dearly Missed
I calculated H0 and then used t = 1/H0 for the age..

that is not the usual estimated age of the universe

it is CLOSE to what astronomers usually estimate

but it is not equal and it is not how they calculate the age or time since beginning of expansion

Maxwells Demon
okay, this isn't professional so I knew it wasn't very precise.. Our teacher told us so as well :)

It's actually a rather difficult exercise to calculate an expression for the age of the Universe. I've worked through calculations umpteen pages long starting with the Friedmann equation. In fact, it's worth noting that almost always it's not possible to obtain an exact expression as often the inevitable integration has no algebraic solution.

(Maxwells- when you cover the Friedmann equation in astronomy, let us know! The above I describe is a very good exercise for covering a good amount in FRW cosmology. (I should think it's also within middle undergraduate level, with a bit of hard work )

If $$H_0$$ is a constant,

then $$T=1/H_0$$ i.e Age of the Universe is a constant, and WOULD be a constant in all context.

What I wanted to express was that, according to Relativity, Speed of a particle cannot be greater than speed of light, So the recession of Galaxies will gradually slow down (As S=D/T), when they approach speed of light according to the equation $$T = T_0 . \sqrt{1 - v^2/c^2}}$$. And the Universe will stop expanding as it is constant

Wallace
What I wanted to express was that, according to Relativity, Speed of a particle cannot be greater than speed of light, So the recession of Galaxies will gradually slow down (As S=D/T), when they approach speed of light according to the equation $$T = T_0 . \sqrt{1 - v^2/c^2}}$$. And the Universe will stop expanding as it is constant

This is a common misconception. There is no limit in general relativity as to the rate of increase between the proper distance of two objects with respect to proper time. None. Nada. :surprised

How can this be the case when we all know that the constancy of the speed of light is so fundamental to Relativity

In special relativity only my first statement is false, i.e. in this case the derivate of the proper distance between any two points with respect to proper time is always less than c. In general relativity however this is not the case. The cut a long story short this is because in special relativity a single Minkowski frame can be created that encompasses the entire universe and all motions can be seen with respect to that frame.

However in a general relativistic universe the presence of the energy within the universe, and particularly the change in that energy ( for instance the reduction in the mean density of matter in the universe at the universe expands ) means that that there is no global Minkowski frame that is common to all parts of the universe at all times. Therefore there is no one frame that all velocities are measured with respect to.

Now this is a very hand waving explanation and only says that proper distance separations can increase at a faster rate than c but dosn't prove that they do. A good reference that is often pointed to on this is "www.mso.anu.edu.au/~charley/papers/DavisLineweaver04.pdf"[/URL] by Davis & Lineweaver. This gives a much fuller and more comprehensive explanation.

In any case, the above is just words used to try and explain how this works. The ultimate proof is that if you solve the general relativistic equations you can easily see these apparently prohibited speeds come out quite naturally. So unless general relativity is wrong you argument does not work.

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marcus
Gold Member
Dearly Missed
Wallace, I concur completely of course.
As a side comment: sometimes we get posters in this forum who invoke the MILNE UNIVERSE picture as if it were a realistic alternative that could fit the observed data.

I never know how to reply. I think the Lineweaver and Davis dispose of the Milne picture, but it would be nice if there were some quick way to explain that the Milne picture obviously doesnt fit what we see---without going to the length they do.

(AFAIK in that picture there is one Minkowski frame and instead of geometric expansion you have matter actually flying apart. It is a vintage 1930s thing, I believe.)

If someone brings up Milne as if it were a serious possibility, could one say something like the following?

Out past z=1.6 things look larger the farther away they are (i.e. as z increases). In Milne case they would look smaller. So Milne picture is in direct contradiction to what we observe.

or is there some problem with this refutation.

Garth
Gold Member
Out past z=1.6 things look larger the farther away they are (i.e. as z increases).

Marcus, do you have a reference for that statement?

You are talking about standard rulers are you not?

Garth

Maxwells Demon
all right, I stepped off :(

marcus
Gold Member
Dearly Missed
Marcus, do you have a reference for that statement?

You are talking about standard rulers are you not?

Garth

there was that paper by Hellaby I cited earlier
and "angular size distance" is written up in Wiki

the point about angular size distance is that it has a MAXIMUM

I think this information is widely dispersed.

But if I had to cite one reference it would be Ned Wright's cosmology calculator. z = 1.6 is where the max ang. size dist. comes.
Or somewhere around there to be determined more precisely by future observations as per Hellaby.

I'd prefer not to argue about authorities and so forth. Let's see what Wallace says. My question was to him "can one say something like the following...or is there some problem with [that way of refuting Milne picture]?"

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Garth
Gold Member
Okay, I was well aware of the theoretical maximum in angular diameter distance, I wondered whether there was observational evidence (such as the angular diameter of quasar radio lobes) to confirm the existence of such a maximum, to determine at what red shift it occurred and that therefore could be used to falsify the Mine model.

Garth

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Wallace
The simplest way to falsify the Milne model would be to ask how it explains the observed acceleration of the expansion?

To take this a step back, if one is skeptical about whether the Universe is truly accelerating given the data we have (which I think is unlikely but is none the less a healthy attitude) you can still ask how the Milne model could possibly explain the supernovae data.

The simplest way to falsify the Milne model would be to ask how it explains the observed acceleration of the expansion?
Well sorry for being perhaps a bit too simple here and without making any particular claim that the Milne model is in any way valid, how difficult is it to imagine an expansion that is decreasingly limited by curvature, as a form of accelerated expansion?

Wallace
I'm not sure about how difficult that might be to imagine, however remember that the Milne model describes test particles moving in a global Minkowski frame so there is no curvature. There is no influence on the expansion rate from anything, be it curvature or otherwise.

The Milne model is a very specific and entirely based on kinematic special relativity so any deviation from constant expansion rate cannot be accounted for within that model.

Taking you suggestion on its own, outside of the context of the Milne model, I'm not sure what you mean by "an expansion that is decreasingly limited by curvature, as a form of accelerated expansion?". Could you clarify this?

I'm not sure about how difficult that might be to imagine, however remember that the Milne model describes test particles moving in a global Minkowski frame so there is no curvature. There is no influence on the expansion rate from anything, be it curvature or otherwise.

The Milne model is a very specific and entirely based on kinematic special relativity so any deviation from constant expansion rate cannot be accounted for within that model.
Obviously, but nothing prevents us to start from the Milne model and then consider the gravitational impact on it, right?

Taking you suggestion on its own, outside of the context of the Milne model, I'm not sure what you mean by "an expansion that is decreasingly limited by curvature, as a form of accelerated expansion?". Could you clarify this?
Start with a Milne model and observe the expansion of its boundary.
Now add mass-energy to this model.

Then there are three conditions that are interesting:

A. Gravitational convergence slows down the expansion of the boundary but is insufficient to stop it.
B. Gravitational convergence compensates for the expansion of the boundary.
C. Gravitational convergence causes the boundary to contract (a trapped surface condition).

In case of A, the mass-energy density decreases over time and thus the convergence will decrease as well, so the expansion of the boundary accelerates over time.
So this means that, in the limit, A, will approach the pure Mile model.

But feel free to explain where I make an error.

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Wallace
There's still something I'm not quite getting here. Starting with the Milne model and then adding gravity gets you to either a flat, open or closed matter only universe depending on how much mass you put in, i.e. standard non Lambda models. Adding vacuum energy gets you to the LCDM model. I'm not sure how you get anything else from the Milne model, which is just an empty universe with test particles. I'm not sure in what way you are suggesting a Milne model + gravity is any different from a regular FRW model, if indeed that is what you are suggesting?

The other thing I'm not sure about is the speculation of what happens at the 'boundary'? The Milne model describes an infinite universe does it not, so what is the boundary?

The Milne model describes an infinite universe does it not, so what is the boundary?
The boundary is simply the surface of the future null cone of the Milne model.

The volumes of all hypersurfaces of constant proper time are obviously infinite.
But is the hypervolume infinite? Did anyone take a crack at that one?

But anyway, the infinity of all hypersurfaces of constant proper time does not preclude space-time from having a boundary or from being closed, as is the case in C.

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Wallace
The boundary is simply the surface of the future null cone of the Milne model.

But who's null cone? If the future light cone of an observer converges then the Universe is closed, which requires the presence of matter (or some kind of w > -1/3 energy at least) to close it. Again I do not comprehend how this is related to the Milne model or some extension of it?

marcus
Gold Member
Dearly Missed
The simplest way to falsify the Milne model would be to ask how it explains the observed acceleration of the expansion?
...

Wallace it's really great having you here! I admire your patience and obvious expertise.

What I have found though is if you get a highly verbal and opinionated person who has an idée fixe about the Milne model then it is very difficult to dislodge.

To ordinary clue-ful people who know some GR it is obvious that Milne is a pathetically bad match to nature---it's just obvious. Open and shut prima facie case!

But in a special case where it has become a kind of Cause you may need a sledge-hammer.

so I am asking about the argument where you use the angular size minimum which is in principle OBSERVABLE.

Hellaby says the next generation of instruments are going to be able to pinpoint the minimum with some useful level of accuracy. I expect it will come around z = 1.6.
The idea that everything out beyond z = 1.6 looks bigger and bigger the farther away it is---this idea seems sufficiently graphic that it might make an impression even on a very obstinate person.

Am I missing something. Any comments on this approach to explaining why Milne is bad?

Wallace
Hellaby says the next generation of instruments are going to be able to pinpoint the minimum with some useful level of accuracy. I expect it will come around z = 1.6.
The idea that everything out beyond z = 1.6 looks bigger and bigger the farther away it is---this idea seems sufficiently graphic that it might make an impression even on a very obstinate person.

I'm not familiar with this kind of test, do you have a useful reference for this? Off the top of my head I would think the biggest problem with this would be finding standard sized objects. If galaxies had the same average size for all cosmic time then this would be easy, but since galaxies evolve, merge and grow in ways we do not fully understand then our measurement of their angular size would be mixed up with our uncertainty over how big in physical size we expect them to be.

On the other hand we would expect galaxies to only get bigger (in physical size) as times goes on, even if the rate at which they do this is unknown. So I guess the presence of the angular size peek should be visible even if galaxy evolution is not completely understood? Considering we have seen galaxies out to z~8-10 via the HUDF surely someone has made a rough measure of this? I havn't thought about this or read anything along these lines before though, so as I say if you have a reference for it I would be interested in reading it!

hellfire
I'm not familiar with this kind of test, do you have a useful reference for this? Off the top of my head I would think the biggest problem with this would be finding standard sized objects.

From Edmund Bertschinger's tutorial http://ocw.mit.edu/NR/rdonlyres/Physics/8-942Fall2001/2F658E61-68A8-40F4-9168-B7AD0E23CA49/0/cosmog.pdf [Broken]:

Unfortunately, good meter-sticks are hard to find in cosmology. Gurvits et al (1999, A&A, 342, 378) have used compact radio sources (typically, relativistic jets of plasma emitting synchrotron radiation, powered by massive black holes in quasars and radio galaxies) but they have not demonstrated rigorously that the objects should have the same physical size. A better test will come from measurements of hot gas in clusters of galaxies by combining X-ray emission with the Sunyaev-Zel’dovich effect (shadowing of the microwave background). Birkinshaw (1999, Phys. Rep., 310, 97) gives a detailed review. Finally, other possibilities have been proposed based on the clustering of quasar absorption lines (Hui et al 1999, ApJ, 511, L5) or high-redshift galaxies (Nair 1999, ApJ, 522, 569). These methods are statistical and are based on comparing clustering in the radial and transverse directions, following a suggestion by Alcock & Paczynski (1979, Nature, 281, 358).

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Garth
Gold Member
The simplest way to falsify the Milne model would be to ask how it explains the observed acceleration of the expansion?

To take this a step back, if one is skeptical about whether the Universe is truly accelerating given the data we have (which I think is unlikely but is none the less a healthy attitude) you can still ask how the Milne model could possibly explain the supernovae data.
It is actually remarkable that the ($\Omega_M$,$\Omega_{\Lambda}$) = (0,0) model fits the SNe Ia very well out to about z~1, see Perlmutter et al's seminal paper Measurements of Omega and Lambda from 42 high-redshift supernovae Figure 2 page 24
The middle solid curve is for ($\Omega_M$,$\Omega_{\Lambda}$) = (0,0). Note that this plot is practically identical to the magnitude residual plot for the best-fit unconstrained cosmology of Fit C, with ($\Omega_M$,$\Omega_{\Lambda}$) = (0.73,1.32).

If the 'empty' universe fits the SNe Ia data then DE would have to produce a total pressure equation of
p = -1/3$\rho$ to give a linearly expanding universe. There is one modified GR theory that does just this.

At greater z the fit is not so good, but this could be explained by the SNe Ia not being standard candles at this early epoch.

We could also note the remarkable cosmological coincidence of the age of the universe (present best value 13.81 Gyrs.) and Hubble time (present best value 13.89 Gyrs.) especially because with an arbitrary amount of DE the age of the universe could be anything from about 10 Gyrs. to infinity. This does look as if the universe has been expanding linearly!!

My original question was asking whether there is any empirical evidence to support the theoretical maximum in angular diameter distance of z ~ 1.6, which would falsify the Milne model as had been previously stated, from hellfire's link paper it looks there is not yet, (unless there is a more recent detection of such a maximum).

Garth

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marcus
Gold Member
Dearly Missed
I'm not familiar with this kind of test, do you have a useful reference for this? ...

Admittedly it is still at the stage of a research recommendation. There was a paper by Hellaby recommending that the "next generation of redshift surveys" get a handle on it. I will look up the reference

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marcus
Gold Member
Dearly Missed
I'm not familiar with this kind of test, do you have a useful reference for this? ...

There was a paper by Hellaby suggesting that the "next generation of redshift surveys" could actually determine the maximum of angular size distance.

To refute Milne one would not have to determine the maximum accurately, which is what Hellaby wants. To refute Milne model one only needs observational evidence that such a maximum exists.

related paper:
http://arxiv.org/astro-ph/0603637 [Broken]
The Mass of the Cosmos
Charles Hellaby
6 pages, 9 graphs in 3 figures. Replacement has very minor changes: puts greek letters on graphs, and adds small corrections made in publication

"We point out that the mass of the cosmos on gigaparsec scales can be measured, owing to the unique geometric role of the maximum in the areal radius. Unlike all other points on the past null cone, this maximum has an associated mass, which can be calculated with very few assumptions about the cosmological model, providing a measurable characteristic of our cosmos. In combination with luminosities and source counts, it gives the bulk mass to light ratio. The maximum is particularly sensitive to the values of the bulk cosmological parameters. In addition, it provides a key reference point in attempts to connect cosmic geometry with observations. We recommend the determination of the distance and redshift of this maximum be explicitly included in the scientific goals of the next generation of reshift surveys. The maximum in the redshift space density provides a secondary large scale characteristic of the cosmos."

for more discussion, the earlier thread was
"got a link for the angular-size redshift relation?"
SpaceTiger weighed in at one point.

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Wallace
It is actually remarkable that the ($\Omega_M$,$\Omega_{\Lambda}$) = (0,0) model fits the SNe Ia very well out to about z~1, see Perlmutter et al's seminal paper Measurements of Omega and Lambda from 42 high-redshift supernovae Figure 2 page 24

It is 'seminal' because it was the first, and by definition has the least data of any supernovae and cosmology paper. Strange that you should point to it then! Although not so strange when you consider that the data obtained since then have far more significantly ruled out the Milne model...

At greater z the fit is not so good, but this could be explained by the SNe Ia not being standard candles at this early epoch.

So you are happy to use SN to support your theory in the region it agree with it and then just when they diverge suggest that this might is where they cease to be standard candles? :zzz:

We could also note the remarkable cosmological coincidence of the age of the universe (present best value 13.81 Gyrs.) and Hubble time (present best value 13.89 Gyrs.) especially because with an arbitrary amount of DE the age of the universe could be anything from about 10 Gyrs. to infinity. This does look as if the universe has been expanding linearly!!

This co-incidence is just that, a coincidence! It means nothing! Think about it for a moment. The age of the universe is not something we can observe it is only something we can derive from our theory, which we obtain by fitting to what we can observe, the data. If two theories give the same derived result for something that is unobservable while one is a good fit to the data and the other is an appalling fit which would you believe?

Take this example. Imagine an Aristotelean physicist and a Newtonian physicist are trying to predict how far a ball thrown at a certain angle will go. The Aristotelean says that the ball will travel in a straight line until a certain point and then will fall vertically downward to the earth. The Newtonian says it will travel on a parabolic path. If they both happen to predict the same final landing point, how can you say which theory is correct? Easy! Watch the ball in flight and observe the parabolic path!

This is just like the universe. We can see the evidence of accelerating and deccelerating eras. Just because you can draw a straight line through an a(t) plot that intersects t=0 and t=1 dosn't mean this is a reasonable model if it does not fit the data!

It's not just about SN. Our models of structure formation work really well for the LCDM model but the Milne model predicts a Universe that looks completely different in terms of the clustering of galaxies etc etc

I find it frustrating that people willfully take one paper detailing one type of measurement and then suggest their pet model fits better than the standard model. The LCDM model is the current standard because it fits all the data with a common set of parameters. It's not perfect, that's for sure, and it might even be completely wrong. However the point is to show how a competing idea fits the whole lot in a better way, not willfully twisting a single out-dated reference into something far more significant than it is!

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Garth
Gold Member
So you are happy to use SN to support your theory in the region it agree with it and then just when they diverge suggest that this might is where they cease to be standard candles? :zzz:
As there is no clear model for SNe Ia is not "happy to use SN to support your theory" just what the standard model does in assuming that their luminosity is constant over cosmological time, because over this time there may have been metallicity evolution?
This co-incidence is just that, a coincidence! It means nothing!
Such as the related coincidental present OOM equality of the densities of baryonic matter, DM and DE in a highly expanded universe? Note: the present derived age of the universe is a function of the exact mix of these components.
Think about it for a moment.
Quite.
The age of the universe is not something we can observe it is only something we can derive from our theory, which we obtain by fitting to what we can observe, the data. If two theories give the same derived result for something that is unobservable while one is a good fit to the data and the other is an appalling fit which would you believe?
I am not advocating the Milne model, although the “Freely Coasting” Cosmology does appear to be more concordant than you allow, however the standard model would be more convincing if Inflation, DM and DE were all verified by laboratory physics. Inflation has explained the coincidental nature of the density, smoothness and flatness problems of old GR cosmology by introducing a new set of non-verified 'entities' with 'coincidental' relative abundances.
Take this example. Imagine an Aristotelean physicist and a Newtonian physicist are trying to predict how far a ball thrown at a certain angle will go. The Aristotelean says that the ball will travel in a straight line until a certain point and then will fall vertically downward to the earth. The Newtonian says it will travel on a parabolic path. If they both happen to predict the same final landing point, how can you say which theory is correct? Easy! Watch the ball in flight and observe the parabolic path!
Apart from the inappropriateness of that example, as we cannot 'observe' the expansion of the universe as one can the trajectory of a ball, the standard model 'R(t)' curve is hardly a simple parabola!

As I said, I am not actually advocating the Milne model, but only pointing out the standard model fits only because of the introduction of components undiscovered in laboratory physics, which until they are discovered, say by the LHC, lead one to keep an open mind on the subject. In the meantime it is intriguing that the simplicity of the Milne model does echo certain features of the universe.

Garth

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Wallace
As there is no clear model for SNe Ia, for example there may be metallicity evolution, is not "happy to use SN to support your theory" just what the standard model does in assuming that their luminosity is constant over cosmological time?

The point is we don't know what evolution effects may or may not exist in the SN we observe, but the best evidence we have suggests that if there are any such effects they are small. Therefore we treat the luminosities as unchanging for all time, for now, until we know any better. This is much more transparent an approach than what you suggest, i.e. requiring that the effect kicks in precisely when it is needed to justify your model.

The key point which you have once again ignored is that the LCDM model is much more than SN. The structure and CMB results are just as crucial and agree with the SN results, hence our guarded confidence in these results.

Such as the related coincidental present OOM equality of the densities of baryonic matter, DM and DE in a highly expanded universe? Note: the present derived age of the universe is a function of the exact mix of these components.

What is the physical significance? So the universe, according to LCDM is 13.8 Gyrs old (or whatever the exact number is). There is nothing significant about this number. It happens to be the inverse of the Hubble time, but there is no significance to this coincidence. It's like saying, wow! my car number plate is 000, what are the odds!

am not advocating the Milne model, although the “Freely Coasting” Cosmology does appear to be more concordant than you allow, however the standard model would be more convincing if Inflation, DM and DE were all verified by laboratory physics. Inflation has explained the coincidental nature of the density, smoothness and flatness problems of old GR cosmology by introducing a new set of coincidental non-verified 'entities'.

I has a quick read of that paper and it dosn't address any issues in structure formation, other than 'structure can form'. I've done simulations of structure in these models before and the clustering statistics are totally different and easily distinguished. Indeed DE and DM are 'non-verified entities' as you suggest, however electrons were also these at some point, as were a host of other physical theories! I wouldn't be that in 10 years we still think these things exist, but inflating the case for alternative models is not the way to go about investigating them.

Apart from the inappropriateness of that example, as we cannot 'observe' the expansion of the universe as one can the trajectory of a ball, the standard model 'R(t)' curve is hardly parabolic!

Why is it inappropriate exactly? We CAN observe the trajectory of R(t)! This is exactly what we do when we study SN! (crossing our fingers that they are standard candles) and also what we do by studying structure at different epochs, since structure is driven by R(t). Of course R(t) is not a parabola, surely you comprehended that this was an analogy, what I'm saying is that is has some curve that we can calculate via our model, and check against observations. The linear coasting (or Milne model) predicts a straight line. The data are incompatible with a straight line and hence we reject the model, even though the staring line passes through the same two end points as the curved one.

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