Expansion Limit of Universe

1. Mar 25, 2007

.ultimate

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

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

Last edited: Mar 25, 2007
2. Mar 25, 2007

marcus

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.

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)

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.

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.

Last edited: Mar 25, 2007
3. Mar 25, 2007

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

4. Mar 25, 2007

Maxwells Demon

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

5. Mar 25, 2007

marcus

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

6. Mar 25, 2007

Maxwells Demon

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

7. Mar 25, 2007

fasterthanjoao

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 )

8. Mar 25, 2007

.ultimate

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

9. Mar 25, 2007

Wallace

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.

Last edited by a moderator: Apr 22, 2017
10. Mar 26, 2007

marcus

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.

11. Mar 26, 2007

Garth

Marcus, do you have a reference for that statement?

You are talking about standard rulers are you not?

Garth

12. Mar 26, 2007

Maxwells Demon

all right, I stepped off :(

13. Mar 26, 2007

marcus

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]?"

Last edited: Mar 26, 2007
14. Mar 26, 2007

Garth

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

Last edited: Mar 26, 2007
15. Mar 26, 2007

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.

16. Mar 26, 2007

MeJennifer

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?

17. Mar 26, 2007

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?

18. Mar 26, 2007

MeJennifer

Obviously, but nothing prevents us to start from the Milne model and then consider the gravitational impact on it, right?

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.

Last edited: Mar 26, 2007
19. Mar 26, 2007

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?

20. Mar 26, 2007

MeJennifer

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.

Last edited: Mar 26, 2007
21. Mar 26, 2007

Wallace

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?

22. Mar 27, 2007

marcus

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?

23. Mar 27, 2007

Wallace

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!

24. Mar 27, 2007

hellfire

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

Last edited by a moderator: May 2, 2017
25. Mar 27, 2007

Garth

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
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

Last edited: Mar 27, 2007