# 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 Expanding Confusions 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: Mar 25, 2007
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