The Density Parameter and R^3 Space

[KingOrdo]

Hi everyone--

I'm trying to get a technical explanation of the expansion of the Universe. I posted something similar in the Cosmology forum, but got a lot of unlettered and wishy-washy responses. I'm hoping here in the SR/GR forum I can find a more mathematically rigorous treatment.

Here's the question: Assuming there is no Dark Energy/Lambda/etc., is there any value for the density parameter Omega such that the Universe will one day reach a maximum volume (or asymptotically approach a maximum volume)?

My understanding was that this would happen if Omega=1; however, some people say that though in this case the Universe will one day stop expanding, it will not ever reach a maximum volume (nor asymptotically approach a maximum volume). This I do not understand.

Thanks!

 

[Garth]

Hi everyone--

I'm trying to get a technical explanation of the expansion of the Universe. I posted something similar in the Cosmology forum, but got a lot of unlettered and wishy-washy responses. I'm hoping here in the SR/GR forum I can find a more mathematically rigorous treatment.

Here's the question: Assuming there is no Dark Energy/Lambda/etc., is there any value for the density parameter Omega such that the Universe will one day reach a maximum volume (or asymptotically approach a maximum volume)?

My understanding was that this would happen if Omega=1; however, some people say that though in this case the Universe will one day stop expanding, it will not ever reach a maximum volume (nor asymptotically approach a maximum volume). This I do not understand.

Thanks!

First of all KingOrdo you have posted again in the Cosmology Forum...
Edit - I've just noticed the thread WAS moved from the S&GR Forum -- my mistake

Secondly I have already answered your question, why won't you accept the answer?

If there is no DE or Cosmological Constant to complicate matters the universe will eventually reach a maximum volume and thereafter contract if $\Omega_{total}$ > 1.

This means there is enough matter and energy in the universe to produce enough gravitational force between the contents of the universe/or enough space-time curvature, whichever you prefer, to stop the expansion and reverse it.

As I said, it is analogous in Newtonian gravitational theory to escape velocity.

If a spacecraft is given escape velocity, or more than escape velocity, the gravitational field of the Earth/wherever will not be strong enough to retain it and it will fly off to infinity.

If however the spacecraft has less than escape velocity then the gravitational field will be able to pull it back to Earth again, this is equivalent to $\Omega_{total}$ > 1 in the parallel case with the expansion of the universe in cosmology.

You do ask a series of questions that suggest you ought to read up on the subject. If nowhere else you can find a lot on the internet at serious web sites such as this one: Welcome to Ned Wright's Cosmology Tutorial

I hope this helps.

And BTW don't be rude to SpaceTiger, he is our Moderator, he knows his stuff and he only wants to help you.

Garth

 

[marcus]

assuming no cosmological constant (Lambda = 0 ) as you have said,

then with Omega = 1
the universe never stops expanding
and never reaches a maximum

but you say

...happen if Omega=1; however, some people say that though in this case the Universe will one day stop expanding, it will not ever reach a maximum volume (nor asymptotically approach a maximum volume).

I doubt that anybody here at PF ever said that in this case "the Universe will one day stop expanding". I suspect someone said something which sounded like that to you, but didn't really mean that. However if not too much trouble, KingOrdo, please find the quote where you think someone said that. We can have a look at the quote.

Before you, I never heard anybody say or read any statement that in the Omega = 1 case the universe eventually stops expanding. As far as I can remember you are the first to present this idea.

As Garth said, in the Omega > 1 case, the universe DOES eventually reach a maximum and stops expanding, and it then begins to contract. (this is always assuming that Lambda = 0, as you said at the start)

 

[KingOrdo]

Sorry, my question was not worded right: what I mean to ask is: Assuming there is no Dark Energy/Lambda/etc., is there any value for the density parameter Omega such that the Universe will one day reach a maximum volume (or asymptotically approach a maximum volume) AND NOT COLLAPSE? (I.e. Get to a maximum volume (or asymptotically approach a maximum volume) and stay there.)

 

[Garth]

Yes, we understood you first time -

The value of the density parameter that ensures there will be a maximum volume and then re-collapse [Edit the word "not" crept in there, following your question, instead of "then" as a typo] if there is no DE/CC is:

$\Omega_{Total}$ > 1.

Is that clear now?

Garth

 

[KingOrdo]

Yes, we understood you first time -

The value of the density parameter that ensures there will be a maximum volume and not re-collapse if there is no DE/CC is:

$\Omega_{Total}$ > 1.

Is that clear now?

Garth

No (cf. marcus: "in the Omega > 1 case, the universe DOES eventually reach a maximum and stops expanding, and it then begins to contract. (this is always assuming that Lambda = 0, as you said at the start)" [my emphasis]).

And indeed, my understanding is that iff Omega > 1 the Universe is spatially S^3 and will inevitably collapse (again, no DE/Lambda).

 

[marcus]

Sorry, my question was not worded right: what I mean to ask is: Assuming there is no Dark Energy/Lambda/etc., is there any value for the density parameter Omega such that the Universe will one day reach a maximum volume (or asymptotically approach a maximum volume) AND NOT COLLAPSE? (I.e. Get to a maximum volume (or asymptotically approach a maximum volume) and stay there.)

The answer is no, there is no value of Omega for which what you describe can happen.

I never heard of a solution to the Einstein equation where it expands to a certain point and then just stays there.
The Einstein equation and the simplified versions derived from it that cosmologists ordinarily use do not have any stable static solutions, as far as I know. Longrange distances are like a ball tossed into the air---they are never static---they are always doing something: increasing forever, or increasing, turning around, decreasing.

the ball can never go up to some maximum height and remains static, suspended there.
the longrange distances can't be static either. static solutions are unstable

maybe that is a bad analogy but i can't think of anything else at the moment. (also I think someone already used it)

 

[Wallace]

The answer is no, there is no value of Omega for which what you describe can happen.

I never heard of a solution to the Einstein equation where it expands to a certain point and then just stays there.

Absolutely right. This is in fact why Einstein introduced a non-zero cosmological constant, since he realized that a static Universe wasn't possible simply with matter, it would have to either be expanding or contracting. By introducing the cosmological constant a static Universe (which was what people thought our Universe was before Hubble turned that idea on its head) was possible. Without the CC or DE stated in the question under discussion, there way to get a static Universe at any point.

I'm sure you are aware of all this marcus, but hopefully that helps to make things just a little clearer for the benefit of others.

 

[KingOrdo]

Many thanks, gents.

That clears everything right up.

 

[Chris Hillman]

Ditto the others, plus: the discussion in the last chapter of D'Inverno, Understanding Einstein's Relativity is highly relevant and you should find it very illuminating.

 

[KingOrdo]

Ditto the others, plus: the discussion in the last chapter of D'Inverno, Understanding Einstein's Relativity is highly relevant and you should find it very illuminating.

When I took GR--many years ago--that was my favorite book. A welcome respite from MTW and Hawking & Ellis.