Is it true that for an isotropic, homogeneous flat universe with dust
and a positive cosmological constant, the universe necessarily expands
forever? The argument may be,

(a_t/a)^2 = (8*pi*G/3)*rho + lambda/3 (Friedmann equation)

where a_t refers to the first derivative of a with respect to t. Now
the right hand side is strictly positive (as rho is positive and
proportional to a^3 for dust), so a_t is always positive.

If this is true, why is it said that in a universe with a positive
cosmological constant, the fate of the universe has no direct relation
with the curvature (k) but depends on the exact proportion of
[(matter+radiation) density] / vacuum energy density?

mathman
For a flat universe, there is a critical density. Current theory is that the universe is at the critical density and that about 70% of it is due to "dark energy", which is explained (in the leading theory) by a small positive cosmological constant.

marcus
Gold Member
Dearly Missed
Originally posted by Wong
Is it true that for an isotropic, homogeneous flat universe with dust
and a positive cosmological constant, the universe necessarily expands
forever? The argument may be,

(a_t/a)^2 = (8*pi*G/3)*rho + lambda/3 (Friedmann equation)

where a_t refers to the first derivative of a with respect to t. Now
the right hand side is strictly positive (as rho is positive and
proportional to a^3 for dust), so a_t is always positive.

If this is true, why is it said that in a universe with a positive
cosmological constant, the fate of the universe has no direct relation
with the curvature (k) but depends on the exact proportion of
[(matter+radiation) density] / vacuum energy density?

Hello Wong if you are still visiting here. I see you posted the question a long time ago. The way I learned the equation it has a "k" term in it.

(a_t/a)^2 = (8*pi*G/3)*rho - k/a^2 + lambda/3

a_tt/a = -(4*pi*G/3)*rho + lambda/3

There are two cosmological equations, you need both. And in the matter-dominated case it is possible for a_t to go negative. So it is possible for the universe to start contracting in this case.

It seems to me that the equations do say that the fate of some kinds of universe depends on curvature (k). And also they say that the fate depends on whether or not
matter+radiation density / vacuum energy density is large
that is what it means to be matter-dominated.

Because if rho is big it can make a_tt negative and it could cause that particular universe to collapse.

So I disagree with you. I think the equations say that the fate of universes does depend on those things you mentioned. If I am wrong please explain why. (If you are still around.)

The Universe only expands forever if you allow it too.

Originally posted by marcus
Hello Wong if you are still visiting here. I see you posted the question a long time ago. The way I learned the equation it has a "k" term in it.

(a_t/a)^2 = (8*pi*G/3)*rho - k/a^2 + lambda/3

a_tt/a = -(4*pi*G/3)*rho + lambda/3

There are two cosmological equations, you need both. And in the matter-dominated case it is possible for a_t to go negative. So it is possible for the universe to start contracting in this case.

It seems to me that the equations do say that the fate of some kinds of universe depends on curvature (k). And also they say that the fate depends on whether or not
matter+radiation density / vacuum energy density is large
that is what it means to be matter-dominated.

Because if rho is big it can make a_tt negative and it could cause that particular universe to collapse.

So I disagree with you. I think the equations say that the fate of universes does depend on those things you mentioned. If I am wrong please explain why. (If you are still around.)

Hi Marcus.

My question refers specifically to the flat universe, so "k" in the above equation should be "0".

The reason I consider only the case k = 0 is that I found if I put rho = k/a^3, where k is a constant, the differential equation admits an exact solution.

(a_t/a)^2 = (8*pi*G/3)*k/a^3 + lambda/3

define C = (8*pi*G*k/3), D = lambda/3,

(a_t/a)^2 = C/a^3 + D

a*(a_t)^2 = Da^3 + C

Assume a_t > 0,

[squr(a)*da]/squr(Da^3+C) = dt (squr refers to "the square root of)

Change the variable from a to x, where x = [squr(D/C)]*a^1.5

E*dx/squr(1-x^2) = dt

where E = 2/[3*squr(D)]

Impose the boundary condition x=0 when t =0,

E*(inverse of hyperbolic sine of x) = t

Then a could be expressed in terms of t,

a = {squr(C/D)*sinh[3*squr(D)*t/2]}^2/3

The interesting thing is when one puts the solution into the second Friedmann Equation (that you mentioned), one found that, in order for the equation to be satisfied, the constant k has to be in a certain proportion to lambda.

I don't know whether when I did is correct, but it surely rests on the assumption that the term rho is inversely proportional to a^3

marcus
Gold Member
Dearly Missed
Hello Wong, the thought just came to me that you might enjoy trying out the Friedmann equation animation at

http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html

It has two windows. You click in the left window to select
a choice of Omega_mass and Omega_lambda

then in the right window you see the evolution of the universe which has those two omegas at the present moment in history.

You should turn the animation on in both windows. and see both curves being traced out at the same time.
The left shows the evolution of (Omega__mass, Omega__lambda)

If you want something resembling our universe you click on
the point (0.3, 0.7) because that is what people think the two omegas are right at the present time. And then you see the past and future history of our universe in the right window. Of course it is just a toy but it is fun to try things with.

You can make histories with no big bang, but rather a kind of bounce. And of course you can make histories that crunch. If you havent already seen something like this then I expect you will enjoy it as I did.