# Homework Help: Find lifetime of closed RD universe

1. Apr 30, 2013

### ck99

1. The problem statement, all variables and given/known data

Integrate field equations for a universe filled with radiation and with k = +1, λ = 0. Find ρ(a) ρ(t) and a(t). Find lifetime of the universe.

2. Relevant equations

Use first Friedmann equation which reduces to

a'2/a2 + a-2 = kρ where k = 8∏/3

3. The attempt at a solution

I have followed my lecture notes to get the following expression for a(t)

a(t) = (k - t2)1/2

ρ(a) = ρ0a-4

ρ(t) = ρ0(k - t2)-2

Hopefully these results are correct! I am not sure how to o the last part though, which is to calculate the lifetime of the universe. Can someone advise where to start with this? Presumably I want t as a function of . . . something. And with a closed universe I think the lifetime ends when a reaches 0 again (the big crunch). But I'm not sure where to go from there . . .

2. Apr 30, 2013

### BruceW

Are you sure you don't mean k=-1 ? Also, I'm guessing in the rest of your post, you are using c=1 ?

3. Apr 30, 2013

### George Jones

Staff Emeritus
Are you sure that this is correct?

4. Apr 30, 2013

### BruceW

aha, whoops, yeah, it should be +k I confused myself a bit there.

Now I've thought about it for a bit, this could be right, if he has scaled it in just the right way, and defined t=0 at just the right time (not at the first singularity). But then this kind of defeats the point, because all this problem is about is finding what that scaling and that time shift are. I think it is this bit that you should go back to.