Find lifetime of closed RD universe

[ck99]

Homework Statement

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.

Homework Equations

Use first Friedmann equation which reduces to

a'²/a² + a^-2 = kρ where k = 8∏/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

In radiation domination we have

ρ(a) = ρ₀a^-4

which leads to

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 . . .

 

[BruceW]

Homework Statement

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.

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

 

[George Jones]

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

a(t) = (k - t²)^1/2

Are you sure that this is correct?

Are you sure you don't mean k=-1 ?

 

[BruceW]

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

a(t) = (k - t²)^1/2

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.

 

[paranormal]

Your calculations for a(t) and ρ(t) look correct. To find the lifetime of the universe, we can set a(t) = 0 and solve for t. This will give us the time at which the universe reaches its maximum size and begins to collapse. This is known as the "big crunch" and marks the end of the universe's lifetime.

So, setting a(t) = 0, we get:

0 = (k - t^2)^(1/2)

t^2 = k

t = ± √k

Since k = 8π/3, t = ± √(8π/3)

Therefore, the lifetime of the closed RD universe is approximately ± √(8π/3) units of time. This is the time it takes for the universe to expand to its maximum size and then collapse back to a singularity.

Note that this calculation assumes that the universe only contains radiation and that there are no other factors (such as dark energy or matter) that could affect the lifetime of the universe.