# Conformal time closed Friedmann universe

1. Homework Statement
A closed Friedmann universe contains a single perfect fluid with an equation of state of the form $$p=w\rho c^2$$. Transforming variables to conformal time $$\tau$$ using $$dt=a(t)d\tau$$, show that the variable $$y=a^{(1+3w)/2}$$ is described by a simple harmonic equation as a function of $$\tau$$. Hence argue that all closed Friedmann models with a given equation of state have the same conformal lifetime.

2. The attempt at a solution
Please give me a starter. I haven't got a clue...

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Dick
Homework Helper
You'll want to start by writing down the Friedmann equations with curvature term +1, changing time derivatives to conformal time derivatives and substitute your energy/pressure relation. Do all this busy work and then think about the problem again.

You'll want to start by writing down the Friedmann equations with curvature term +1, changing time derivatives to conformal time derivatives and substitute your energy/pressure relation. Do all this busy work and then think about the problem again.
So first the Friedmann eq. is

$$\left(\frac{\dot{a}}{a_0}\right)^2=H^2_0 \Omega_{0w} \left(\frac{a_0}{a}\right)^{1+3w}$$

and changing the time derivative like

$$\frac{da}{dt}=\frac{1}{a}\frac{da}{d\tau}$$

giving

$$\left(\frac{da}{d\tau}\right)^2=a^2a_0^2H_0^2\Omega_{0w}\left(\frac{a_0}{a}\right)^{1+3w}=H_0^2\Omega_{0w}a^{1-3w}a_0^{3+3w}=H_0^2\Omega_{0w}a^{1-3p/\rho c^2}a_0^{3+3p/\rho c^2}$$.

I still don't get it.

Dick
Homework Helper
I don't get it either. That doesn't look like the first Friedmann equation to me. Where's rho? Where's the curvature term? You've already done a bunch of substitutions.

I have used the Friedmann equation

$$\left(\frac{\dot{a}}{a_0}\right)^2 - \frac{8\pi}{3}G\rho\left(\frac{a}{a_0}\right)^2 = H_0^2\left(1-\frac{\rho_0}{\rho_{0c}}\right) = H_0^2(1-\Omega_0) = -\frac{Kc^2}{a_0^2}$$

and

$$\rho a^{3(1+w)} = const. = \rho_{0w}a_0^{3(1+w)}$$

to get

$$\left(\frac{\dot{a}}{a_0}\right)^2=H^2_0 \left[\Omega_{0w} \left(\frac{a_0}{a}\right)^{1+3w} + (1-\Omega_{0w}) \right]$$

Then, for a curved model, the last term is negligible so

$$\left(\frac{\dot{a}}{a_0}\right)^2=H^2_0 \Omega_{0w} \left(\frac{a_0}{a}\right)^{1+3w}$$

This is all from Coles Cosmology.

Dick
Homework Helper
You are assuming the evolution is a power law in a(t) and it's not. The curvature term is not negligible. You are going to want to find an equation of the form b'=Kb where ' denotes the derivative wrt conformal time.

Well, then I'm back to zero and still stuck.

Dick
Homework Helper
Ok, try this. Go back to the REAL Friedmann equation for H^2. We know this is a closed universe so it will reach a point of maximum expansion and then collapse. So pick t_0 to be the time of maximum expansion, so H_0=0, may as well take a_0=1 to make a choice of scale. Now put in your dependence of rho on and fix the constants at t_0. Now what does the equation look like?

Dick
Homework Helper
Well. I actually worked through this in detail for practice. It does work. Substitute p=w*rho. Solve the H^2 eqn for rho and substitute it into the a'' one. Express the t derivatives in terms of conformal derivatives. Finally substitute y^(2/(3*w+1)) for a. You will get a y'' term, a (y')^2 term and a y^2 term. If you have actually managed to do all of the math correctly (it took me a few times), you will find the (y')^2 terms magically cancel (hence the choice of exponent). The remaining equation looks like a harmonic oscillator. Yahooo!

Uhm, what equations are H^2 and a''?

Dick