Hubble's law and expansion question

[tortin]

Hi, I was looking through past exams for one of my courses, and I came across a question I wasn't quite sure how to do, so I was wondering if anyone could give some hints (or possibly the rationale for solution), thanks!

Homework Statement

Observers in three universes A, B, and C measure identical constants. However, universe A is a critical universe (i.e. p = pc (with c being critical density), Universe B is an accelerating universe, and Universe C is close to being an empty universe (i.e. p<<pc or p~0). Rank the ages of the three universes as measured by these observers from largest to smallest, and explain your reasoning.

Homework Equations

None really...

The Attempt at a Solution

I'm not quite sure what you can infer about the age of A if it's a critical universe, because it always stays flat with p = pc. C is probably the oldest universe, since it's emptying out, and I'd guess B is younger than C, since it's accelerating so not yet emptied out.

However, I was also wondering what the relevance of the identical Hubble constants would give in this problem.Thanks for anyone who can give some help / hints!

edit - maybe this should be in advanced physics forum?

 

[Kurdt]

I think you're going to have to solve the Friedman and fluid equations for each case and compare the estimation of the ages you get in terms of the Hubble constant.

 

[astrorob]

As Kurdt said, you should start with your basic Friedmann equation and solve to find the dependence of scale factor $$\dot{R}$$ with time, t.

Hints:

Since the scale factor directly relates to the Hubble constant you can work out the age for each universe using:

$$H(t)$$ = $$\frac{\dot{R}}R$$

For the universe of critical density we then use our critical density relation:

$$\rho_c = \frac{3 H^2}{8 \pi G}$$

which applies for a k=0, or a flat, universe.

For an accelerating universe $$\Lambda$$, the cosmological constant, is now non zero.

Edit: Shouldn't this be in advanced physics?

 

[tortin]

As Kurdt said, you should start with your basic Friedmann equation and solve to find the dependence of scale factor $$\dot{R}$$ with time, t.

Hints:

Since the scale factor directly relates to the Hubble constant you can work out the age for each universe using:

$$H(t)$$ = $$\frac{\dot{R}}R$$

For the universe of critical density we then use our critical density relation:

$$\rho_c = \frac{3 H^2}{8 \pi G}$$

which applies for a k=0, or a flat, universe.

For an accelerating universe $$\Lambda$$, the cosmological constant, is now non zero.

Edit: Shouldn't this be in advanced physics?

Hmm, so for Universe A (the critical one), we would have:
(removing the constants 8piG/3)

$$\Omega \propto \frac{\rho}{H(t)^{2}}$$
However, $$H(t) = \frac{\dot{R}}R$$, so we can replace it such that:
$$\dot{R}^{2}\Omega \propto \rho$$
$$\rho \propto \Omega z^{2}$$ (z being redshift)
So for universe A, $$\Omega$$ is 1, z would be larger than universe B, since $$\Omega$$ is close to 0, so A is older than B.

for universe C, addition of a cosmological constant would make z larger than A, giving the age as: C>A>B (in order of age)

would that be correct? (but it's assuming different Hubble parameters for each universe..)

I'm trying to do this more intuitively, since it's a first year course (and the Freidmann equation wasn't given on the exam), and although we did cover the equation, it seems a bit unlikely we'd have to calculate anything...

thanks!

 

[Kurdt]

It depends of course on the class you're doing. I'd check the notes very carefully, but I'd be surprised if for a first year course they hadn't given you some sort of graph depicting the density parameter at various values. You should be able to work out the answer from this. I was perhaps being a bit too zealous when I suggested my original post.