# Hubble's law and expansion question

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!

1. 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.

2. Homework Equations
None really...

3. 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?

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Kurdt
Staff Emeritus
Gold Member
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.

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?

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
Staff Emeritus