[Cosmology] Scale Factor Values

1. May 12, 2011

ajclarke

Hello.

I have been working through some questions and answers to do with cosmology. One of them asks you to consider a model where:

$$\Omega_{MO}=3$$
$$\Omega_{\Lambda O}=0.01$$
$$\Omega_{RO}=0$$
and asks you to show mathematically that the model re-collapses.

Following through the math, I get three values of a: -14.87,1.51 and 13.36.

Clearly the first can be disregarded and unphysical since a cannot be negative, but I can't decide whats the significance between the second two which allows me to isolate the value corresponding to collapse.

Cheers.

2. May 12, 2011

zhermes

What 'math' are you following through with?

3. May 12, 2011

BillSaltLake

If a is normalized time, then it may have zero diameter 14.87 time units in the past, first collapse 1.51 in the future, and a "recollapse" later. Not sure if that's correct though.

4. May 12, 2011

cristo

Staff Emeritus
What is the definition of $\Omega_{s0}$ for some species $s$? What is $\Omega_{\rm total 0}$ in the universe you are studying?

5. May 13, 2011

Chalnoth

Make use of the second Friedmann equation to make sure that when $H(a)$ goes to zero, $dH/da$ is negative.

6. May 13, 2011

ajclarke

I used the equation for the Hubble Parameter as a function of redshift, then changed this over to be a function of scale factor instead.

$$\Omega_{total 0} = 1$$

I don't understand the first bitof the question I'm sorry.

I'm uncertain as to how that determines which of the two remaining parameters is the recollapsing universe?

7. May 13, 2011

Chalnoth

If the derivative of the Hubble parameter is negative, then it's recollapsing.

8. May 13, 2011

BillSaltLake

Don't you have Ωtotal0 equal to 3.01, instead of unity?

9. May 13, 2011

George Jones

Staff Emeritus
Use the second derivative test from elementary calculus. $a\left(t\right)$ has a local maximum at $t = t_1$ if $da/dt \left(t_1 \right) = 0$ and $d^2 a/dt^2 \left(t_1 \right) < 0$. To find $d^2 a/dt^2$, differentiate the Friedmann equation.