[Cosmology] Scale Factor Values

1. ajclarke

35
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.

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3. zhermes

What 'math' are you following through with?

4. BillSaltLake

184
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.

5. cristo

8,394
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?

6. Chalnoth

5,151
Make use of the second Friedmann equation to make sure that when $H(a)$ goes to zero, $dH/da$ is negative.

7. ajclarke

35
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?

8. Chalnoth

5,151
If the derivative of the Hubble parameter is negative, then it's recollapsing.

9. BillSaltLake

184
Don't you have Ωtotal0 equal to 3.01, instead of unity?

10. George Jones

6,363
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.