[Cosmology] Scale Factor Values

[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 what's the significance between the second two which allows me to isolate the value corresponding to collapse.

Cheers.
Adam

 

[zhermes]

What 'math' are you following through with?

 

[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.

 

[cristo]

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

 

[Chalnoth]

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

 

[ajclarke]

What 'math' are you following through with?

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.

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

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

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

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

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

 

[Chalnoth]

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

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

 

[BillSaltLake]

Don't you have Ω_total0 equal to 3.01, instead of unity?

 

[George Jones]

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.