[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:

[tex]\Omega_{MO}=3 [/tex]
[tex]\Omega_{\Lambda O}=0.01 [/tex]
[tex]\Omega_{RO}=0 [/tex]
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.
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 [itex]\Omega_{s0}[/itex] for some species [itex]s[/itex]? What is [itex]\Omega_{\rm total 0}[/itex] in the universe you are studying?

Chalnoth

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

ajclarke

Quote by zhermes

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.

Quote by cristo

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

[tex]\Omega_{total 0} = 1[/tex]

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

Quote by Chalnoth

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

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

Chalnoth

Quote by ajclarke

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. [itex]a\left(t\right)[/itex] has a local maximum at [itex]t = t_1[/itex] if [itex]da/dt \left(t_1 \right) = 0[/itex] and [itex]d^2 a/dt^2 \left(t_1 \right) < 0[/itex]. To find [itex]d^2 a/dt^2 [/itex], differentiate the Friedmann equation.

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BillSaltLake Recognitions: Gold Member	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 Mentor	[Cosmology] Scale Factor Values What is the definition of [itex]\Omega_{s0}[/itex] for some species [itex]s[/itex]? What is [itex]\Omega_{\rm total 0}[/itex] in the universe you are studying?

Chalnoth Recognitions: Science Advisor	Make use of the second Friedmann equation to make sure that when [itex]H(a)[/itex] goes to zero, [itex]dH/da[/itex] is negative.

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