Homework Help: Cosmology - determining if a model universe would recollapse

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1. Mar 24, 2017

James McKeets

1. The problem statement, all variables and given/known data
Show mathematically that a model with:
Ω_M0 = 3
Ω_Λ0 = 0.01
Ω_R0 = 0
Ω_T0 = 3.01
is a model that re-collapses in the future. Be certain to indicate at what value of the scale factor 'a' the expansion reverses and becomes contraction.

2. Relevant equations
It's hinted pretty strongly that we should probably be using:
( (d(a)/dt ) / a ) = H^2 {Ω_M0 a^{-3} + Ω_Λ0 - (Ω_T0 - 1) a^{-2}}
And that we should be solving a cubic somewhere along the way

3. The attempt at a solution
So my first plan (I've spent many many hours on this) was to move the 'a' over to the RHS, and then expand out the (d(a)/dt ) using the Friedman equations. This allowed me to reduce the data and find a cubic eq in 'a' such that:

3a^3+0.01a-3=0

However, when I plot this, I get an exponentially increasing line. I was expecting kinda an arch, which would tell me that scale factor has increased, then decreased back to 0.
Any ideas?

Many thanks,

James

Last edited: Mar 24, 2017
2. Mar 24, 2017

Buzz Bloom

Hi James:

I have not seen these Ω subscripts before, and I am not sure what they represent. Are you working with the Lambda-CDM model Friedman equation of the form below with
Ωm = Ωmatter = ΩM,
Ωr = Ωcurvature= Ωk, and
ΩΛ = Ωdarkenergy?​

If you are not, then I think that is your problem. I am not sure, but I think the term that is zero should be the curvature term. On the other hand, you may want to set radiation to zero and have a positive curvature.

Also, usually the four Ω coefficients should sum to unity. If they don't, then a0 is not 1.

Good luck, and I hope this helps.

Regards,
Buzz

3. Mar 25, 2017

James McKeets

Hello,

I agree, the format is strange, however it's what we were provided with.
I believe your conversion of subscripts is correct.

I will continue working and post a solution if I find one. Thank you again for your help

Best,
James

4. Mar 25, 2017

Buzz Bloom

ΩHi James:

I suggest you compare your form of the Friedman equation with the one I quoted. I think you left out a "^2". You also need to use a/a0 rather than a, and you need to calculate a0 using the sum of the four terms Ωsubscript × a0n equals 1.

You do not need to solve the differential equation for da/dt. Since da/dt is a velocity what do you need to find out to see if the velocity changes to become negative.

Also, note that the value of H0 is not specified. Therefore da/dt must always become negative for any specific value of H0.

If this approach shows da/dt stays > 0, then you need to assume radiation is zero, and curvature is not zero.

I noticed that
ΩR + ΩM + ΩΛ = ΩT.​
This suggests that the the curvature term
Ωk = 1-ΩT.​
This means that a0 = 1. See The the end of the "Density parameter" section of
Having a negative curvature makes it easy to show eventual collapse.

Regards,
Buzz

Last edited: Mar 25, 2017
5. Mar 26, 2017

Buzz Bloom

Hi @James McKeets:

I added something to my previous post which you may have missed.

I had one more clarifying idea.

You have an expression H2 = f(a). Taking the square root you get
H = (+/-)√f(a).​
This means that for every value of a where f(a)>0, f(a) can also be <0.
If there is no value of a for which f(a)=0, then the negative values are interpreted as time running backwards.
If there is a value of a for which f(a) = 0, say amax, then there is a corresponding value of t, say tamax.

For this case, think about what da(t-tamax)/dt and a(t-tamax) look like. Are there any symmetries with respect to tamax?

Regards,
Buzz