Cosmology - determining if a model universe would recollapse

[James McKeets]

Homework Statement

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.

Homework 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

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?

If you need more information about the problem, I'd be happy to help, including further information about what I've tried so far.

Many thanks,

James

 

[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

Ω_t = Ω_radiation = Ω_R,
Ω_m = Ω_matter = Ω_M,
Ω_r = Ω_curvature= Ω_k, and
Ω_Λ = Ω_darkenergy?​

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

 

[James McKeets]

Hello,

Thank you so much for your reply.

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 helpJames

 

[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 Ω_subscript × a₀ⁿ equals 1.

I also just thought another hint that might help you.

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.

ADDED
I noticed that

Ω_R + Ω_M + Ω_Λ = Ω_T.​

This suggests that the the curvature term

Ω_k = 1-Ω_T.​

This means that a₀ = 1. See The the end of the "Density parameter" section of

https://en.wikipedia.org/wiki/Friedmann_equations.​

Having a negative curvature makes it easy to show eventual collapse.

Regards,
Buzz

 

[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 a_max, then there is a corresponding value of t, say t_amax.

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

Regards,
Buzz