1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Mar 24, 2017 #1
    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:


    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,

    Last edited: Mar 24, 2017
  2. jcsd
  3. Mar 24, 2017 #2

    Buzz Bloom

    User Avatar
    Gold Member

    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
    Ωt = Ωradiation = ΩR,
    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.

  4. Mar 25, 2017 #3

    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 help

  5. Mar 25, 2017 #4

    Buzz Bloom

    User Avatar
    Gold Member

    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.

    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.

    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.

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

    Buzz Bloom

    User Avatar
    Gold Member

    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?

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted