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Homework Help: Computing the age of the universe at the big crunch

  1. Nov 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a universe where there are two types of matter: dust and curvature. The density of dust evolves with time as ρdust(t) = ρ0R(t)^(−3) and curvature evolves as ρcurvature = −kR(t)^(-2) where ρ0 and k are constants.

    The Friedmann equation is:
    H^2 = [(8∏G)/3][ρdust + ρcurvature] (where H^2 = (Rdot/R)^2 )

    Consider the case where k is positive (this describes a closed sphere). Imagine that the universe starts out with zero size at time t = 0 – i.e. at the big bang – and begins expanding. Show that at some point the universe will reach a maximum size, before proceeding to collapse to zero size in a ”big crunch.” Compute the age of the universe at the big crunch.

    2. Relevant equations

    I know that ρdust + ρcurvature > ρcritical where ρcritical = 3H^2/(8∏G) other than that I am unsure if any other equations would be useful, and whether even this one is.

    3. The attempt at a solution

    OK I have come up with an integral expression for the amount of time t, it takes for the universe to evolve from a size R1 to a size R2.
    this integral is:

    t = 1/sqrt((8∏G)/3) ∫ dR/sqrt(ρ0/R -k) . Is this expression useful? It is a difficult integral to solve... I think if I evaluated this integral from 0 to Rmax I would be able to find the time t of the universe at the "big crunch"

    Is there an easier way to go about this or am I on the correct path?
  2. jcsd
  3. Nov 13, 2012 #2
    Can you find the time (or redshift, or scale factor) when expansion stops and collapsing starts? That might be of some use.
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