Computing the age of the universe at the big crunch

[uppiemurphy]

Homework Statement

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.

Homework 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.

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?

 

[clamtrox]

Can you find the time (or redshift, or scale factor) when expansion stops and collapsing starts? That might be of some use.