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?