Assume that the universe today is flat with both matter and a cosmological constant, the latter with energy density which remains constant with time. Integrate Eq. (1.2) to find the present age of the universe. That is, rewrite Eq. (1.2) as:
dt = H0-1 da / a [ΩΛ + (1 - ΩΛ ) / a3 ]-1/2
where ΩΛ is the ratio of energy density in the cosmological constant to the critical density.
H2(t) = 8πG/3 [ρ(t) + (ρcr - ρ0) / a2(t)]
ρcr ≡ 3H02 / 8πG
The Attempt at a Solution
I'm having a little difficulty getting equation 1.2 into the format desired above. My idea was to factor out a ρcr from the right hand side, which leads me to the following:
H2(t) = 8πG/3 * ρcr [ρ(t)/ρcr + (ρcr - ρ0) / a2(t)*ρcr]
= 8πG/3 * (3H02/8πG) [ρ(t)/ρcr + (ρcr - ρ0) / a2(t)*ρcr]
= H02 [ρ(t)/ρcr + (ρcr - ρ0) / a2(t)*ρcr]
At this point, I run into a little difficulty switching things up inside the brackets like I need to.
Using some things I've gleaned from various sources online, I've come to the following idea:
ρ(t)/ρcr = Ω
Where Ω = Ωmatter + ΩΛ
I'm not really quite sure what the difference ρcr - ρ0 is supposed to represent, so my last step looks like the following:
H2(t) = H02 [Ω + 1 / a2 - ρ0 / (ρcr * a2)]
Without actually telling me what to do, could someone provide a hint as to what I may be missing? Thank you very much.