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## Homework Statement

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 = H

_{0}

^{-1}da / a [Ω

_{Λ}+ (1 - Ω

_{Λ}) / a

^{3}]

^{-1/2}

where Ω

_{Λ}is the ratio of energy density in the cosmological constant to the critical density.

## Homework Equations

[/B]

Equation 1.2:

H

^{2}(t) = 8πG/3 [ρ(t) + (ρ

_{cr}- ρ

_{0}) / a

^{2}(t)]

ρ

_{cr}≡ 3H

_{0}

^{2}/ 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 ρ

_{cr}from the right hand side, which leads me to the following:

H

^{2}(t) = 8πG/3 * ρ

_{cr}[ρ(t)/ρ

_{cr}+ (ρ

_{cr}- ρ

_{0}) / a

^{2}(t)*ρ

_{cr}]

= 8πG/3 * (3H

_{0}

^{2}/8πG) [ρ(t)/ρ

_{cr}+ (ρ

_{cr}- ρ

_{0}) / a

^{2}(t)*ρ

_{cr}]

= H

_{0}

^{2}[ρ(t)/ρ

_{cr}+ (ρ

_{cr}- ρ

_{0}) / a

^{2}(t)*ρ

_{cr}]

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)/ρ

_{cr}= Ω

Where Ω = Ω

_{matter}+ Ω

_{Λ}

I'm not really quite sure what the difference ρ

_{cr}- ρ

_{0}is supposed to represent, so my last step looks like the following:

H

^{2}(t) = H

_{0}

^{2}[Ω + 1 / a

^{2}- ρ

_{0}/ (ρ

_{cr}* a

^{2})]

Without actually telling me what to do, could someone provide a hint as to what I may be missing? Thank you very much.