First Friedmann Equation forms

[dillingershaw]

Homework Statement

the first friedmann equation is:
(\frac{\dot{a}}{a})^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}+\frac{\Lambda}{3}
In the case of a closed Universe (k > 0) containing only non-relativistic matter and no cosmological constant, write the Friedmann equation in terms of, H(a), H₀, a and Ω₀ (where Ω₀ is the current matter density parameter). Assume that the current scale factor, a₀ = 1

Homework Equations

The Attempt at a Solution

so far what I have is:
H(a)²=H₀²Ω₀-\frac{kc^2}{a^2}+\frac{\Lambda}{3}
I've seen things like
H(a)²=H₀²[\Omega_0\frac{a}{a_0}+1-\Omega_0]

but I have no explanation for this and so can't tell if it's right.

 

[gavman]

usyd student? if so try lecture 7 slide 12 ;)
I'm not sure about solving the fluid equation for p=0 - apparently its a seperable integral or something, but i think you can use what's provided on slides without proof. From there I think you are just trying to draw a relationship between the flat universe when k=0 and this closed universe for some k>0. For me, it is very time consuming to understand and show working.

I don't fully understand how we can create a single form of Friedmann for a closed universe by using flat universe equations for density, etc. Very confusing.