Friedmann's 1st equation and density parameters

[tomwilliam2]

Homework Statement

I'm trying to work out how the expression:
$$H^2 = H_0^2 \left [ \Omega_0 \frac{a}{a_0} + 1 - \Omega_0\right]$$
can be deduced from Friedmann's first equation:
$$H^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{R^2}$$
And I have a number of questions.

Firstly, I've often seen the 1st Friedmann equation written with a $\frac{\Lambda c^2}{3}$ term in it, but my textbook gives it as above. I guess they are equivalent, but I can't see how immediately. I'd like to know how you get the first expression above from Friedmann's equation so that I can work out whether it is valid for a spatially flat universe (k=0).
I also note that there is no $\Omega(t)$ term, so the $a$ in the square brackets provides the time-dependent element. Is it possible to write the $\Omega_0\frac{a}{a_0}$ in terms of $\Omega$ instead?

Thanks in advance!

 

[member 553083]

Homework EquationsFriedmann's First Equation:$$H^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{R^2}$$The Attempt at a SolutionI think the first part of my question is answered by noting that we can rewrite Friedmann's equation as follows:$$H^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{R^2} + \frac{\Lambda c^2}{3}$$where $\Lambda$ is the cosmological constant. Then, setting $\Lambda$ equal to zero and rearranging gives us:$$H^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{R^2} + \frac{\Omega_0 c^2}{3}\left ( \frac{a}{a_0}\right )$$where $\Omega_0$ is the current value of density parameter, and $a_0$ is the current scale factor. This can be further rearranged to give us the desired expression:$$H^2 = H_0^2 \left [ \Omega_0 \frac{a}{a_0} + 1 - \Omega_0\right]$$where $H_0$ is the current value of the Hubble parameter. As for the second part of my question, I'm not sure if it is possible to write the $\Omega_0\frac{a}{a_0}$ term in terms of $\Omega$ instead, but I'm guessing that it is not possible since the density parameter is a function of time, whereas the scale factor is a function of time and space.