What Is the FRW Metric and How Is It Applied in General Relativity?

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Homework Statement

(a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H.
(b) Express density parameter as a function of a where density dominates and find values of w.
(c) If curvature is negligible, what values must w be to prevent a singularity? Find a.
(d) Find an expression for the deceleration parameter and redshift.

Homework Equations

The Attempt at a Solution

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Part(a)
The metric is given by
ds^2 = c^2 dt^2 - a(t)^2 \left[ \frac{dr^2}{1-kr^2} + r^2(d\theta^2 + sin^2 \theta d\phi^2) \right]
The FRW equations are
\left( \frac{\dot a}{a} \right)^2 = \frac{8 \pi G \rho_I}{3} + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2 (t)}
\ddot a(t) = -\frac{4\pi G}{3} \left(\rho_I + \frac{3P}{c^2} \right) a(t) + \frac{1}{3} \Lambda c^2 a(t)
Density parameter is given by
\Omega = \frac{8\pi G}{3H^2}\left( \rho_I + \frac{\Lambda c^2}{8 \pi G} - \frac{3 kc^2}{8 \pi G} \right)

How do I express it in terms of $a$ and $H$ only? I know that $\rho_I \propto a^{-3(1+w_I)} = \rho_I(0) a^{-3(1+w_I)}$

This is as far as I can go. Would appreciate any input, many thanks!

 

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Managed some progress with part (a)! I think they are looking for $H^2 = \frac{8 \pi G}{3}\rho = H_0a^{-3(1+w_I)}$.

For part (b), I think they want $\Omega = \frac{8 \pi G}{3 H_0^2} \rho_I a^{-3(1+w_I)}$.

I suppose the confusion was that I thought $\rho_I = \rho_{I,0} a^{-3(1+w_I)}$ when in fact they use $\rho_I$ as the density of today.