Is the Critical Density Related to the Friedmann Equation?

[HawkEye5220]

Homework Statement

For a κ=0 universe with no cosmological constant, show that H(z)=H₀(1+z)^3/2

Homework Equations

Friedmann equation: H²=\frac{8*\pi*g}{3c^2}-\frac{κc^2}{r^2}*\frac{1}{a(t)^2}

The Attempt at a Solution

I know that R(z)=R₀/(1+z) but I do not know where this comes from. Following this, I should be able to take a density ρ(z)=ρ(now)*(1+z)^3 and input it into the Friedman equation but I am not sure how to proceed

 

[cepheid]

The Friedmann equation should be $H^2 = \frac{8\pi G}{3}\rho$, where rho is the total mass density of the universe (in this case considered to be entirely due to matter). You can ignore the second term with kappa entirely, since it's zero (flat universe). You are correct that you can write $\rho = \rho_0 (1+z)^3$ for ordinary matter. The trick now is figuring out how to rewrite the $(8\pi G)/3$ pre-factor in terms of something else. Hint: what is the expression for the critical density?