Scale Factor & ##\Omega##: Finding the Relation

[AHSAN MUJTABA]

Homework Statement

I have to plot density parameters of radiation, matter and vacuum with initial or current values $1*10^-4, 0.3,0.7$ respectively against $log(a(t))$ with values a=$10^-35 to 10^35$ in any plotting software.

Relevant Equations

1)$\rho_i=\rho_io *a^{-n}_i$ (density evolves as power law) where n_i=3,4,0 for matter, radiation and vacuum respectively.
2) Friedmann equation(first kind)
3) Space time and geometry by Sean Caroll

I am trying to develop a relation between scale factor (a(t)) and $\Omega$. The relation came out to be evolve as $\Omega_i=\Omega_io * a^{-n}$ but my graph isn't right it's giving values of $a(t)$ to higher extent.
I consulted my instructor he only added that I should include $H_o$ somewhere.
I am confused about which relation I should get to plot these in a right way.

 

[Nate810]

The relation between the scale factor and the density parameter (Ω) is given by: Ω = Ωo * a^(-3n),where Ωo is the present-day density parameter, a is the scale factor, and n is the equation of state parameter. The equation of state is related to the Hubble parameter H0 through the following expression: n = (2/3) * (1 + (w*H0^2)/(ρc))where w is the equation of state parameter, ρc is the critical density, and H0 is the Hubble constant. Therefore, the relation between the scale factor and the density parameter can be written as: Ω = Ωo * a^(-3(2/3)*(1 + (w*H0^2)/(ρc)))This relation allows you to plot the scale factor (a(t)) against the density parameter (Ω).