# Solving the Friedmann Equation for a specific universe

Suppose instead of the cosmological constant there is a quantum field with equation state parameter $w = -1/2$. Assume also it is a flat universe with only this quantum field ($Q$) and with non relativistic matter ($w=0$). Currently $\Omega(m,0) <$ or $= 1$ and $\Omega(Q,0) = 1 - \Omega(m,0)$.

How would you solve the Friedmann Equation to find $a(t)$ for this universe?

I know that the Friedmann Equation for a universe with only matter can be written as $(\dot a)^2/(H_0(t))^2 = \Omega_0/a + (1-\Omega_0)$ but how can I find it for a universe with both matter and $Q$?

The particular calculation you're looking for is the lookback time (light travel distance in the parlance of Wikipedia's article). That calculation gives you $t_L(z)$, where $z$ is the observed cosmological redshift of a source at that epoch. If you want $a(t)$ you'll have to invert three relationships: \begin{align} t_L & \equiv t_{\mathrm{now}} - t, \\ t_L(z) & = \int_0^z \frac{t_{\mathrm{Hubble}}}{(1+z')\, E(z')} \operatorname{d}z', and \\ \frac{1}{1+z} & = \frac{a}{a_{\mathrm{now}}}, \end{align} where $E(z)$ is defined by $$E(z) \equiv \frac{\dot{a}(z)}{a(z)} \times \frac{a(0)}{\dot{a}(0)},$$ and can be worked out from the Friedmann equations.