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 $Q$?