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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 $Q$?

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It pretty much has to be done numerically. One software package that wraps the numerical integrals in question is astropy. You can find the relevant information in the Astropy Cosmology package documentation.

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.

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